# Special functions

## Special functions Topics

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### Delta amplitude

Given a Jacobi amplitude and a elliptic modulus in an elliptic integral,

### Abelian integral

An Abelian integral, are also called a hyperelliptic integral, is an integral of the formwhere is a polynomial of degree .

### Mathieu differential equation

(1)(Abramowitz and Stegun 1972; Zwillinger 1997, p. 125), having solution(2)where and are Mathieu functions. The equation arises in separation of variables of the Helmholtz differential equation in elliptic cylindrical coordinates. Whittaker and Watson (1990) use a slightly different form to define the Mathieu functions.The modified Mathieu differential equation(3)(Iyanaga and Kawada 1980, p. 847; Zwillinger 1997, p. 125) arises in separation of variables of the Helmholtz differential equation in elliptic cylindrical coordinates, and has solutions(4)The associated Mathieu differential equation is given by(5)(Ince 1956, p. 403; Zwillinger 1997, p. 125).

### Third kind

In the theory of special functions, a class of functions is said to be "of the third kind" if it is similar to but distinct from previously defined functions already defined to be of the first and second kinds. The only common functions of the third kind are the elliptic integral of the third kind and the Bessel function of the third kind (more commonly called the Hankel function).

### Rogers mod 14 identities

The Rogers mod 14 identities are a set of three Rogers-Ramanujan-likeidentities given by(1)(2)(3)(4)(5)(6)(7)(8)(9)(OEIS A105780, A105781,and A105782).The -identity was found by Rogers (1894) and appears as formula 61 in the list of Slater (1952). The - and -identities were found by Rogers (1917) and appeared as formulas 60 and 59 respectively in Slater (1952).

### Ramanujan theta functions

Ramanujan's two-variable theta function is defined by(1)for (Berndt 1985, p. 34; Berndt et al. 2000). It satisfies(2)and(3)(4)(Berndt 1985, pp. 34-35; Berndt et al. 2000), where is a q-Pochhammer symbol, i.e., a q-series.A one-argument form of is also defined by(5)(6)(7)(OEIS A010815; Berndt 1985, pp. 36-37; Berndt et al. 2000), where is a q-Pochhammer symbol. The identities above are equivalent to the pentagonal number theorem.The function also satisfies(8)(9)Ramanujan's -function is defined by(10)(11)(12)(13)(14)(OEIS A000122), where is a Jacobi theta function (Berndt 1985, pp. 36-37). is a generalization of , with the two being connected by(15)Special values of include(16)(17)where is a gamma function.Ramanujan's -function is defined by(18)(19)(20)(21)(22)(23)(OEIS A010054; Berndt 1985, p. 37).Ramanujan's -function is defined by(24)(25)(26)(OEIS A000700; Berndt 1985, p. 37).A..

### Fine's equation

The q-series identity(1)(2)(3)(4)where is a q-Pochhammer symbol, is the number of divisors of that are congruent to 1, 5, 7, and 11 (mod 24) minus the number of divisors of congruent to , , , and (mod 24), and is a Kronecker symbol.The coefficients of the first few powers of starting with , 1, ... are 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, ... (OEIS A000377).

### Ramanujan psi sum

A sum which includes both the Jacobi triple product and the q-binomial theorem as special cases. Ramanujan's sum iswhere the notation denotes q-series. For , this becomes the q-binomial theorem.

### Dyson mod 27 identities

The Dyson mod 27 identities are a set of four Rogers-Ramanujan-likeidentities given by(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(OEIS A104501, A104502,A104503, and A104504).Bailey (1947) systematically studied and generalized Rogers's work on Rogers-Ramanujan type identities in a paper submitted in late 1943. At the time, G. H. Hardy was the editor of the Proceedings of the London Mathematical Society and Hardy had recently taught the young Freeman Dyson in one of his undergraduate classes at Cambridge. He was therefore aware of Dyson's interest in Ramanujan-Rogers-type identities through his rediscovery of the Rogers-Selberg identities. Ignoring the usual convention of keeping the referee anonymous (since as far as Hardy knew, Bailey and Dyson were the only two people in all of England who were interested in Rogers-Ramanujan type identities at the time) and thinking that they would like to be in contact with each..

### Dedekind eta function

Min Max The Dedekind eta function is defined over the upper half-plane by(1)(2)(3)(4)(5)(6)(OEIS A010815), where is the square of the nome , is the half-period ratio, and is a q-series (Weber 1902, pp. 85 and 112; Atkin and Morain 1993; Berndt 1994, p. 139).The Dedekind eta function is implemented in the WolframLanguage as DedekindEta[tau].Rewriting the definition in terms of explicitly in terms of the half-period ratio gives the product(7) Min Max Re Im It is illustrated above in the complex plane. is a modular form first introduced by Dedekind in 1877, and is related to the modular discriminant of the Weierstrass elliptic function by(8)(Apostol 1997, p. 47).A compact closed form for the derivative is given by(9)where is the Weierstrass zeta function and and are the invariants corresponding to the half-periods . The derivative of satisfies(10)where is an Eisenstein series, and(11)A special value is given by(12)(13)(OEIS..

### Cauchy binomial theorem

(1)(2)where is a q-binomial coefficient.

### Quintuple product identity

The quintuple product identity, also called the Watson quintuple product identity, states(1)It can also be written(2)or(3)The quintuple product identity can be written in q-seriesnotation as(4)where and (Gasper and Rahman 1990, p. 134; Leininger and Milne 1999).Using the notation of the Ramanujantheta function (Berndt 1985, p. 83),(5)

### Borwein conjectures

Use the definition of the q-series(1)and define(2)Then P. Borwein has conjectured that (1) the polynomials , , and defined by(3)have nonnegative coefficients, (2) the polynomials , , and defined by(4)have nonnegative coefficients, (3) the polynomials , , , , and defined by(5)have nonnegative coefficients, (4) the polynomials , , and defined by(6)have nonnegative coefficients, (5) for odd and , consider the expansion(7)with(8)then if is relatively prime to and , the coefficients of are nonnegative, and (6) given and , consider(9)the generating function for partitions inside an rectangle with hook difference conditions specified by , , and . Let and be positive rational numbers and an integer such that and are integers. then if (with strict inequalities for ) and , then has nonnegative coefficients...

### Bailey mod 9 identities

The Bailey mod 9 identities are a set of three Rogers-Ramanujan-like identities appearing as equations (1.6), (1.8), and (1.7) on p. 422 of Bailey (1947) given by(1)(2)(3)(4)(5)(6)(7)(8)(9)(OEIS A104467, A104468,and A104469).Unfortunately, Bailey used non-standard (and essentially unreadable) notation in the paper where these identities first appeared. All three of these identities appear in the list of Slater (1952) as equations (42), (41), and (40) in that order. However, all three contain misprints.In one sense, these identities are the next logical step in the following sequence: 1. The two Rogers-Ramanujan identities (triple product on mod 5 over ). 2. The three Rogers-Selberg identities (triple product on mod 7 over ). 3. The (sort of) four Bailey mod 9 identities (triple product on mod 9 over ). Here, "sort of" refers to the fact that between and , there is an "identity" in which the product side contains..

A number of the form , where is a positive rational number which is not the square of another rational number is called a pure quadratic surd. A number of the form , where is rational and is a pure quadratic surd is sometimes called a mixed quadratic surd (Hardy 1967, p. 20).Quadratic surds are sometimes also called quadratic irrationals.In 1770, Lagrange proved that any quadratic surd has a regular continued fraction which is periodic after some point. This result is known as Lagrange's continued fraction theorem.

### Power

A power is an exponent to which a given quantity is raised. The expression is therefore known as " to the th power." A number of powers of are plotted above (cf. Derbyshire 2004, pp. 68 and 73).The power may be an integer, real number, or complex number. However, the power of a real number to a non-integer power is not necessarily itself a real number. For example, is real only for .A number other than 0 taken to the power 0 is defined to be 1, which followsfrom the limit(1)This fact is illustrated by the convergence of curves at in the plot above, which shows for , 0.4, ..., 2.0. It can also be seen more intuitively by noting that repeatedly taking the square root of a number gives smaller and smaller numbers that approach one from above, while doing the same with a number between 0 and 1 gives larger and larger numbers that approach one from below. For square roots, the total power taken is , which approaches 0 as is large, giving in the limit that..

### Hyperfactorial

The hyperfactorial (Sloane and Plouffe 1995) is the function defined by(1)(2)where is the K-function.The hyperfactorial is implemented in the WolframLanguage as Hyperfactorial[n].For integer values , 2, ... are 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... (OEIS A002109).The hyperfactorial can also be generalized to complex numbers, as illustrated above.The Barnes G-function and hyperfactorial satisfy the relation(3)for all complex .The hyperfactorial is given by the integral(4)and the closed-form expression(5)for , where is the Riemann zeta function, its derivative, is the Hurwitz zeta function, and(6) also has a Stirling-like series(7)(OEIS A143475 and A143476). has the special value(8)(9)(10)where is the Euler-Mascheroni constant and is the Glaisher-Kinkelin constant.The derivative is given by(11)..

### Bessel polynomial

Krall and Fink (1949) defined the Bessel polynomials as the function(1)(2)where is a modified Bessel function of the second kind. They are very similar to the modified spherical bessel function of the second kind . The first few are(3)(4)(5)(6)(7)(OEIS A001497). These functions satisfy thedifferential equation(8)Carlitz (1957) subsequently considered the related polynomials(9)This polynomial forms an associated Sheffer sequencewith(10)This gives the generating function(11)The explicit formula is(12)(13)where is a double factorial and is a confluent hypergeometric function of the first kind. The first few polynomials are(14)(15)(16)(17)(OEIS A104548).The polynomials satisfy the recurrence formula(18)

### Jinc function

The jinc function is defined as(1)where is a Bessel function of the first kind, and satisfies . The derivative of the jinc function is given by(2)The function is sometimes normalized by multiplying by a factor of 2 so that (Siegman 1986, p. 729).The first real inflection point of the function occurs when(3)namely 2.29991033... (OEIS A133920).The unique real fixed point occurs at 0.48541702373... (OEIS A133921).

### Weber's formula

where , , and , , is a Bessel function of the first kind, and is a modified Bessel function of the first kind.

### Poisson's bessel function formula

For ,where is a Bessel function of the first kind, and is the gamma function.

### Hankel's integral

where is a Bessel function of the first kind and is the gamma function. Hankel's integral can be derived from Sonine's integral.

### Weber's discontinuous integrals

(1)(2)where is a zeroth order Bessel function of the first kind.

### Hankel function of the second kind

(1)where is a Bessel function of the first kind and is a Bessel function of the second kind. Hankel functions of the second kind is implemented in the Wolfram Language as HankelH2[n, z].Hankel functions of the second kind can be representedas a contour integral using(2)The derivative of is given by(3)The plots above show the structure of in the complex plane.

### Watson's theorem

where is a generalized hypergeometric function and is the gamma function (Bailey 1935, p. 16; Koepf 1998, p. 32).

### Nicholson's formula

Let be a Bessel function of the first kind, a Bessel function of the second kind, and a modified Bessel function of the first kind. Thenfor , where denotes the real part of .

### Hankel function of the first kind

The Hankel functions of the first kind are defined as(1)where is a Bessel function of the first kind and is a Bessel function of the second kind. Hankel functions of the first kind is implemented in the Wolfram Language as HankelH1[n, z].Hankel functions of the first kind can be represented as a contour integral over the upper half-plane using(2)The derivative of is given by(3)The plots above show the structure of in the complex plane.

### Hankel function

There are two types of functions known as Hankel functions. The more common one is a complex function (also called a Bessel function of the third kind, or Weber Function) which is a linear combination of Bessel functions of the first and second kinds.Another type of Hankel function is defined by the contourintegralfor , , , where is a Hankel contour. The Riemann zeta function can be expressed in terms of asfor and , where is the gamma function (Krantz 1999, p. 160).

### Watson's formula

Let be a Bessel function of the first kind, a Bessel function of the second kind, and a modified Bessel function of the first kind. Also let and require . ThenThe fourth edition of Gradshteyn and Ryzhik (2000), Iyanaga and Kawada (1980), and Ito (1987) erroneously give the exponential with a plus sign. A related integral is given byfor .

### Modified spherical bessel function of the second kind

A modified spherical Bessel function of the second kind, also called a "spherical modified Bessel function of the first kind" (Arfken 1985) or (regrettably) a "modified spherical Bessel function of the third kind" (Abramowitz and Stegun 1972, p. 443), is the second solution to the modified spherical Bessel differential equation, given by(1)where is a modified Bessel function of the second kind (Arfken 1985, p. 633)For positive , the first few values for small nonnegative integer indices are(2)(3)(4)(5)(6)(OEIS A001498).Writing(7)the are given by the recurrence equation(8)together with(9)(10)(Abramowitz and Stegun 1972, p. 444). has no definite parity (Arfken 1985, p. 633). is related to the spherical Hankel function of the first kind by(11)for and integer (Arfken 1985, p. 633).They also satisfy the differential identities(12)(13)and the recurrence relations(14)(15)(Arfken..

### Modified spherical bessel function of the first kind

A modified spherical Bessel function of the first kind (Abramowitz and Stegun 1972), also called a "spherical modified Bessel function of the first kind" (Arfken 1985), is the first solution to the modified spherical Bessel differential equation, given by(1)where is a modified Bessel function of the first kind (Arfken 1985, p. 633).For positive , the first few values for small nonnegative integer indices are(2)(3)(4)(5)(6)(OEIS A094674 and A094675).Writing(7)the are given by the recurrence equation(8)together with(9)(10)(Abramowitz and Stegun 1972, p. 443).The parity of is (Arfken 1985, p. 633). is related to the spherical Bessel function of the first kind by(11)for and integer (Arfken 1985, p. 633).They also satisfy the differential identities(12)(13)and the recurrence relations(14)(15)(Arfken 1985, p. 634)...

### Debye's asymptotic representation

Debye's asymptotic representation is an asymptotic expansion for a Hankel function of the first kind with . For , , , and ,(1)For , , , , and ,(2)Finally, for , , and ,(3)

### Modified bessel function of the second kind

The modified bessel function of the second kind is the function which is one of the solutions to the modified Bessel differential equation. The modified Bessel functions of the second kind are sometimes called the Basset functions, modified Bessel functions of the third kind (Spanier and Oldham 1987, p. 499), or Macdonald functions (Spanier and Oldham 1987, p. 499; Samko et al. 1993, p. 20). The modified Bessel function of the second kind is implemented in the Wolfram Language as BesselK[nu, z]. is closely related to the modified Bessel function of the first kind and Hankel function ,(1)(2)(3)(Watson 1966, p. 185). A sum formula for is(4)where is the digamma function (Abramowitz and Stegun 1972). An integral formula is(5)which, for , simplifies to(6)Other identities are(7)for and(8)(9)The special case of gives as the integrals(10)(11)(Abramowitz and Stegun 1972, p. 376)...

### Catalan integrals

Special cases of general formulas due to Bessel.where is a Bessel function of the first kind. Now, let and . Then

### Modified bessel function of the first kind

A function which is one of the solutions to the modified Bessel differential equation and is closely related to the Bessel function of the first kind . The above plot shows for , 2, ..., 5. The modified Bessel function of the first kind is implemented in the Wolfram Language as BesselI[nu, z].The modified Bessel function of the first kind can be defined by the contour integral(1)where the contour encloses the origin and is traversed in a counterclockwise direction (Arfken 1985, p. 416).In terms of ,(2)For a real number , the function can be computed using(3)where is the gamma function. An integral formula is(4)which simplifies for an integer to(5)(Abramowitz and Stegun 1972, p. 376).A derivative identity for expressing higher order modified Bessel functions in terms of is(6)where is a Chebyshev polynomial of the first kind.The special case of gives as the series(7)..

### Spherical hankel function of the second kind

The spherical Hankel function of the second kind is defined by(1)(2)where is the Hankel function of the second kind and and are the spherical Bessel functions of the first and second kinds.It is implemented in Wolfram Language Version 6 as SphericalHankelH2[n, z].Explicitly, the first few are given by(3)(4)(5)(6)The derivative is given by(7)The plot above shows the real and imaginary parts of on the real axis for , 1, ..., 5.The plots above shows the real and imaginary parts of in the complex plane.

### Mehler's bessel function formula

For ,(1)(2)where is a zeroth order Bessel function of the first kind and is a zeroth order Bessel function of the second kind.

### Bourget function

The function defined by the contour integralwhere denotes the contour encircling the point once in a counterclockwise direction. It is equal to(Watson 1966, p. 326).

### Spherical hankel function of the first kind

The spherical Hankel function of the first kind is defined by(1)(2)where is the Hankel function of the first kind and and are the spherical Bessel functions of the first and second kinds.It is implemented in the Wolfram Language as SphericalHankelH1[n, z].Explicitly, the first few are(3)(4)(5)(6)The derivative is given by(7)The plot above shows the real and imaginary parts of on the real axis for , 1, ..., 5.The plots above shows the real and imaginary parts of in the complex plane.

### Lommel polynomial

The Lommel polynomials arise from the equation(1)where is a Bessel function of the first kind and is a complex number (Watson 1966, p. 294). The function is given for by(2)(3)where is a gamma function, is a Bessel function of the first kind, and is a generalized hypergeometric function. Since (1) must reduce to the usual recurrence formula for Bessel functions, it follows that(4)(5)

### Spherical bessel function of the second kind

The spherical Bessel function of the second kind, denoted or , is defined by(1)where is a Bessel function of the second kind and, in general, and are complex numbers.The spherical Bessel function of the second kind is implemented in the Wolfram Language as SphericalBesselY[n, z].The function is most commonly encountered in the case an integer, in which case it is given by(2)(3)(4)(5)where is a Bessel function of the first kind.Specific cases for small nonnegative are given by(6)(7)(8)

### Lommel's integrals

The two integrals involving Bessel functionsof the first kind given byandwhere is a Bessel function of the first kind.

### Lommel function

There are several functions called "Lommel functions." One type of Lommel function appear in the solution to the Lommel differential equation and are given by(1)where and are generalized and confluence hypergeometric functions, respectively and is typically denoted just as . is also given by(2)where and are Bessel functions of the first and second kinds (Watson 1966, p. 346; Gradshteyn and Ryzhik 2000, pp. 936-937).If a minus sign precedes the term in the general form of Lommel differential equation, then the solution is(3)where and are modified Bessel functions of the first and second kinds.A function closely related to is also sometimes defined (Gradshteyn and Ryzhik 2000, p. 936).Lommel functions of two variables are related to the Bessel function of the first kind and arise in the theory of diffraction (Chandrasekhar 1960, p. 369) and, in particular, Mie scattering (Watson 1966, p. 537),(4)(5)These..

### Spherical bessel function

A solution to the spherical Bessel differential equation. The two types of solutions are denoted (spherical Bessel function of the first kind) or (spherical Bessel function of the second kind).

### Bessel function

A function defined by the recurrence relations(1)and(2)The Bessel functions are more frequently defined as solutions to the differentialequation(3)There are two classes of solution, called the Bessel function of the first kind and Bessel function of the second kind . (A Bessel function of the third kind, more commonly called a Hankel function, is a special combination of the first and second kinds.) Several related functions are also defined by slightly modifying the defining equations.

### Sonine's integral

where is a Bessel function of the first kind and is the gamma function.

### Sommerfeld's formula

There are (at least) two equations known as Sommerfeld's formula. The first iswhere is a Bessel function of the first kind. The second states that under appropriate restrictions,

### Sinc function

Min Max The sinc function , also called the "sampling function," is a function that arises frequently in signal processing and the theory of Fourier transforms. The full name of the function is "sine cardinal," but it is commonly referred to by its abbreviation, "sinc." There are two definitions in common use. The one adopted in this work defines(1)where is the sine function, plotted above.This has the normalization(2)This function is implemented in the WolframLanguage as Sinc[x]. Min Max Re Im When extended into the complex plane, is illustrated above.An interesting property of is that the set of local extrema of corresponds to its intersections with the cosine function , as illustrated above.The derivative is given by(3)and the indefinite integral by(4)where is the sine integral.Woodward (1953), McNamee et al. (1971), and Bracewell (1999, p. 62) adoptthe alternative definition(5)(6)The latter..

### Kelvin functions

Kelvin defined the Kelvin functions bei and beraccording to(1)(2)(3)(4)where is a Bessel function of the first kind and is a modified Bessel function of the first kind. These functions satisfy the Kelvin differential equation.Similarly, the functions kei and kerby(5)where is a modified Bessel function of the second kind. For the special case ,(6)(7)

### Schl&auml;fli's formula

For ,where is a Bessel function of the first kind.

### Kei

The function is defined as the imaginary part of(1)where is a modified Bessel function of the second kind. Therefore,(2)where is the imaginary part.It is implemented as KelvinKei[nu,z]. has a complicated series given by Abramowitz and Stegun (1972, p. 380).The special case is commonly denoted and has the plot shown above. has the series expansion(3)where is the digamma function (Abramowitz and Stegun 1972, p. 380).

### Perfect power

A perfect power is a number of the form , where is a positive integer and . If the prime factorization of is , then is a perfect power iff .Including duplications (i.e., taking all numbers up to some cutoff and taking all their powers) and taking , the first few are 4, 8, 9, 16, 16, 25, 27, 32, 36, 49, 64, 64, 64, ... (OEIS A072103). Here, 16 is duplicated since(1)As shown by Goldbach, the sum of reciprocals of perfect powers (excluding 1) withduplications converges,(2)The first few numbers that are perfect powers in more than one way are 16, 64, 81,256, 512, 625, 729, 1024, 1296, 2401, 4096, ... (OEIS A117453).The first few perfect powers without duplications are 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, ... (OEIS A001597). Even more amazingly, the sum of the reciprocals of these numbers (excluding 1) is given by(3)(OEIS A072102), where is the Möbius function and is the Riemann zeta function.The numbers of perfect powers without duplications..

### Odd power

An odd power is a number of the form for an integer and a positive odd integer. The first few odd powers are 1, 8, 27, 32, 64, 125, 128, 216, 243, 343, 512, ... (OEIS A070265). Amazingly, the double series of reciprocals of the odd powers that are congruent to 3 (mod 4) is given by

### Lorentzian function

Min Max The Lorentzian function is the singly peaked function given by(1)where is the center and is a parameter specifying the width. The Lorentzian function is normalized so that(2)It has a maximum at , where(3)Its value at the maximum is(4)It is equal to half its maximum at(5)and so has full width at half maximum . The function has inflection points at(6)giving(7)where(8) Min Max Re Im The Lorentzian function extended into the complex planeis illustrated above.The Lorentzian function gives the shape of certain types of spectral lines and is the distribution function in the Cauchy distribution. The Lorentzian function has Fourier transform(9)The Lorentzian function can also be used as an apodization function, although its instrument function is complicated to express analytically...

### Gaussian function

Min Max Min Max Re Im In one dimension, the Gaussian function is the probabilitydensity function of the normal distribution,(1)sometimes also called the frequency curve. The full width at half maximum (FWHM) for a Gaussian is found by finding the half-maximum points . The constant scaling factor can be ignored, so we must solve(2)But occurs at , so(3)Solving,(4)(5)(6)(7)The full width at half maximum is thereforegiven by(8)In two dimensions, the circular Gaussian function is the distribution function for uncorrelated variates and having a bivariate normal distribution and equal standard deviation ,(9)The corresponding elliptical Gaussian function corresponding to is given by(10)The Gaussian function can also be used as an apodizationfunction(11)shown above with the corresponding instrumentfunction. The instrument function is(12)which has maximum(13)As , equation (12) reduces to(14)The hypergeometric function is also..

### Principal square root

The unique nonnegative square root of a nonnegative real number. For example, the principal square root of 9 is 3, although both and 3 are square roots of 9.The concept of principal square root cannot be extended to real negative numbers since the two square roots of a negative number cannot be distinguished until one of the two is defined as the imaginary unit, at which point and can then be distinguished. Since either choice is possible, there is no ambiguity in defining as "the" square root of .

### Jacobi theta functions

The Jacobi theta functions are the elliptic analogs of the exponential function, and may be used to express the Jacobi elliptic functions. The theta functions are quasi-doubly periodic, and are most commonly denoted in modern texts, although the notations and (Borwein and Borwein 1987) are sometimes also used. Whittaker and Watson (1990, p. 487) gives a table summarizing notations used by various earlier writers.The theta functions are given in the Wolfram Language by EllipticTheta[n, z, q], and their derivatives are given by EllipticThetaPrime[n, z, q].The translational partition function for an ideal gas can be derived using elliptic theta functions (Golden 1961, pp. 119 and 133; Melzak 1973, p. 122; Levine 2002, p. 838).The theta functions may be expressed in terms of the nome , denoted , or the half-period ratio , denoted , where and and are related by(1)Let the multivalued function be interpreted to stand..

### Jacobi elliptic functions

The Jacobi elliptic functions are standard forms of elliptic functions. The three basic functions are denoted , , and , where is known as the elliptic modulus. They arise from the inversion of the elliptic integral of the first kind,(1)where , is the elliptic modulus, and is the Jacobi amplitude, giving(2)From this, it follows that(3)(4)(5)(6)(7)(8)These functions are doubly periodic generalizations of the trigonometric functions satisfying(9)(10)(11)In terms of Jacobi theta functions,(12)(13)(14)(Whittaker and Watson 1990, p. 492), where (Whittaker and Watson 1990, p. 464) and the elliptic modulus is given by(15)Ratios of Jacobi elliptic functions are denoted by combining the first letter of the numerator elliptic function with the first of the denominator elliptic function. The multiplicative inverses of the elliptic functions are denoted by reversing the order of the two letters. These combinations give a total..

### Weber functions

Although Bessel functions of the second kind are sometimes called Weber functions, Abramowitz and Stegun (1972) define a separate Weber function as(1)These function may also be written as(2)where is a regularized hypergeometric function.This function is implemented in the Wolfram Language as WeberE[nu, z] and is an analog of the Anger function.Special values for real include(3)(4)(5)(6)where is a Struve function.Letting be a root of unity, another set of Weber functions is defined as(7)(8)(9)(10)(11)(Weber 1902, Atkin and Morain 1993), where is the Dedekind eta function and is the half-period ratio. These functions are related to the Ramanujan g- and G-functions and the elliptic lambda function.The Weber functions satisfy the identities(12)(13)(14)(15)(16)(17)(Weber 1902, Atkin and Morain 1993)...

### Versine

The versine, also known as the "versed sine," is a little-used trigonometric function defined by(1)(2)where is the sine and is the cosine.The versine can be extended to the complex planeas illustrated above.Its derivative is given by(3)and its indefinite integral by(4)

### Hyperbolic cosine

Min Max Min Max Re Im The hyperbolic cosine is defined as(1)The notation is sometimes also used (Gradshteyn and Ryzhik 2000, p. xxix). This function describes the shape of a hanging cable, known as the catenary. It is implemented in the Wolfram Language as Cosh[z].Special values include(2)(3)where is the golden ratio.The derivative is given by(4)where is the hyperbolic sine, and the indefinite integral by(5)where is a constant of integration.The hyperbolic cosine has Taylor series(6)(7)(OEIS A010050).

### Coversine

The coversine is a little-used entire trigonometricfunction defined by(1)(2)where is the versine and is the sine.The coversine can be extended to the complex planeas illustrated above.Its derivative is given by(3)and its indefinite integral by(4)

### Vercosine

The vercosine, written and also known as the "versed cosine," is a little-used trigonometric function defined by(1)(2)where is the cosine.

### Hyperbolic cosecant

The hyperbolic cosecant is defined as(1)It is implemented in the Wolfram Languageas Csch[z].It is related to the hyperbolic cotangentthough(2)The derivative is given by(3)where is the hyperbolic cotangent, and the indefinite integral by(4)where is a constant of integration.It has Taylor series(5)(6)(7)(OEIS A036280 and A036281), where is a Bernoulli polynomial and is a Bernoulli number.Sums include(8)(9)(OEIS A110191; Berndt 1977).The plot above shows a bifurcation diagram for .

### Covercosine

The covercosine, also called the coversed cosine, is a little-used trigonometric function defined by(1)(2)where is the vercosine and is the sine.

### Struve function

The Struve function, denoted or occasionally , is defined as(1)where is the gamma function (Abramowitz and Stegun 1972, pp. 496-499). Watson (1966, p. 338) defines the Struve function as(2)The Struve function is implemented as StruveH[n,z].The Struve function and its derivatives satisfy(3)For integer , the Struve function gives the solution to(4)where is the double factorial.The Struve function arises in the problem of the rigid-piston radiator mounted in an infinite baffle, which has radiation impedance given by(5)where(6)(7)where is the piston radius, is the wavenumber , is the density of the medium, is the speed of sound, is the first order Bessel function of the first kind and is the Struve function of the first kind.The illustrations above show the values of the Struve function in the complex plane.For integer orders,(8)(9)(10)(11)(OEIS A001818 and A079484).A simple approximation of for real is given by(12)with..

### Cosine

The cosine function is one of the basic functions encountered in trigonometry (the others being the cosecant, cotangent, secant, sine, and tangent). Let be an angle measured counterclockwise from the x-axis along the arc of the unit circle. Then is the horizontal coordinate of the arc endpoint.The common schoolbook definition of the cosine of an angle in a right triangle (which is equivalent to the definition just given) is as the ratio of the lengths of the side of the triangle adjacent to the angle and the hypotenuse, i.e.,(1)A convenient mnemonic for remembering the definition of the sine, cosine, and tangent is SOHCAHTOA (sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, tangent equals opposite over adjacent).As a result of its definition, the cosine function is periodic with period . By the Pythagorean theorem, also obeys the identity(2) Min Max Re Im The definition of the cosine function can be extended to..

### Spherical bessel function of the first kind

The spherical Bessel function of the first kind, denoted , is defined by(1)where is a Bessel function of the first kind and, in general, and are complex numbers.The function is most commonly encountered in the case an integer, in which case it is given by(2)(3)(4)Equation (4) shows the close connection between and the sinc function .Spherical Bessel functions are implemented in the Wolfram Language as SphericalBesselJ[nu, z] using the definition(5)which differs from the "traditional version" along the branch cut of the square root function, i.e., the negative real axis (e.g., at ), but has nicer analytic properties for complex (Falloon 2001).The first few functions are(6)(7)(8)which includes the special value(9)

### Hacoversine

The hacoversine, also known as the hacoversed sine and cohaversine, is a little-used trigonometric function defined by(1)(2)where is the coversine and is the sine.

### Bessel function of the second kind

A Bessel function of the second kind (e.g, Gradshteyn and Ryzhik 2000, p. 703, eqn. 6.649.1), sometimes also denoted (e.g, Gradshteyn and Ryzhik 2000, p. 657, eqn. 6.518), is a solution to the Bessel differential equation which is singular at the origin. Bessel functions of the second kind are also called Neumann functions or Weber functions. The above plot shows for , 1, 2, ..., 5. The Bessel function of the second kind is implemented in the Wolfram Language as BesselY[nu, z].Let be the first solution and be the other one (since the Bessel differential equation is second-order, there are two linearly independent solutions). Then(1)(2)Take (1) minus (2),(3)(4)so , where is a constant. Divide by ,(5)(6)Rearranging and using gives(7)(8)where is the so-called Bessel function of the second kind. can be defined by(9)(Abramowitz and Stegun 1972, p. 358), where is a Bessel function of the first kind and, for an integer..

### Sine

The sine function is one of the basic functions encountered in trigonometry (the others being the cosecant, cosine, cotangent, secant, and tangent). Let be an angle measured counterclockwise from the x-axis along an arc of the unit circle. Then is the vertical coordinate of the arc endpoint, as illustrated in the left figure above.The common schoolbook definition of the sine of an angle in a right triangle (which is equivalent to the definition just given) is as the ratio of the lengths of the side of the triangle opposite the angle and the hypotenuse, i.e.,(1)A convenient mnemonic for remembering the definition of the sine, as well as the cosine and tangent, is SOHCAHTOA (sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, tangent equals opposite over adjacent).As a result of its definition, the sine function is periodic with period . By the Pythagorean theorem, also obeys the identity(2) Min Max Re Im The definition..

### Hacovercosine

The hacovercosine, also known as the hacoversed cosine and cohavercosine, is a little-used trigonometric function defined by(1)(2)where is the covercosine and is the sine.

### Bessel function of the first kind

The Bessel functions of the first kind are defined as the solutions to the Bessel differential equation(1)which are nonsingular at the origin. They are sometimes also called cylinder functions or cylindrical harmonics. The above plot shows for , 1, 2, ..., 5. The notation was first used by Hansen (1843) and subsequently by Schlömilch (1857) to denote what is now written (Watson 1966, p. 14). However, Hansen's definition of the function itself in terms of the generating function(2)is the same as the modern one (Watson 1966, p. 14). Bessel used the notation to denote what is now called the Bessel function of the first kind (Cajori 1993, vol. 2, p. 279).The Bessel function can also be defined by the contour integral(3)where the contour encloses the origin and is traversed in a counterclockwise direction (Arfken 1985, p. 416).The Bessel function of the first kind is implemented in the Wolfram Language as BesselJ[nu,..

### Shi

Min Max Min Max Re Im The hyperbolic sine integral, often called the "Shi function" for short, is defined by(1)The function is implemented in the WolframLanguage as the function SinhIntegral[z].It has Maclaurin series(2)(3)(OEIS A061079).It has derivative(4)and indefinite integral(5)

### Generalized hypergeometric function

The generalized hypergeometric function is given by a hypergeometricseries, i.e., a series for which the ratio of successive terms can be written(1)(The factor of in the denominator is present for historical reasons of notation.) The resulting generalized hypergeometric function is written(2)(3)where is the Pochhammer symbol or rising factorial(4)A generalized hypergeometric function therefore has parameters of type 1 and parameters of type 2.A number of generalized hypergeometric functions has special names. is called a confluent hypergeometric limit function, and is implemented in the Wolfram Language as Hypergeometric0F1[b, z]. (also denoted ) is called a confluent hypergeometric function of the first kind, and is implemented in the Wolfram Language as Hypergeometric1F1[a, b, z]. The function is often called "the" hypergeometric function or Gauss's hypergeometric function, and is implemented in the Wolfram..

### Ber

The function is defined through the equation(1)where is a Bessel function of the first kind, so(2)where is the real part.The function is implemented in the Wolfram Language as KelvinBer[nu, z].The function has the series expansion(3)where is the gamma function (Abramowitz and Stegun 1972, p. 379), which can be written in closed form as(4)where is a modified Bessel function of the first kind.The special case , commonly denoted , corresponds to(5)where is the zeroth order Bessel function of the first kind. The function has the series expansion(6)which can be written in closed form as(7)(8)

### Bei

The function is defined through the equation(1)where is a Bessel function of the first kind, so(2)where is the imaginary part.It is implemented in the Wolfram Language as KelvinBei[nu, z]. has the series expansion(3)where is the gamma function (Abramowitz and Stegun 1972, p. 379), which can be written in closed form as(4)where is a modified Bessel function of the first kind.The special case , commonly denoted , corresponds to(5)where is the zeroth order Bessel function of the first kind. The function has the series expansion(6)Closed forms include(7)(8)

### Neville theta functions

The functions(1)(2)(3)(4)where and are the Jacobi theta functions and is the complete elliptic integral of the first kind.The Neville theta functions are implemented in the Wolfram Language as NevilleThetaC[z, m], NevilleThetaD[z, m], NevilleThetaN[z, m], and NevilleThetaS[z, m].

### Anger function

An entire function which is a generalization of the Bessel function of the first kind defined byAnger's original function had an upper limit of , but the current notation was standardized by Watson (1966).The Anger function may also be written aswhere is a regularized hypergeometric function.If is an integer , then , where is a Bessel function of the first kind.The Anger function is implemented in the Wolfram Language as AngerJ[nu, z].

### Modified struve function

(1)(2)where is the gamma function. is related to the ordinary Struve function by(3)(Abramowitz and Stegun 1972, p. 498).The Struve function is implemented in the Wolfram Language as StruveL[n, z].The plots above show in the complex plane.

### Exponential function

Min Max Min Max Re Im The exponential function is the entire functiondefined by(1)where e is the solution of the equation so that . is also the unique solution of the equation with .The exponential function is implemented in the WolframLanguage as Exp[z].It satisfies the identity(2)If ,(3)The exponential function satisfies the identities(4)(5)(6)(7)where is the Gudermannian (Beyer 1987, p. 164; Zwillinger 1995, p. 485).The exponential function has Maclaurin series(8)and satisfies the limit(9)If(10)then(11)(12)(13)The exponential function has continued fraction(14)(Wall 1948, p. 348). Min Max Re Im The above plot shows the function (Trott 2004, pp. 165-166).Integrals involving the exponential function include(15)(16)(Borwein et al. 2004, p. 55)...

### Inverse trigonometric functions

The inverse trigonometric functions are the inverse functions of the trigonometric functions, written , , , , , and .Alternate notations are sometimes used, as summarized in the following table.alternate notations (Spanier and Oldham 1987, p. 333; Gradshteyn and Ryzhik 2000, p. 207) (Spanier and Oldham 1987, p. 333), (Spanier and Oldham 1987, p. 333; Gradshteyn and Ryzhik 2000, p. 208; Jeffrey 2000, p. 127) (Spanier and Oldham 1987, p. 333), (Spanier and Oldham 1987, p. 333; Gradshteyn and Ryzhik 2000, p. 207) (Spanier and Oldham 1987, p. 333; Gradshteyn and Ryzhik 2000, p. 209) (Spanier and Oldham 1987, p. 333; Gradshteyn and Ryzhik 2000, p. 207) (Spanier and Oldham 1987, p. 333), (Spanier and Oldham 1987, p. 333; Gradshteyn and Ryzhik 2000, p. 208; Jeffrey 2000, p. 127)The inverse trigonometric functions are multivalued...

### Parabolic cylinder function

The parabolic cylinder functions are a class of functions sometimes called Weber functions. There are a number of slightly different definitions in use by various authors.Whittaker and Watson (1990, p. 347) define the parabolic cylinder functions as solutions to the Weber differential equation(1)The two independent solutions are given by and , where(2)(3)which, in the right half-plane , is equivalent to(4)where is the Whittaker function (Whittaker and Watson 1990, p. 347; Gradshteyn and Ryzhik 2000, p. 1018) and is a confluent hypergeometric function of the first kind.This function is implemented in the Wolfram Language as ParabolicCylinderD[nu, z].For a nonnegative integer , the solution reduces to(5)(6)where is a Hermite polynomial and is a modified Hermite polynomial. Special cases include(7)(8)for , where is an modified Bessel function of the second kind.Plots of the function in the complex plane are shown..

### Siegel theta function

The Siegel theta function is a -invariant meromorphic function on the space of all symmetric complex matrices with positive definite imaginary part. It is defined bywhere is a complex -vector, is an integer -vector that ranges over the entire -D lattice of integers, and denotes a matrix (or vector) transpose.The Siegel theta function is implemented in the Wolfram Language as SiegelTheta[Omega, s].This function was investigated by many of the luminaries of nineteenth century mathematics, Riemann, Weierstrass, Frobenius, Poincaré. Umemura has expressed the roots of an arbitrary polynomial in terms of Siegel theta functions (Mumford 1984).

### Riemann theta function

The Riemann theta function is a complex function of complex variables that occurs in the construction of quasi-periodic solutions of various equations in mathematical physics (Deconinck et al. 2004). Any Abelian function can be expressed as a ratio of homogeneous polynomials of the Riemann theta function (Igusa 1972, Deconinck et al. 2004).Let the imaginary part of a matrix be positive definite, and be a row vector with coefficients in . Then the Riemann theta function is defined byRiemann (1857) first considered these functions associated with Riemann surfaces, and the most general form of the Riemann theta function defined above was first considered by Wirtinger (1895).An overview of the properties of the Riemann theta function is given by Mumford (1983, 1984, 1991), and algorithms for numeric computations have been developed by Deconinck et al. (2004)...

### Nome

Given a Jacobi theta function, the nomeis defined as(1)(2)(3)(Borwein and Borwein 1987, pp. 41, 109 and 114), where is the half-period ratio, is the complete elliptic integral of the first kind, and is the elliptic modulus. The nome is implemented in the Wolfram Language as EllipticNomeQ[m].Extreme care is needed when consulting the literature, as it is common in the theory of modular functions (and in particular the Dedekind eta function) to use the symbol to denote , i.e., the square of the usual nome (e.g., Berndt 1993, p. 139). In this work, the modular version of is denoted(4)(e.g., Borwein and Borwein 1987, p. 118).The nome in plotted above in the complex -plane.Various notations for Jacobi theta functionsinvolving the nome include(5)where is the half-period ratio (Whittaker and Watson 1990, p. 464) and(6)Special values include(7)(8)(9)The nome has Maclaurin series in parameter given by(10)(OEIS A002639..

### Mordell integral

The integralwhich is related to the Jacobi theta functions, mock theta functions, Riemann zeta function, and Siegel theta function.

### Landen's formula

where are Jacobi theta functions. This transformation was used by Gauss to show that elliptic integrals could be computed using the arithmetic-geometric mean.

### Jacobi triple product

The Jacobi triple product is the beautiful identity(1)In terms of the Q-functions, (1)is written(2)which is one of the two Jacobi identities. In q-series notation, the Jacobi triple product identity is written(3)for and (Gasper and Rahman 1990, p. 12; Leininger and Milne 1999). Another form of the identity is(4)(Hirschhorn 1999).Dividing (4) by and letting gives the limiting case(5)(6)(Jacobi 1829; Hardy and Wright 1979; Hardy 1999, p. 87; Hirschhorn 1999; Leininger and Milne 1999).For the special case of , (◇) becomes(7)(8)(9)(10)where is a Jacobi elliptic function. In terms of the two-variable Ramanujan theta function , the Jacobi triple product is equivalent to(11)(Berndt et al. 2000).One method of proof for the Jacobi identity proceeds by defining the function(12)(13)Then(14)Taking (14) (13),(15)(16)which yields the fundamental relation(17)Now define(18)(19)Using (17), (19) becomes(20)(21)so(22)Expand..

### Gegenbauer polynomial

The Gegenbauer polynomials are solutions to the Gegenbauer differential equation for integer . They are generalizations of the associated Legendre polynomials to -D space, and are proportional to (or, depending on the normalization, equal to) the ultraspherical polynomials .Following Szegö, in this work, Gegenbauer polynomials are given in terms of the Jacobi polynomials with by(1)(Szegö 1975, p. 80), thus making them equivalent to the Gegenbauer polynomials implemented in the Wolfram Language as GegenbauerC[n, lambda, x]. These polynomials are also given by the generating function(2)The first few Gegenbauer polynomials are(3)(4)(5)(6)In terms of the hypergeometric functions,(7)(8)(9)They are normalized by(10)for .Derivative identities include(11)(12)(13)(14)(15)(16)(17)(18)(Szegö 1975, pp. 80-83).A recurrence relation is(19)for , 3, ....Special double- formulas also exist(20)(21)(22)(23)Koschmieder..

### Zernike polynomial

The Zernike polynomials are a set of orthogonal polynomials that arise in the expansion of a wavefront function for optical systems with circular pupils. The odd and even Zernike polynomials are given by(1)where the radial function is defined for and integers with by(2)Here, is the azimuthal angle with and is the radial distance with (Prata and Rusch 1989). The even and odd polynomials are sometimes also denoted(3)(4)Zernike polynomials are implemented in the Wolfram Language as ZernikeR[n, m, rho].Other closed forms for include(5)for odd and , where is the gamma function and is a hypergeometric function. This can also be written in terms of the Jacobi polynomial as(6)The first few nonzero radial polynomials are(7)(8)(9)(10)(11)(12)(13)(14)(15)(Born and Wolf 1989, p. 465).The radial functions satisfy the orthogonality relation(16)where is the Kronecker delta, and are related to the Bessel function of the first kind by(17)(Born..

### Eberlein polynomial

The Eberlein polynomials of degree and variable are the orthogonal polynomials arising in the Johnson scheme that may be defined by(1)(2)

### Legendre polynomial

The Legendre polynomials, sometimes called Legendre functions of the first kind, Legendre coefficients, or zonal harmonics (Whittaker and Watson 1990, p. 302), are solutions to the Legendre differential equation. If is an integer, they are polynomials. The Legendre polynomials are illustrated above for and , 2, ..., 5. They are implemented in the Wolfram Language as LegendreP[n, x].The associated Legendre polynomials and are solutions to the associated Legendre differential equation, where is a positive integer and , ..., .The Legendre polynomial can be defined by the contour integral(1)where the contour encloses the origin and is traversed in a counterclockwise direction (Arfken 1985, p. 416).The first few Legendre polynomials are(2)(3)(4)(5)(6)(7)(8)When ordered from smallest to largest powers and with the denominators factored out, the triangle of nonzero coefficients is 1, 1, , 3, , 5, 3, , ... (OEIS A008316)...

### Chebyshev polynomial of the second kind

A modified set of Chebyshev polynomials defined by a slightly different generating function. They arise in the development of four-dimensional spherical harmonics in angular momentum theory. They are a special case of the Gegenbauer polynomial with . They are also intimately connected with trigonometric multiple-angle formulas. The Chebyshev polynomials of the second kind are denoted , and implemented in the Wolfram Language as ChebyshevU[n, x]. The polynomials are illustrated above for and , 2, ..., 5.The first few Chebyshev polynomials of the second kind are(1)(2)(3)(4)(5)(6)(7)When ordered from smallest to largest powers, the triangle of nonzero coefficients is 1; 2; , 4; , 8; 1, , 16; 6, , 32; ... (OEIS A053117).The defining generating function of the Chebyshevpolynomials of the second kind is(8)(9)for and . To see the relationship to a Chebyshev polynomial of the first kind , take of equation (9) to obtain(10)(11)Multiplying (◇)..

### Laguerre polynomial

The Laguerre polynomials are solutions to the Laguerre differential equation with . They are illustrated above for and , 2, ..., 5, and implemented in the Wolfram Language as LaguerreL[n, x].The first few Laguerre polynomials are(1)(2)(3)(4)When ordered from smallest to largest powers and with the denominators factored out, the triangle of nonzero coefficients is 1; , 1; 2, , 1; , 18, 1; 24, , ... (OEIS A021009). The leading denominators are 1, , 2, , 24, , 720, , 40320, , 3628800, ... (OEIS A000142).The Laguerre polynomials are given by the sum(5)where is a binomial coefficient.The Rodrigues representation for theLaguerre polynomials is(6)and the generating function for Laguerre polynomialsis(7)(8)A contour integral that is commonly taken asthe definition of the Laguerre polynomial is given by(9)where the contour encloses the origin but not the point (Arfken 1985, pp. 416 and 722).The Laguerre polynomials satisfy the recurrencerelations(10)(Petkovšek..

### Chebyshev polynomial of the first kind

The Chebyshev polynomials of the first kind are a set of orthogonal polynomials defined as the solutions to the Chebyshev differential equation and denoted . They are used as an approximation to a least squares fit, and are a special case of the Gegenbauer polynomial with . They are also intimately connected with trigonometric multiple-angle formulas. The Chebyshev polynomials of the first kind are denoted , and are implemented in the Wolfram Language as ChebyshevT[n, x]. They are normalized such that . The first few polynomials are illustrated above for and , 2, ..., 5.The Chebyshev polynomial of the first kind can be defined by the contour integral(1)where the contour encloses the origin and is traversed in a counterclockwise direction (Arfken 1985, p. 416).The first few Chebyshev polynomials of the first kind are(2)(3)(4)(5)(6)(7)(8)When ordered from smallest to largest powers, the triangle of nonzero coefficients is 1; 1; , 2;..

### Krawtchouk polynomial

Let be a step function with the jump(1)at , 1, ..., , where , and . Then the Krawtchouk polynomial is defined by(2)(3)(4)for , 1, ..., . The first few Krawtchouk polynomials are(5)(6)(7)Koekoek and Swarttouw (1998) define the Krawtchouk polynomial without the leading coefficient as(8)The Krawtchouk polynomials have weighting function(9)where is the gamma function, recurrence relation(10)and squared norm(11)It has the limit(12)where is a Hermite polynomial.The Krawtchouk polynomials are a special case of the Meixnerpolynomials of the first kind.

### Charlier polynomial

The orthogonal polynomials defined by(1)(2)where is the Pochhammer symbol (Koekoek and Swarttouw 1998). The first few are given by(3)(4)(5)

### Pollaczek polynomial

Let , and write(1)Then define by the generating function(2)The generating function may also be written(3)where is a Chebyshev polynomial of the second kind.Pollaczek polynomials satisfy the recurrence relation(4)for , 3, ... with(5)(6)In terms of the hypergeometric function ,(7)They obey the orthogonality relation(8)where is the Kronecker delta, for , 1, ..., with the weighting function(9)

### Jacobi polynomial

The Jacobi polynomials, also known as hypergeometric polynomials, occur in the study of rotation groups and in the solution to the equations of motion of the symmetric top. They are solutions to the Jacobi differential equation, and give some other special named polynomials as special cases. They are implemented in the Wolfram Language as JacobiP[n, a, b, z].For , reduces to a Legendre polynomial. The Gegenbauer polynomial(1)and Chebyshev polynomial of thefirst kind can also be viewed as special cases of the Jacobi polynomials.Plugging(2)into the Jacobi differential equationgives the recurrence relation(3)for , 1, ..., where(4)Solving the recurrence relation gives(5)for . They form a complete orthogonal system in the interval with respect to the weighting function(6)and are normalized according to(7)where is a binomial coefficient. Jacobi polynomials can also be written(8)where is the gamma function and(9)Jacobi polynomials..

### Associated legendre polynomial

The associated Legendre polynomials and generalize the Legendre polynomials and are solutions to the associated Legendre differential equation, where is a positive integer and , ..., . They are implemented in the Wolfram Language as LegendreP[l, m, x]. For positive , they can be given in terms of the unassociated polynomials by(1)(2)where are the unassociated Legendre polynomials. The associated Legendre polynomials for negative are then defined by(3)There are two sign conventions for associated Legendre polynomials. Some authors (e.g., Arfken 1985, pp. 668-669) omit the Condon-Shortley phase , while others include it (e.g., Abramowitz and Stegun 1972, Press et al. 1992, and the LegendreP[l, m, z] command in the Wolfram Language). Care is therefore needed in comparing polynomials obtained from different sources. One possible way to distinguish the two conventions is due to Abramowitz and Stegun (1972, p. 332), who..

### Hermite polynomial

The Hermite polynomials are set of orthogonal polynomials over the domain with weighting function , illustrated above for , 2, 3, and 4. Hermite polynomials are implemented in the Wolfram Language as HermiteH[n, x].The Hermite polynomial can be defined by the contour integral(1)where the contour encloses the origin and is traversed in a counterclockwise direction (Arfken 1985, p. 416).The first few Hermite polynomials are(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)When ordered from smallest to largest powers, the triangle of nonzero coefficientsis 1; 2; -2, 4; -12, 8; 12, -48, 16; 120, -160, 32; ... (OEIS A059343).The values may be called Hermite numbers.The Hermite polynomials are a Sheffer sequencewith(13)(14)(Roman 1984, p. 30), giving the exponentialgenerating function(15)Using a Taylor series shows that(16)(17)Since ,(18)(19)Now define operators(20)(21)It follows that(22)(23)(24)(25)(26)so(27)and(28)(Arfken..

### Associated laguerre polynomial

Solutions to the associated Laguerre differential equation with and an integer are called associated Laguerre polynomials (Arfken 1985, p. 726) or, in older literature, Sonine polynomials (Sonine 1880, p. 41; Whittaker and Watson 1990, p. 352). Associated Laguerre polynomials are implemented in the Wolfram Language as LaguerreL[n, k, x]. In terms of the unassociated Laguerre polynomials,(1)The Rodrigues representation for theassociated Laguerre polynomials is(2)(3)(4)(5)where is a Whittaker function.The associated Laguerre polynomials are a Sheffersequence with(6)(7)giving the generating function(8)(9)where the usual factor of in the denominator has been suppressed (Roman 1984, p. 31). Many interesting properties of the associated Laguerre polynomials follow from the fact that (Roman 1984, p. 31).The associated Laguerre polynomials are given explicitly by the formula(10)where..

### Hahn polynomial

The orthogonal polynomials defined by(1)where is the Pochhammer symbol and is a generalized hypergeometric function (Koepf 1998). The first few are given by(2)(3)Koekoek and Swarttouw (1998) define another Hahn polynomial(4)the dual Hahn polynomial(5)the continuous Hahn polynomial(6)and the continuous dual Hahn polynomial(7)for , 1, ..., , and where(8)

### Legendre duplication formula

Gamma functions of argument can be expressed in terms of gamma functions of smaller arguments. From the definition of the beta function,(1)Now, let , then(2)and , so and(3)(4)(5)(6)Now, use the beta function identity(7)to write the above as(8)Solving for and using then gives(9)(10)

### Trigamma function

Min Max Re Im A special function corresponding to a polygamma function with , given by(1)An alternative function(2)is sometimes called the trigamma function, where(3)Sums and differences of for small integral and can be expressed in terms of Catalan's constant and . For example,(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)

### Lanczos approximation

An approximation for the gamma function with is given by(1)where is an arbitrary constant such that ,(2)where is a Pochhammer symbol and(3)and(4)(5)with (Lanczos 1964; Luke 1969, p. 30). satisfies(6)and if is a positive integer, then satisfies the identity(7)(Luke 1969, p. 30).A similar result is given by(8)(9)(10)where is a Pochhammer symbol, is a factorial, and(11)The first few values of are(12)(13)(14)(15)(16)(OEIS A054379 and A054380; Whittaker and Watson 1990, p. 253). Note that Whittaker and Watson incorrectly give as 227/60.Yet another related result gives(17)(Whittaker and Watson 1990, p. 261), where is a Hurwitz zeta function and is a polygamma function.

### Kramp's symbol

The symbol defined by(1)(2)(3)where is the Pochhammer symbol and is the gamma function. Note that the definition by Erdélyi et al. (1981, p. 52) incorrectly gives the prefactor of (3) as .

### Dixon's identity

where is a binomial coefficient and is a gamma function.

### Thomae's theorem

Thomae's theorem, also called Thomae's transformation, is the generalizedhypergeometric function identity(1)where is the gamma function, is a generalized hypergeometric function,(2)and (Bailey 1935, p. 14). It is a generalization of Dixon's theorem (Slater 1966, p. 52).An equivalent formulation is given by(3)(Hardy 1999, p. 104). The symmetry of this form was used by Ramanujan in his proof of the identity, which is essentially the same as Thomae's. Interestingly, this is one of the few cases in which Ramanujan gives an explicit proof of one of his propositions (Hardy 1999, p. 104).A special case of the theorem is given by(4)(J. Sondow, pers. comm., May 25, 2003).

### Digamma function

Min Max Re Im A special function which is given by the logarithmic derivative of the gamma function (or, depending on the definition, the logarithmic derivative of the factorial).Because of this ambiguity, two different notations are sometimes (but not always) used, with(1)defined as the logarithmic derivative of the gamma function , and(2)defined as the logarithmic derivative of the factorial function. The two are connected by the relationship(3)The th derivative of is called the polygamma function, denoted . The notation(4)is therefore frequently used for the digamma function itself, and Erdélyi et al. (1981) use the notation for . The digamma function is returned by the function PolyGamma[z] or PolyGamma[0, z] in the Wolfram Language, and typeset using the notation .The digamma function arises in simple sums such as(5)(6)where is a Lerch transcendent.Special cases are given by(7)(8)(9)(10)Gauss's digamma theorem states..

### Incomplete gamma function

The "complete" gamma function can be generalized to the incomplete gamma function such that . This "upper" incomplete gamma function is given by(1)For an integer (2)(3)where is the exponential sum function. It is implemented as Gamma[a, z] in the Wolfram Language.The special case of can be expressed in terms of the subfactorial as(4)The incomplete gamma function has continued fraction(5)(Wall 1948, p. 358).The lower incomplete gamma function is given by(6)(7)(8)where is the confluent hypergeometric function of the first kind. For an integer ,(9)(10)It is implemented as Gamma[a,0, z] in the Wolfram Language.By definition, the lower and upper incomplete gamma functions satisfy(11)The exponential integral is closely related to the incomplete gamma function by(12)Therefore, for real ,(13)..

### Central beta function

Min Max Re Im The central beta function is defined by(1)where is the beta function. It satisfies the identities(2)(3)(4)(5)With , the latter gives the Wallis formula. For , 2, ... the first few values are 1, 1/6, 1/30, 1/140, 1/630, 1/2772, ... (OEIS A002457), which have denominators .When ,(6)where(7)The central beta function satisfies(8)(9)(10)(11)For an odd positive integer, the central beta function satisfies the identity(12)

### Regularized gamma function

The regularized gamma functions are defined by(1)(2)where and are incomplete gamma functions and is a complete gamma function. The function is implemented in the Wolfram Language as GammaRegularized[a, 0, z], and is implemented as GammaRegularized[a, z]. and satisfy the identity(3)The derivatives of and are(4)(5)and the second derivatives are(6)(7)The integrals are(8)(9)

### Incomplete beta function

A generalization of the complete beta function definedby(1)sometimes also denoted . The so-called Chebyshev integral is given by(2)The incomplete beta function is implemented in the Wolfram Language as Beta[z, a, b].It is given in terms of hypergeometric functionsby(3)(4)It is also given by the series(5)where is a Pochhammer symbol.The incomplete beta function reduces to the usual beta function when ,(6)It has derivative(7)and indefinite integral(8)

### Regularized beta function

The regularized beta function is defined bywhere is the incomplete beta function and is the (complete) beta function. The regularized beta function is sometimes also denoted and is implemented in the Wolfram Language as BetaRegularized[z, a, b]. The four-argument version BetaRegularized[z1, z2, a, b] is equivalent to .

### Hankel's symbol

The symbol defined by(1)(2)where is the gamma function. If is an integer, then this simplifies to(3)given incorrectly by Erdélyi et al. (1981, p. 52).

Letwhere is a Pochhammer symbol, and let be a negative integer. Thenwhere is the gamma function.

### Polygamma function

A special function mostly commonly denoted , , or which is given by the st derivative of the logarithm of the gamma function (or, depending on the definition, of the factorial ). This is equivalent to the th normal derivative of the logarithmic derivative of (or ) and, in the former case, to the th normal derivative of the digamma function . Because of this ambiguity in definition, two different notations are sometimes (but not always) used, namely(1)(2)(3)which, for can be written as(4)(5)where is the Hurwitz zeta function.The alternate notation(6)is sometimes used, with the two notations connected by(7)Unfortunately, Morse and Feshbach (1953) adopt a notation no longer in standard use in which Morse and Feshbach's "" is equal to in the usual notation. Also note that the function is equivalent to the digamma function and is sometimes known as the trigamma function. is implemented in the Wolfram Language as PolyGamma[n, z] for positive..

### Binet's log gamma formulas

Binet's first formula for , where is a gamma function, is given byfor (Erdélyi et al. 1981, p. 21; Whittaker and Watson 1990, p. 249).Binet's second formula isfor (Erdélyi et al. 1981, p. 22; Whittaker and Watson 1990, pp. 250-251).

### Gauss's hypergeometric theorem

for , where is a (Gauss) hypergeometric function. If is a negative integer , this becomeswhich is known as the Chu-Vandermonde identity.

### Nu function

(1)(2)where is the gamma function (Erdélyi et al. 1981, p. 217; Prudnikov et al. 1990, p. 799; Gradshteyn and Ryzhik 2000, p. 1109).The notation is also sometimes used to denote the divisor function giving the number of divisors or the number of distinct prime factors of a positive integer .

### Mu function

There are two functions commonly denoted , each of which is defined in terms of integrals. Another unrelated mathematical function represented using the Greek letter is the Möbius function.The two-argument -function is defined by the definite integralwhere is the gamma function (Erdélyi et al. 1981a, p. 388; Prudnikov et al. 1990, p. 798; Gradshteyn and Ryzhik 2000, p. 1109), while the three-argument -function is defined by(Prudnikov et al. 1990, p. 798; Gradshteyn and Ryzhik 2000, p. 1109).

### Morley's formula

(1)(2)where is a Pochhammer symbol and is the gamma function. This is a special case of the identity(3)

### Barnes' lemma

If a contour in the complex plane is curved such that it separates the increasing and decreasing sequences of poles, thenwhere is the gamma function (Bailey 1935, p. 7).Barnes' second lemma states thatprovided that (Bailey 1935, pp. 42-43).

### Mellin's formula

where is the digamma function and is the gamma function.

### Malmst&eacute;n's formula

The integral representation of by(1)(2)where is the log gamma function and is the digamma function.

### Bailey's theorem

Let be the gamma function and denote a double factorial, thenWriting the sums explicitly, Bailey's theorem states

### Walsh function

The Walsh functions consist of trains of square pulses (with the allowed states being and 1) such that transitions may only occur at fixed intervals of a unit time step, the initial state is always , and the functions satisfy certain other orthogonality relations. In particular, the Walsh functions of order are given by the rows of the Hadamard matrix when arranged in so-called "sequency" order (Thompson et al. 1986, p. 204; Wolfram 2002, p. 1073). There are Walsh functions of length , illustrated above for , 2, and 3.Walsh functions were used by electrical engineers such as Frank Fowle in the 1890s to find transpositions of wires that minimized crosstalk and were introduced into mathematics by Walsh (1923; Wolfram 2002, p. 1073).Amazingly, concatenating the Walsh functions (while simultaneously replacing s by 0s), where is the ceiling function, gives the Thue-Morse sequence (Wolfram 2002, p. 1073)...

### Sal

An odd Walsh function with sequency defined by

### Cal

An even Walsh function with sequency defined by

### Boxcar function

The functionwhich is equal to 1 for and 0 otherwise. Here is the Heaviside step function. The special case gives the unit rectangle function.

### Dirichlet integrals

There are several types of integrals which go under the name of a "Dirichlet integral." The integral(1)appears in Dirichlet's principle.The integral(2)where the kernel is the Dirichlet kernel, gives the th partial sum of the Fourier series.Another integral is denoted(3)for , ..., .There are two types of Dirichlet integrals which are denoted using the letters , , , and . The type 1 Dirichlet integrals are denoted , , and , and the type 2 Dirichlet integrals are denoted , , and .The type 1 integrals are given by(4)(5)where is the gamma function. In the case ,(6)where the integration is over the triangle bounded by the x-axis, y-axis, and line and is the beta function.The type 2 integrals are given for -D vectors and , and ,(7)(8)(9)where(10)(11)and are the cell probabilities. For equal probabilities, . The Dirichlet integral can be expanded as a multinomial series as(12)For small , and can be expressed analytically either partially or..

### Glasser function

The interesting function defined by the definiteintegralillustrated above (Glasser 1990). The integral cannot be done in closed form, but has a number of remarkable properties, the foremost of which is that the first "hump" has a single subhump, the second hump has two subhumps, and so on. The function is asymptotic toThe derivatives of the Glasser function are illustrated above.

### Debye functions

The first Debye function is defined by(1)(2)for , , and are Bernoulli numbers. Particular values are given by(3)(4)(5)where is a polylogarithm and is the Riemann zeta function. Abramowitz and Stegun (1972, p. 998) tabulate numerical values of for to 4 and to 10.The second Debye function is defined by(6)(7)for and .The sum of these two integrals is(8)(9)where is the Riemann zeta function.

### Exponential integral

Min Max Re Im Let be the En-function with ,(1)(2)Then define the exponential integral by(3)where the retention of the notation is a historical artifact. Then is given by the integral(4)This function is implemented in the WolframLanguage as ExpIntegralEi[x].The exponential integral is closely related to the incomplete gamma function by(5)Therefore, for real ,(6)The exponential integral of a purely imaginarynumber can be written(7)for and where and are cosine and sine integral.Special values include(8)(OEIS A091725).The real root of the exponential integral occurs at 0.37250741078... (OEIS A091723), which is , where is Soldner's constant (Finch 2003).The quantity (OEIS A073003) is known as the Gompertz constant.The limit of the following expression can be given analytically(9)(10)(OEIS A091724), where is the Euler-Mascheroni constant.The Puiseux series of along the positive real axis is given by(11)where the denominators..

### Eulerian integral of the second kind

For and ,(1)(2)(3)where is the Pochhammer symbol and is the beta function.

### Cosine integral

The most common form of cosine integral is(1)(2)(3)(4)where is the exponential integral, is the En-function, and is the Euler-Mascheroni constant. is returned by the Wolfram Language command CosIntegral[x], and is also commonly denoted . has the series expansion(5)(Havil 2003, p. 106; after inserting a minus sign in the definition).The derivative is(6)and the integral is(7) has zeros at 0.616505, 3.38418, 6.42705, .... Extrema occur when(8)or , or , , , ..., which are alternately maxima and minima. At these points, equals 0.472001, , 0.123772, .... Inflection points occur when(9)which simplifies to(10)which has solutions 2.79839, 6.12125, 9.31787, ....The related function(11)(12)is sometimes also defined.To find a closed form for an integral power of a cosine function, use integrationby parts to obtain(13)(14)Now, if is even so , then(15)(16)On the other hand, if is odd so , then(17)Now let ,(18)The general result is then(19)The..

### Chi

Min Max Min Max Re Im The hyperbolic cosine integral, often called the "Chi function" for short, is defined by(1)where is the Euler-Mascheroni constant. The function is given by the Wolfram Language command CoshIntegral[z].The Chi function has a unique real root at (OEIS A133746).The derivative of is(2)and the integral is(3)

### Logarithmic integral

The logarithmic integral (in the "American" convention; Abramowitz and Stegun 1972; Edwards 2001, p. 26), is defined for real as(1)(2)Here, PV denotes Cauchy principal value of the integral, and the function has a singularity at .The logarithmic integral defined in this way is implemented in the WolframLanguage as LogIntegral[x].There is a unique positive number(3)(OEIS A070769; Derbyshire 2004, p. 114) known as Soldner's constant for which , so the logarithmic integral can also be written as(4)for .Special values include(5)(6)(7)(8)(OEIS A069284), where is Soldner's constant (Edwards 2001, p. 34). Min Max Re Im The definition can also be extended to the complex plane,as illustrated above.Its derivative is(9)and its indefinite integral is(10)where is the exponential integral. It also has the definite integral(11)where (OEIS A002162) is the natural logarithm of 2.The logarithmic integral obeys(12)where..

### Beta function

The beta function is the name used by Legendre and Whittaker and Watson (1990) for the beta integral (also called the Eulerian integral of the first kind). It is defined by(1)The beta function is implemented in the Wolfram Language as Beta[a, b].To derive the integral representation of the beta function, write the product oftwo factorials as(2)Now, let , , so(3)(4)Transforming to polar coordinates with , (5)(6)(7)(8)The beta function is then defined by(9)(10)Rewriting the arguments then gives the usual form for the beta function,(11)(12)By symmetry,(13)The general trigonometric form is(14)Equation (14) can be transformed to an integral over polynomials by letting ,(15)(16)(17)(18)For any with ,(19)(Krantz 1999, p. 158).To put it in a form which can be used to derive the Legendre duplication formula, let , so and , and(20)(21)To put it in a form which can be used to develop integral representations of the Bessel functions and hypergeometric..

### Log gamma function

Min Max Min Max Re Im The plots above show the values of the function obtained by taking the natural logarithm of the gamma function, . Note that this introduces complicated branch cut structure inherited from the logarithm function. Min Max Re Im For this reason, the logarithm of the gamma function is sometimes treated as a special function in its own right, and defined differently from . This special "log gamma" function is implemented in the Wolfram Language as LogGamma[z], plotted above. As can be seen, the two definitions have identical real parts, but differ markedly in their imaginary components. Most importantly, although the log gamma function and are equivalent as analytic multivalued functions, they have different branch cut structures and a different principal branch, and the log gamma function is analytic throughout the complex -plane except for a single branch cut discontinuity along the negative real axis. In particular,..

### Ein function

(1)(2)where is the Euler-Mascheroni constant and is the En-function with .

### Beta exponential function

Another "beta function" defined in termsof an integral is the "exponential" beta function, given by(1)(2)If is an integer, then(3)where is the En-Function. The exponential beta function satisfies the recurrence relation(4)The values for , 1, and 2 are(5)(6)(7)

### Alpha function

(1)(2)It is equivalent to(3)where is the En-function.

### Ultraprimorial

The function defined by , where is a prime number and is a primorial. The values for , 3, ..., are 4, 46656, 205891132094649000000000000000000000000000000, ... (OEIS A165812).

### Triangle coefficient

The triangle coefficient is function of three variables written and defined by(Shore and Menzel 1968, p. 273), where is a factorial. It arises for example in the definition of the Wigner 6j-symbol.

### Multifactorial

A generalization of the factorial and doublefactorial,(1)(2)(3)etc., where the products run through positive integers.The following table lists the values of the first few multifactorials for , 2, ....A0001421, 2, 6, 24, 120, 720, ...A0068821, 2, 3, 8, 15, 48, 105, ...A0076611, 2, 3, 4, 10, 18, 28, 80, 162, 280, ...A0076621, 2, 3, 4, 5, 12, 21, 32, 45, 120, ...

### Superfactorial

The superfactorial of is defined by Pickover (1995) as(1)The first two values are 1 and 4, but subsequently grow so rapidly that already has a huge number of digits.Sloane and Plouffe (1995) define the superfactorial by(2)(3)which is equivalent to the integral values of the Barnes G-function. The values for , 2, ... are 1, 1, 2, 12, 288, 34560, ... (OEIS A000178). This function has an unexpected connection with Bell numbers.

### Double factorial

The double factorial of a positive integer is a generalization of the usual factorial defined by(1)Note that , by definition (Arfken 1985, p. 547).The origin of the notation appears not to not be widely known and is not mentioned in Cajori (1993).For , 1, 2, ..., the first few values are 1, 1, 2, 3, 8, 15, 48, 105, 384, ... (OEIS A006882). The numbers of decimal digits in for , 1, ... are 1, 4, 80, 1285, 17831, 228289, 2782857, 32828532, ... (OEIS A114488).The double factorial is implemented in the WolframLanguage as n!! or Factorial2[n].The double factorial is a special case of the multifactorial.The double factorial can be expressed in terms of the gammafunction by(2)(Arfken 1985, p. 548).The double factorial can also be extended to negative odd integers using the definition(3)(4)for , 1, ... (Arfken 1985, p. 547). Min Max Re Im Similarly, the double factorial can be extended to complex arguments as(5)There are many identities..

### Falling factorial

The falling factorial , sometimes also denoted (Graham et al. 1994, p. 48), is defined by(1)for . Is also known as the binomial polynomial, lower factorial, falling factorial power (Graham et al. 1994, p. 48), or factorial power.The falling factorial is related to the rising factorial (a.k.a. Pochhammer symbol) by(2)The falling factorial is implemented in the Wolfram Language as FactorialPower[x, n].A generalized version of the falling factorial can defined by(3)and is implemented in the Wolfram Language as FactorialPower[x, n, h].The usual factorial is related to the falling factorialby(4)(Graham et al. 1994, p. 48).In combinatorial usage, the falling factorial is commonly denoted and the rising factorial is denoted (Comtet 1974, p. 6; Roman 1984, p. 5; Hardy 1999, p. 101), whereas in the calculus of finite differences and the theory of special functions, the falling factorial is denoted..

### Factorion

A factorion is an integer which is equal to the sum of factorials of its digits. There are exactly four such numbers:(1)(2)(3)(4)(OEIS A014080; Gardner 1978, Madachy 1979, Pickover 1995). Obviously, the factorion of an -digit number cannot exceed .

### Rising factorial

The rising factorial , sometimes also denoted (Comtet 1974, p. 6) or (Graham et al. 1994, p. 48), is defined by(1)This function is also known as the rising factorial power (Graham et al. 1994, p. 48) and frequently called the Pochhammer symbol in the theory of special functions. The rising factorial is implemented in the Wolfram Language as Pochhammer[x, n].The rising factorial is related to the gamma function by(2)where(3)and is related to the falling factorial by(4)The usual factorial is therefore related to the risingfactorial by(5)for nonnegative integers (Graham et al. 1994, p. 48).Note that in combinatorial usage, the falling factorial is denoted and the rising factorial is denoted (Comtet 1974, p. 6; Roman 1984, p. 5; Hardy 1999, p. 101), whereas in the calculus of finite differences and the theory of special functions, the falling factorial is denoted and the rising factorial is denoted..

### Zonal harmonic

A zonal harmonic is a spherical harmonic of the form , i.e., one which reduces to a Legendre polynomial (Whittaker and Watson 1990, p. 302). These harmonics are termed "zonal" since the curves on a unit sphere (with center at the origin) on which vanishes are parallels of latitude which divide the surface into zones (Whittaker and Watson 1990, p. 392).Resolving into factors linear in , multiplied by when is odd, then replacing by allows the zonal harmonic to be expressed as a product of factors linear in , , and , with the product multiplied by when is odd (Whittaker and Watson 1990, p. 1990).

### Tesseral harmonic

A tesseral harmonic is a spherical harmonic of the form . These harmonics are so named because the curves on which they vanish are parallels of latitude and meridians, which divide the surface of a sphere into quadrangles whose angles are right angles (Whittaker and Watson 1990, p. 392).Resolving into factors linear in , multiplied by when is odd, then replacing by allows the tesseral harmonics to be expressed as products of factors linear in , , and multiplied by one of 1, , , , , , , and (Whittaker and Watson 1990, p. 536).

### Surface harmonic

Any linear combination of real sphericalharmonicsfor fixed whose sum is not premultiplied by a factor (Whittaker and Watson 1990, p. 392).

### Spherical harmonic closure relations

The sum of the absolute squares of the spherical harmonics over all values of is(1)The double sum over and is given by(2)(3)where is the delta function.

### Spherical harmonic addition theorem

A formula also known as the Legendre addition theorem which is derived by finding Green's functions for the spherical harmonic expansion and equating them to the generating function for Legendre polynomials. When is defined by(1)The Legendre polynomial of argument is given by(2)(3)(4)Another version of the formula can be given as(5)(O. Marichev, pers. comm., Jan. 15, 2008).

### Spherical harmonic

The spherical harmonics are the angular portion of the solution to Laplace's equation in spherical coordinates where azimuthal symmetry is not present. Some care must be taken in identifying the notational convention being used. In this entry, is taken as the polar (colatitudinal) coordinate with , and as the azimuthal (longitudinal) coordinate with . This is the convention normally used in physics, as described by Arfken (1985) and the Wolfram Language (in mathematical literature, usually denotes the longitudinal coordinate and the colatitudinal coordinate). Spherical harmonics are implemented in the Wolfram Language as SphericalHarmonicY[l, m, theta, phi].Spherical harmonics satisfy the spherical harmonic differential equation, which is given by the angular part of Laplace's equation in spherical coordinates. Writing in this equation gives(1)Multiplying by gives(2)Using separation of variables by equating the -dependent..

### Sectorial harmonic

A spherical harmonic ofthe formor

### Ellipsoidal harmonic of the first kind

The first solution to Lamé's differential equation, denoted for , ..., . They are also called Lamé functions. The product of two ellipsoidal harmonics of the first kind is a spherical harmonic. Whittaker and Watson (1990, pp. 536-537) write(1)(2)and give various types of ellipsoidal harmonics and their highest degree terms as 1. 2. 3. 4. .A Lamé function of degree may be expressed as(3)where or 1/2, are real and unequal to each other and to , , and , and(4)Byerly (1959) uses the recurrence relations to explicitly compute some ellipsoidal harmonics, which he denoted by , , , and ,(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)

### Multidimensional polylogarithm

A multidimensional polylogarithm is a generalization of the usual polylogarithmtowith positive integers (Borwein et al. 2004, p. 27).

### Harmonic logarithm

For all integers and nonnegative integers , the harmonic logarithms of order and degree are defined as the unique functions satisfying 1. , 2. has no constant term except , 3. , where the "Roman symbol" is defined by(1)(Roman 1992). This gives the special cases(2)(3)where is a harmonic number. The harmonic logarithm has the integral(4)The harmonic logarithm can be written(5)where is the differential operator, (so is the th integral). Rearranging gives(6)This formulation gives an analog of the binomial theorem called the logarithmic binomial theorem. Another expression for the harmonic logarithm is(7)where is a Pochhammer symbol and is a two-index harmonic number (Roman 1992).

### Common logarithm

Min Max The common logarithm is the logarithm to base 10. The notation is used by physicists, engineers, and calculator keypads to denote the common logarithm. However, mathematicians generally use the same symbol to mean the natural logarithm ln, . Worse still, in Russian literature the notation is used to denote a base-10 logarithm, which conflicts with the use of the symbol lg to indicate the logarithm to base 2. To avoid all ambiguity, it is best to explicitly specify when the logarithm to base 10 is intended. In this work, , is used for the natural logarithm, and is used for the logarithm to the base 2.The situation is complicated even more by the fact that number theorists (e.g., Ivić 2003) commonly use the notation to denote the nested natural logarithm .The common logarithm is implemented in the Wolfram Language as Log[10, x] and Log10[x].Hardy and Wright (1979, p. 8) assert that the common logarithm has "no mathematical interest."Common..

### Logarithm

The logarithm for a base and a number is defined to be the inverse function of taking to the power , i.e., . Therefore, for any and ,(1)or equivalently,(2)For any base, the logarithm function has a singularity at . In the above plot, the blue curve is the logarithm to base 2 (), the black curve is the logarithm to base (the natural logarithm ), and the red curve is the logarithm to base 10 (the common logarithm, i.e., ).Note that while logarithm base 10 is denoted in this work, on calculators, and in elementary algebra and calculus textbooks, mathematicians and advanced mathematics texts uniformly use the notation to mean , and therefore use to mean the common logarithm. Extreme care is therefore needed when consulting the literature.The situation is complicated even more by the fact that number theorists (e.g., Ivić 2003) commonly use the notation to denote the nested natural logarithm .In the Wolfram Language, the logarithm to the base is implemented..

### Log sine function

The log sine function, also called the logsine function, is defined by(1)The first few cases are given by(2)(3)(4)where is the Riemann zeta function.The log sine function is related to the log cosinefunction by(5)and the two are equal if the range of integration for is restricted from 0 to to 0 to .

### Log cosine function

By analogy with the log sine function, definethe log cosine function by(1)The first few cases are given by(2)(3)(4)where is the Riemann zeta function.The log cosine function is related to the log sinefunction by(5)and the two are equal if the range of integration for is restricted from 0 to to 0 to .Oloa (2011) computed an exact value of the log cosine integral(6)where is the Riemann zeta function, is the Euler-Mascheroni constant, is a multivariate zeta function, and denotes . A closed form for in terms of more elementary functions is not known as of Apr. 2011, but it is numerically given by(7)(Oloa 2011; OEIS A189272).

### Binary logarithm

The binary logarithm is the logarithm to base 2.The notation is sometimes used to denote this function in number theoretic literature. However, because Russian and German literature use the symbol to denote the base-10 logarithm and since this is the use recommended by the United States Department of Commerce (Taylor 1995, p. 33), this practice is discouraged. (To confuse matters even more, some German literature uses the notation to mean the binary logarithm.)The binary logarithm is implemented in the Wolfram Language as Log[2, z] and Log2[z].When information theoretic functions (like entropy) are computed using , the units of information are obtained in bits. When is used instead, the units of information are known as "nats."

### Natural logarithm

The natural logarithm is the logarithm having base e, where(1)This function can be defined(2)for .This definition means that e is the unique number with the property that the area of the region bounded by the hyperbola , the x-axis, and the vertical lines and is 1. In other words,(3)The notation is used in physics and engineering to denote the natural logarithm, while mathematicians commonly use the notation . In this work, denotes a natural logarithm, whereas denotes the common logarithm.There are a number of notational conventions in common use for indication of a power of a natural logarithm. While some authors use (i.e., using a trigonometric function-like convention), it is also common to write .Common and natural logarithms can be expressed in terms of each other as(4)(5)The natural logarithm is especially useful in calculusbecause its derivative is given by the simple equation(6)whereas logarithms in other bases have the more complicated..

### Ln

is the notation used in physics and engineering to denote the logarithm to base e, also called the natural logarithm, i.e.,The United States Department of Commerce recommends that the notation be used in this way to refer to the natural logarithm (Taylor 1995, p. 33).Unfortunately, mathematicians in the United States commonly use the symbol to refer to the natural logarithm, as does TraditionalForm typesetting in the Wolfram Language. The use of for different purposes by different mathematical communities causes considerable confusion, so extreme care is needed in determining if the symbol found in the wild refers to or .The natural logarithm is implemented in the Wolfram Language as Log[x], which is equivalent to Log[E, x].

### Napierian logarithm

The first definition of the logarithm was constructed by Napier and popularized through his posthumous pamphlet (Napier 1619). It this pamphlet, Napier sought to reduce the operations of multiplication, division, and root extraction to addition and subtraction. To this end, he defined the "logarithm" of a number by(1)written .This definition leads to the remarkable relations(2)(3)(4)which give the identities(5)(6)(7)(Havil 2003, pp. 8-9). While Napier's definition is different from the modern one (in particular, it decreases with increasing , but also fails to satisfy a number of properties of the modern logarithm), it provides the desired property of transforming multiplication into addition.The Napierian logarithm can be given in terms of the modern logarithm by solving equation (1) for , giving(8)Because a ratio of logarithms appears in this expression, any logarithm base can be used as long as the same value..

### Lg

There are several conflicting meanings associated with the notation . In German and Russian literature, the notation is used to mean the common logarithm . This is also the usage recommended by the United States Department of Commerce (Taylor 1995, p. 33).However, is sometimes identified with the binary logarithm in some number theoretic literature (and here, mean the base-2 logarithm, not the nested natural logarithm as defined by Ivić 2003).Great care is therefore needed to determine the intended definition for when it is encountered in the wild.

### Antilogarithm

The inverse function of the logarithm,defined such thatThe antilogarithm in base of is therefore .

### Pochhammer symbol

The Pochhammer symbol(1)(2)(Abramowitz and Stegun 1972, p. 256; Spanier 1987; Koepf 1998, p. 5) for is an unfortunate notation used in the theory of special functions for the rising factorial, also known as the rising factorial power (Graham et al. 1994, p. 48) or ascending Factorial (Boros and Moll 2004, p. 16). The Pochhammer symbol is implemented in the Wolfram Language as Pochhammer[x, n].In combinatorics, the notation (Roman 1984, p. 5), (Comtet 1974, p. 6), or (Graham et al. 1994, p. 48) is used for the rising factorial, while or denotes the falling factorial (Graham et al. 1994, p. 48). Extreme caution is therefore needed in interpreting the notations and .The first few values of for nonnegative integers are(3)(4)(5)(6)(7)(OEIS A054654).In closed form, can be written(8)where is a Stirling number of the first kind.The Pochhammer symbol satisfies(9)the dimidiation formulas(10)(11)and..

### Gamma function

The (complete) gamma function is defined to be an extension of the factorial to complex and real number arguments. It is related to the factorial by(1)a slightly unfortunate notation due to Legendre which is now universally used instead of Gauss's simpler (Gauss 1812; Edwards 2001, p. 8).It is analytic everywhere except at , , , ..., and the residue at is(2)There are no points at which .The gamma function is implemented in the WolframLanguage as Gamma[z].There are a number of notational conventions in common use for indication of a power of a gamma functions. While authors such as Watson (1939) use (i.e., using a trigonometric function-like convention), it is also common to write .The gamma function can be defined as a definite integral for (Euler's integral form)(3)(4)or(5)The complete gamma function can be generalized to the upper incomplete gamma function and lower incomplete gamma function . Min Max Re Im Plots of the real and imaginary..

### Rayleigh's formulas

The formulas(1)(2)for , 1, 2, ..., where is a spherical Bessel function of the first kind and is a spherical Bessel function of the second kind.

### Weyrich's formula

For and real, with and ,where is a Hankel function of the first kind.

### Ramanujan's integral

where is a Bessel function of the first kind.

### Heun's differential equation

A natural extension of the Riemannp-differential equation given bywhere

### Poisson integral

There are at least two integrals called the Poisson integral. The first is also known as Bessel's second integral,(1)where is a Bessel function of the first kind and is a gamma function. It can be derived from Sonine's integral. With , the integral becomes Parseval's integral.In complex analysis, let be a harmonic function on a neighborhood of the closed disk , then for any point in the open disk ,(2)In polar coordinates on ,(3)where and is the Poisson kernel. For a circle,(4)For a sphere,(5)where(6)

### Cubed

A number taken to the power 3 is said to be cubed, so is called " cubed." This terminology derives from the fact that the volume of a cube of edge length is given by , hence the operation taking a cube's edge length to its volume is known as "cubing" and the quantity derived from by cubing is known as " cubed."

### Squared

A number taken to the power 2 is said to be squared, so is called " squared." This terminology derives from the fact that the area of a square of edge length is given by , hence the operation taking a square's edge length to its area is known as "squaring" and the quantity derived from by squaring is known as " squared."Not that the term "squaring" also refers to the geometric construction, using only compass and straightedge, of a square which has the same area as a given geometric figure.

### Watson's identities

Let , , and be the roots of the cubic equation(1)then the Rogers L-function satisfies(2)(3)(4)

### Inverse tangent integral

The inverse tangent integral is defined in terms of the dilogarithm by(1)(Lewin 1958, p. 33). It has the series(2)and gives in closed form the sum(3)that was considered by Ramanujan (Lewin 1958, p. 39). The inverse tangent integralcan be expressed in terms of the dilogarithm as(4)in terms of Legendre's chi-function as(5)in terms of the Lerch transcendent by(6)and as the integral(7) has derivative(8)It satisfies the identities(9)where(10)is the generalized inverse tangent function. has the special value(11)where is Catalan's constant, and the functional relationships(12)the two equivalent identities(13)(14)and(15)(Lewin 1958, p. 39). The triplication formula is given by(16)which leads to(17)and the algebraic form(18)(Lewin 1958, p. 41).

### Trilogarithm

Min Max The trilogarithm , sometimes also denoted , is special case of the polylogarithm for . Note that the notation for the trilogarithm is unfortunately similar to that for the logarithmic integral .The trilogarithm is implemented in the Wolfram Language as PolyLog[3, z]. Min Max Re Im Plots of in the complex plane are illustrated above.Functional equations for the trilogarithm include(1)Analytic values for include(2)where is Apéry's constant and is the golden ratio.Bailey et al. showed that(3)

### Lobachevsky's function

The name Lobachevsky's function is sometimes given to the functionalso denoted , where is Clausen's integral.

### Spence's integral

Min Max Min Max Re Im (1)(2)where is the dilogarithm.

### Spence's function

Min Max Min Max Re Im (1)(2)where is the dilogarithm.

### Clausen function

Define(1)(2)then the Clausen functions are defined by(3)sometimes also written as (Arfken 1985, p. 783).Then the Clausen function can be given symbolically in terms of the polylogarithm as(4)For , the function takes on the special form(5)and for , it becomes Clausen's integral(6)The symbolic sums of opposite parity are summable symbolically, and the first few are given by(7)(8)(9)(10)(11)for (Abramowitz and Stegun 1972).

### Jonqui&egrave;re's relation

Jonquière's relation, sometimes also spelled "Joncquière's relation"(Erdélyi et al. 1981, p. 31), statesErdélyi et al. (1981, p. 31), where is a polylogarithm, is the gamma function, and is the Hurwitz zeta function, and is not a member of the real interval .The most general form of the identity valid everywhere in the complexplane is

### Nielsen generalized polylogarithm

A generalization of the polylogarithm functiondefined byThe function reduces to the usual polylogarithmfor the caseThe Nielsen generalized polylogarithm is implemented as PolyLog[n,p, z].

### Hyperbolic tangent

Min Max Min Max Re Im By way of analogy with the usual tangent(1)the hyperbolic tangent is defined as(2)(3)(4)where is the hyperbolic sine and is the hyperbolic cosine. The notation is sometimes also used (Gradshteyn and Ryzhik 2000, p. xxix). is implemented in the Wolfram Language as Tanh[z].Special values include(5)(6)where is the golden ratio.The derivative of is(7)and higher-order derivatives are given by(8)where is an Eulerian number.The indefinite integral is given by(9) has Taylor series(10)(11)(OEIS A002430 and A036279).As Gauss showed in 1812, the hyperbolic tangent can be written using a continuedfraction as(12)(Wall 1948, p. 349; Olds 1963, p. 138). This continued fraction is also known as Lambert's continued fraction (Wall 1948, p. 349).The hyperbolic tangent satisfies the second-order ordinary differential equation(13)together with the boundary conditions and ...

### Hyperbolic sine

Min Max Min Max Re Im The hyperbolic sine is defined as(1)The notation is sometimes also used (Gradshteyn and Ryzhik 2000, p. xxix). It is implemented in the Wolfram Language as Sinh[z].Special values include(2)(3)where is the golden ratio.The value(4)(OEIS A073742) has Engel expansion 1, 6, 20, 42, 72, 110, ... (OEIS A068377), which has closed form for .The derivative is given by(5)where is the hyperbolic cosine, and the indefinite integral by(6)where is a constant of integration. has the Taylor series(7)(8)(OEIS A009445).

### Hyperbolic secant

Min Max Min Max Re Im The hyperbolic secant is defined as(1)(2)where is the hyperbolic cosine. It is implemented in the Wolfram Language as Sech[z].On the real line, it has a maximum at and inflection points at (OEIS A091648). It has a fixed point at (OEIS A069814).The derivative is given by(3)where is the hyperbolic tangent, and the indefinite integral by(4)where is a constant of integration. has the Taylor series(5)(6)(OEIS A046976 and A046977), where is an Euler number and is a factorial.Equating coefficients of , , and in the Ramanujan cos/cosh identity(7)gives the amazing identities(8)

### Hyperbolic lemniscate function

By analogy with the lemniscate functions,hyperbolic lemniscate functions can also be defined(1)(2)(3)(4)where is a hypergeometric function.Let and , and write(5)(6)where is the constant obtained by setting and , which is given by(7)(8)with is a complete elliptic integral of the first kind. Ramanujan showed that(9)(10)and(11)(Berndt 1994).

### Inverse hyperbolic tangent

Min Max Re Im The inverse hyperbolic tangent (Zwillinger 1995, p. 481; Beyer 1987, p. 181), sometimes called the area hyperbolic tangent (Harris and Stocker 1998, p. 267), is the multivalued function that is the inverse function of the hyperbolic tangent.The function is sometimes denoted (Jeffrey 2000, p. 124) or (Gradshteyn and Ryzhik 2000, p. xxx). The variants or (Harris and Stocker 1998, p. 263) are sometimes used to refer to explicit principal values of the inverse hyperbolic tangent, although this distinction is not always made. Worse yet, the notation is sometimes used for the principal value, with being used for the multivalued function (Abramowitz and Stegun 1972, p. 87). Note that in the notation , is the hyperbolic tangent and the superscript denotes an inverse function, not the multiplicative inverse.The principal value of is implemented in the Wolfram Language as ArcTanh[z] and..

### Inverse hyperbolic sine

Min Max Min Max Re Im The inverse hyperbolic sine (Beyer 1987, p. 181; Zwillinger 1995, p. 481), sometimes called the area hyperbolic sine (Harris and Stocker 1998, p. 264) is the multivalued function that is the inverse function of the hyperbolic sine.The variants or (Harris and Stocker 1998, p. 263) are sometimes used to refer to explicit principal values of the inverse hyperbolic sine, although this distinction is not always made. Worse yet, the notation is sometimes used for the principal value, with being used for the multivalued function (Abramowitz and Stegun 1972, p. 87). The notations (Jeffrey 2000, p. 124) and (Gradshteyn and Ryzhik 2000, p. xxx) are sometimes also used. Note that in the notation , is the hyperbolic sine and the superscript denotes an inverse function, not the multiplicative inverse.Its principal value of is implemented in the Wolfram Language as ArcSinh[z] and in the..

### Hyperbolic functions

The hyperbolic functions , , , , , (hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic cosecant, hyperbolic secant, and hyperbolic cotangent) are analogs of the circular functions, defined by removing s appearing in the complex exponentials. For example,(1)so(2)Note that alternate notations are sometimes used, as summarized in the following table.alternate notations (Gradshteyn and Ryzhik 2000, p. xxvii) (Gradshteyn and Ryzhik 2000, p. xxvii) (Gradshteyn and Ryzhik 2000, p. xxvii) (Gradshteyn and Ryzhik 2000, p. xxvii)The hyperbolic functions share many properties with the corresponding circular functions. In fact, just as the circle can be represented parametrically by(3)(4)a rectangular hyperbola (or, more specifically,its right branch) can be analogously represented by(5)(6)where is the hyperbolic cosine and is the hyperbolic sine.The hyperbolic functions arise in many..

### Inverse hyperbolic secant

Min Max Re Im The inverse hyperbolic secant (Beyer 1987, p. 181; Zwillinger 1995, p. 481), sometimes called the area hyperbolic secant (Harris and Stocker 1998, p. 271) and sometimes also denoted (Jeffrey 2000, p. 124), is the multivalued function that is the inverse function of the hyperbolic secant. The variants or (Harris and Stocker 1998, p. 263) are sometimes used to refer to explicit principal values of the inverse hyperbolic secant, although this distinction is not always made. Worse yet, the notation is sometimes used for the principal value, with being used for the multivalued function (Abramowitz and Stegun 1972, p. 87). Note that in the notation , is the hyperbolic secant and the superscript denotes an inverse function, not the multiplicative inverse.The principal value of is implemented in the Wolfram Language as ArcSech[z].The inverse hyperbolic secant is a multivalued function and..

### Hyperbolic cotangent

Min Max Re Im The hyperbolic cotangent is defined as(1)The notation is sometimes also used (Gradshteyn and Ryzhik 2000, p. xxix). It is implemented in the Wolfram Language as Coth[z].The hyperbolic cotangent satisfies the identity(2)where is the hyperbolic cosecant.It has a unique real fixed point where(3)at (OEIS A085984), which is related to the Laplace limit in the solution of Kepler's equation.The derivative is given by(4)where is the hyperbolic cosecant, and the indefinite integral by(5)where is a constant of integration.The Laurent series of is given by(6)(7)(OEIS A002431 and A036278), where is a Bernoulli number and is a Bernoulli polynomial. An asymptotic series about infinity on the real line is given by(8)

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