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Klein's absolute invariant

Min Max Min Max Re Im Let and be periods of a doubly periodic function, with the half-period ratio a number with . Then Klein's absolute invariant (also called Klein's modular function) is defined as(1)where and are the invariants of the Weierstrass elliptic function with modular discriminant(2)(Klein 1877). If , where is the upper half-plane, then(3)is a function of the ratio only, as are , , and . Furthermore, , , , and are analytic in (Apostol 1997, p. 15).Klein's absolute invariant is implemented in the WolframLanguage as KleinInvariantJ[tau].The function is the same as the j-function, modulo a constant multiplicative factor.Every rational function of is a modular function, and every modular function can be expressed as a rational function of (Apostol 1997, p. 40).Klein's invariant can be given explicitly by(4)(5)(Klein 1878-1879, Cohn 1994), where is the elliptic lambda function(6) is a Jacobi theta function, the are..

Continued fraction map

Min Max Re Im for , where is the floor function. The natural invariant of the map is

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..

Sign

Min Max Re Im The sign of a real number, also called sgn or signum, is for a negative number (i.e., one with a minus sign ""), 0 for the number zero, or for a positive number (i.e., one with a plus sign ""). In other words, for real ,(1)For real , this can be written(2)and satisfies(3) for real can also be defined as(4)where is the Heaviside step function.The sign function is implemented in the Wolfram Language for real as Sign[x]. For nonzero complex numbers, Sign[z] returns , where is the complex modulus of . can also be interpreted as an unspecified point on the unit circle in the complex plane (Rich and Jeffrey 1996).

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..

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..

Köbe function

Min Max Re Im The function(1)defined on the unit disk . For , the Köbe function is a schlicht function(2)with for all (Krantz 1999, p. 149). For ,(3)illustrated above.

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).

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..

Sine integral

Min Max Min Max Re Im The most common "sine integral" is defined as(1) is the function implemented in the Wolfram Language as the function SinIntegral[z]. is an entire function.A closed related function is defined by(2)(3)(4)(5)where is the exponential integral, (3) holds for , and(6)The derivative of is(7)where is the sinc function and the integral is(8)A series for is given by(9)(Havil 2003, p. 106).It has an expansion in terms of sphericalBessel functions of the first kind as(10)(Harris 2000).The half-infinite integral of the sinc functionis given by(11)To compute the integral of a sine function times a power(12)use integration by parts. Let(13)(14)so(15)Using integration by parts again,(16)(17)(18)Letting , so(19)General integrals of the form(20)are related to the sinc function and can be computedanalytically...

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..

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)

Fresnel integrals

There are a number of slightly different definitions of the Fresnel integrals. In physics, the Fresnel integrals denoted and are most often defined by(1)(2)so(3)(4)These Fresnel integrals are implemented in the Wolfram Language as FresnelC[z] and FresnelS[z]. and are entire functions. Min Max Re Im Min Max Re Im The and integrals are illustrated above in the complex plane.They have the special values(5)(6)(7)and(8)(9)(10)An asymptotic expansion for gives(11)(12)Therefore, as , and . The Fresnel integrals are sometimes alternatively defined as(13)(14)Letting so , and (15)(16)In this form, they have a particularly simple expansion in terms of sphericalBessel functions of the first kind. Using(17)(18)(19)where is a spherical Bessel function of the second kind(20)(21)(22)(23)(24)Related functions , , , and are defined by(25)(26)(27)(28)..

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)...

Fractional part

The function giving the fractional (noninteger) part of a real number . The symbol is sometimes used instead of (Graham et al. 1994, p. 70; Havil 2003, p. 109), but this notation is not used in this work due to possible confusion with the set containing the element .Unfortunately, there is no universal agreement on the meaning of for and there are two common definitions. Let be the floor function, then the Wolfram Language command FractionalPart[x] is defined as(1)(left figure). This definition has the benefit that , where is the integer part of . Although Spanier and Oldham (1987) use the same definition as the Wolfram Language, they mention the formula only very briefly and then say it will not be used further. Graham et al. (1994, p. 70), and perhaps most other mathematicians, use the different definition(2)(right figure). Min Max Re Im The fractional part function can also be extended to the complexplane as(3)as illustrated..

Riemann sum

Graph the Riemann sum of as x goes from to using rectangles taking samples at the  Maximum Minimum Left Right Midpoint Print estimated and actual areas? Rectangle Color Plot Color Light Gray Dark Gray Black White Red Orange Yellow Green Blue Purple Light Gray Dark Gray Black White Red Orange Yellow Green Blue Purple Replot Let a closed interval be partitioned by points , where the lengths of the resulting intervals between the points are denoted , , ..., . Let be an arbitrary point in the th subinterval. Then the quantityis called a Riemann sum for a given function and partition, and the value is called the mesh size of the partition.If the limit of the Riemann sums exists as , this limit is known as the Riemann integral of over the interval . The shaded areas in the above plots show the lower and upper sums for a constant mesh size...

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)

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..

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)

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..

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..

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,..

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..

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..

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..

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..

Nearest integer function

The nearest integer function, also called nint or the round function, is defined such that is the integer closest to . While the notation is sometimes used to denote the nearest integer function (Hastad et al. 1988), this notation is rather cumbersome and is not recommended. Also note that while is sometimes used to denote the nearest integer function, is also commonly used to denote the floor function (including by Gauss in his third proof of quadratic reciprocity in 1808), so this notational use is also discouraged.Since the definition is ambiguous for half-integers, the additional rule that half-integers are always rounded to even numbers is usually added in order to avoid statistical biasing. For example, , , , , etc. This convention is followed in the C math.h library function rint, as well as in the Wolfram Language, where the nearest integer function is implemented as Round[x].Since usage concerning fractional part/value and integer..

Floor function

The floor function , also called the greatest integer function or integer value (Spanier and Oldham 1987), gives the largest integer less than or equal to . The name and symbol for the floor function were coined by K. E. Iverson (Graham et al. 1994).Unfortunately, in many older and current works (e.g., Honsberger 1976, p. 30; Steinhaus 1999, p. 300; Shanks 1993; Ribenboim 1996; Hilbert and Cohn-Vossen 1999, p. 38; Hardy 1999, p. 18), the symbol is used instead of (Graham et al. 1994, p. 67). In fact, this notation harks back to Gauss in his third proof of quadratic reciprocity in 1808. However, because of the elegant symmetry of the floor function and ceiling function symbols and , and because is such a useful symbol when interpreted as an Iverson bracket, the use of to denote the floor function should be deprecated. In this work, the symbol is used to denote the nearest integer function since it naturally..

Ceiling function

The function which gives the smallest integer , shown as the thick curve in the above plot. Schroeder (1991) calls the ceiling function symbols the "gallows" because of the similarity in appearance to the structure used for hangings. The name and symbol for the ceiling function were coined by K. E. Iverson (Graham et al. 1994). Min Max Re Im The ceiling function is implemented in the Wolfram Language as Ceiling[z], where it is generalized to complex values of as illustrated above.Although some authors used the symbol to denote the ceiling function (by analogy with the older notation for the floor function), this practice is strongly discouraged (Graham et al. 1994, p. 67). Also strongly discouraged is the use of the symbol to denote the ceiling function (e.g., Harary 1994, pp. 91, 93, and 118-119), since this same symbol is more commonly used to denote the fractional part of .Since usage concerning fractional..

Complex modulus

Min Max Min Max Re Im The modulus of a complex number , also called the complex norm, is denoted and defined by(1)If is expressed as a complex exponential (i.e., a phasor), then(2)The complex modulus is implemented in the Wolfram Language as Abs[z], or as Norm[z].The square of is sometimes called the absolute square.Let and be two complex numbers. Then(3)(4)so(5)Also,(6)(7)so(8)and, by extension,(9)The only functions satisfying identities of the form(10)are , , and (Robinson 1957).

Absolute value

Min Max The absolute value of a real number is denoted and defined as the "unsigned" portion of ,(1)(2)where is the sign function. The absolute value is therefore always greater than or equal to 0. The absolute value of for real is plotted above. Min Max Re Im The absolute value of a complex number , also called the complex modulus, is defined as(3)This form is implemented in the Wolfram Language as Abs[z] and is illustrated above for complex .Note that the derivative (read: complex derivative) does not exist because at every point in the complex plane, the value of the derivative of depends on the direction in which the derivative is taken (so the Cauchy-Riemann equations cannot and do not hold). However, the real derivative (i.e., restricting the derivative to directions along the real axis) can be defined for points other than as(4)As a result of the fact that computer algebra languages such as the Wolfram Language generically deal with..

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)

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.

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)

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..

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)

Inverse hyperbolic cotangent

Min Max Re Im The inverse hyperbolic cotangent (Beyer 1987, p. 181; Zwillinger 1995, p. 481), sometimes called the area hyperbolic cotangent (Harris and Stocker 1998, p. 267), is the multivalued function that is the inverse function of the hyperbolic cotangent.The variants and (Harris and Stocker 1998, p. 263) are sometimes used to refer to explicit principal values of the inverse hyperbolic cotangent, 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 function is sometimes denoted (Jeffrey 2000, p. 124) or (Gradshteyn and Ryzhik 2000, p. xxx). 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 ArcCoth[z]The..

Inverse hyperbolic cosine

Min Max Re Im The inverse hyperbolic cosine (Beyer 1987, p. 181; Zwillinger 1995, p. 481), sometimes called the area hyperbolic cosine (Harris and Stocker 1998, p. 264) is the multivalued function that is the inverse function of the hyperbolic cosine.The variants and (Harris and Stocker 1998, p. 263) are sometimes used to refer to explicit principal values of the inverse cotangent, 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 function is sometimes denoted (Abramowitz and Stegun 1972, p. 87; Jeffrey 2000, p. 124) or (Gradshteyn and Ryzhik 2000, p. xxx). Note that in the notation , is the hyperbolic cosine and the superscript denotes an inverse function, not the multiplicative inverse.The principal value of is implemented in the Wolfram..

Inverse hyperbolic cosecant

Min Max Min Max Re Im The inverse hyperbolic cosecant (Zwillinger 1995, p. 481), sometimes called the area hyperbolic cosecant (Harris and Stocker 1998, p. 271) and sometimes denoted (Beyer 1987, p. 181) or (Abramowitz and Stegun 1972, p. 87; Jeffrey 2000, p. 124), is the multivalued function that is the inverse function of the hyperbolic cosecant. The variants (Abramowitz and Stegun 1972, p. 87) and (Harris and Stocker 1998, p. 263) are sometimes used to refer to explicit principal values. 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 cosecant and the superscript denotes an inverse function, not the multiplicative inverse.The inverse hyperbolic cosecant is a multivalued function and hence requires a branch cut in the complex plane, which..

Inverse sine

Min Max Re Im The inverse sine is the multivalued function (Zwillinger 1995, p. 465), also denoted (Abramowitz and Stegun 1972, p. 79; Harris and Stocker 1998, p. 307; Jeffrey 2000, p. 124), that is the inverse function of the sine. The variants (e.g., Bronshtein and Semendyayev, 1997, p. 69) and are sometimes used to refer to explicit principal values of the inverse sine, although this distinction is not always made (e.g,. Zwillinger 1995, p. 466). Worse yet, the notation is sometimes used for the principal value, with being used for the multivalued function (Abramowitz and Stegun 1972, p. 80). Note that in the notation (commonly used in North America and in pocket calculators worldwide), is the sine and the superscript denotes the inverse function, not the multiplicative inverse.The principal value of the inverse sine is implemented as ArcSin[z] in the Wolfram Language. In the GNU C library,..

Inverse secant

Min Max Re Im The inverse secant (Zwillinger 1995, p. 465), also denoted (Abramowitz and Stegun 1972, p. 79; Harris and Stocker 1998, p. 315; Jeffrey 2000, p. 124), is the inverse function of the secant. The variants (Beyer 1987, p. 141) and are sometimes used to indicate the principal value, although this distinction is not always made (e.g., Zwillinger 1995, p. 466). Worse yet, the notation is sometimes used for the principal value, with being used for the multivalued function (Abramowitz and Stegun 1972, p. 80). In the notation (commonly used in North America and in pocket calculators worldwide), is the secant and the superscript denotes the inverse function, not the multiplicative inverse.The principal value of the inverse secant is implemented as ArcSec[z] in the Wolfram Language.The inverse secant is a multivalued function and hence requires a branch cut in the complex plane, which the..

Cotangent

Min Max Re Im The cotangent function is the function defined by(1)(2)(3)where is the tangent. The cotangent is implemented in the Wolfram Language as Cot[z].The notations (Erdélyi et al. 1981, p. 7; Jeffrey 2000, p. 111) and (Gradshteyn and Ryzhik 2000, p. xxix) are sometimes used in place of . Note that the cotangent is not in as widespread use in Europe as are , , and , although it does appear explicitly in various German and Russian handbooks (e.g., Gradshteyn and 2000, p. 28). Interestingly, is treated on par with the other trigonometric functions in most tabulations (Gellert et al. 1989, p. 222; Gradshteyn and Ryzhik 2000, p. 28), while and are sometimes not (Gradshteyn and Ryzhik 2000, p. 28).An important identity connecting the cotangent with the cosecantis given by(4)The cotangent has smallest real fixed point such at 0.8603335890... (OEIS A069855; Bertrand 1865, p. 285).The..

Inverse cotangent

Min Max Re Im The inverse cotangent is the multivalued function (Zwillinger 1995, p. 465), also denoted (Abramowitz and Stegun 1972, p. 79; Harris and Stocker 1998, p. 311; Jeffrey 2000, p. 124) or (Spanier and Oldham 1987, p. 333; Gradshteyn and Ryzhik 2000, p. 208; Jeffrey 2000, p. 127), that is the inverse function of the cotangent. The variants (e.g., Beyer 1987, p. 141; Bronshtein and Semendyayev, 1997, p. 70) and are sometimes used to refer to explicit principal values of the inverse cotangent, although this distinction is not always made (e.g., Zwillinger 1995, p. 466). Worse yet, the notation is sometimes used for the principal value, with being used for the multivalued function (Abramowitz and Stegun 1972, p. 80). Note that in the notation (commonly used in North America and in pocket calculators worldwide), is the cotangent and the superscript denotes an inverse..

Inverse cosine

Min Max Re Im The inverse cosine is the multivalued function (Zwillinger 1995, p. 465), also denoted (Abramowitz and Stegun 1972, p. 79; Harris and Stocker 1998, p. 307; Jeffrey 2000, p. 124), that is the inverse function of the cosine. The variants (e.g., Beyer 1987, p. 141; Bronshtein and Semendyayev, 1997, p. 69) and are sometimes used to refer to explicit principal values of the inverse cosine, although this distinction is not always made (e.g,. Zwillinger 1995, p. 466). Worse yet, the notation is sometimes used for the principal value, with being used for the multivalued function (Abramowitz and Stegun 1972, p. 80) Note that the notation (commonly used in North America and in pocket calculators worldwide), is the cosine and the superscript denotes the inverse function, not the multiplicative inverse.The principal value of the inverse cosine is implemented in the Wolfram Language..

Inverse cosecant

Min Max Re Im The inverse cosecant is the multivalued function (Zwillinger 1995, p. 465), also denoted (Abramowitz and Stegun 1972, p. 79; Spanier and Oldham 1987, p. 332; Harris and Stocker 1998, p. 315; Jeffrey 2000, p. 125), that is the inverse function of the cosecant. The variants (e.g., Beyer 1987, p. 141; Bronshtein and Semendyayev, 1997, p. 70) and are sometimes used to refer to explicit principal values of the inverse cosecant, although this distinction is not always made (e.g,. Zwillinger 1995, p. 466). Worse yet, the notation is sometimes used for the principal value, with being used for the multivalued function (Abramowitz and Stegun 1972, p. 80). Note that in the notation (commonly used in North America and in pocket calculators worldwide), is the cosecant and the superscript denotes an inverse function, not the multiplicative inverse.The principal value of the inverse..

Cosecant

Min Max Re Im The cosecant is the function defined by(1)(2)where is the sine. The cosecant is implemented in the Wolfram Language as Csc[z].The notation is sometimes also used (Gellert et al. 1989, p. 222; Gradshteyn and Ryzhik 2000, p. xxix). Note that the cosecant does not appear to be in consistent widespread use in Europe, although it does appear explicitly in various German and Russian handbooks (e.g., Gellert et al. 1989, p. 222; Gradshteyn and Ryzhik 2000, pp. xxix and p. 43). Interestingly, while is treated on par with the other trigonometric functions in some tabulations (Gellert et al. 1989, p. 222), it is not in others (Gradshteyn and Ryzhik 2000, who do not list it in their table of "basic functional relations" on p. 28, but do give identities involving it on p. 43).Harris and Stocker (1998, p. 300) call secant and cosecant "rarely used functions," but..

Secant

Min Max Re Im is the trigonometric function defined by(1)(2)where is the cosine. The secant is implemented in the Wolfram Language as Sec[z].Note that the secant does not appear to be in consistent widespread use in Europe, although it does appear explicitly in various German and Russian handbooks (e.g., Gradshteyn and Ryzhik 2000, p. 43). Interestingly, while is treated on a par with the other trigonometric functions in some tabulations (Gellert et al. 1989, p. 222), it is not in others (Gradshteyn and Ryzhik 2000, who do not list it in their table of "basic functional relations" on p. 28, but do give identities involving it on p. 43).Tropfke states, "The history of the secant function begins almost contemporaneously with that of the tangent, but ended after discovery of logarithmic calculation in the first half of the 17th century" (Tropfke 1923, pp. 28) and, "The secant naturally..

Sigmoid function

Min Max Re Im The sigmoid function, also called the sigmoidal curve (von Seggern 2007, p. 148) or logistic function, is the function(1)It has derivative(2)(3)(4)and indefinite integral(5)(6)It has Maclaurin series(7)(8)(9)where is an Euler polynomial and is a Bernoulli number.It has an inflection point at , where(10)It is also the solution to the ordinarydifferential equation(11)with initial condition .

Gudermannian

Min Max The Gudermannian function is the odd function denoted either or which arises in the inverse equations for the Mercator projection. expresses the latitude in terms of the vertical position in this projection, so the Gudermannian function is defined by(1)(2)For real , this definition is also equal to(3)(4)The Gudermannian is implemented in the WolframLanguage as Gudermannian[z].The derivative of the Gudermannian is(5)and its indefinite integral is(6)where is the dilogarithm.It has Maclaurin series(7)(OEIS A091912 and A136606).The Gudermannian connects the trigonometricand hyperbolic functions via(8)(9)(10)(11)(12)(13)The Gudermannian is related to the exponentialfunction by(14)(15)(16)(Beyer 1987, p. 164; Zwillinger 1995, p. 485).Other fundamental identities are(17)(18)(Zwillinger 1995, p. 485).If , then(19)(20)(21)(22)(Beyer 1987, p. 164; Zwillinger 1995, p. 530),..

Square root

Min Max A square root of is a number such that . When written in the form or especially , the square root of may also be called the radical or surd. The square root is therefore an nth root with .Note that any positive real number has two square roots, one positive and one negative. For example, the square roots of 9 are and , since . Any nonnegative real number has a unique nonnegative square root ; this is called the principal square root and is written or . For example, the principal square root of 9 is , while the other square root of 9 is . In common usage, unless otherwise specified, "the" square root is generally taken to mean the principal square root. The principal square root function is the inverse function of for . Min Max Re Im Any nonzero complex number also has two square roots. For example, using the imaginary unit i, the two square roots of are . The principal square root of a number is denoted (as in the positive real case) and is returned by the..

Elliptic lambda function

Min Max Re Im The elliptic lambda function is a -modular function defined on the upper half-plane by(1)where is the half-period ratio, is the nome(2)and are Jacobi theta functions.The elliptic lambda function is essentially the same as the inverse nome, the difference being that elliptic lambda function is a function of the half-period ratio , while the inverse nome is a function of the nome , where is itself a function of .It is implemented as the Wolfram Languagefunction ModularLambda[tau].The elliptic lambda function satisfies the functional equations(3)(4) has the series expansion(5)(OEIS A115977), and has the series expansion(6)(OEIS A029845; Conway and Norton 1979; Borweinand Borwein 1987, p. 117). gives the value of the elliptic modulus for which the complementary and normal complete elliptic integrals of the first kind are related by(7)i.e., the elliptic integral singular value for . It can be computed from(8)where(9)and..

Airy functions

There are four varieties of Airy functions: , , , and . Of these, and are by far the most common, with and being encountered much less frequently. Airy functions commonly appear in physics, especially in optics, quantum mechanics, electromagnetics, and radiative transfer. and are entire functions.A generalization of the Airy function was constructed by Hardy.The Airy function and functions are plotted above along the real axis.The and functions are defined as the two linearly independent solutions to(1)(Abramowitz and Stegun 1972, pp. 446-447; illustrated above), written in the form(2)where(3)(4)where is a confluent hypergeometric limit function. These functions are implemented in the Wolfram Language as AiryAi[z] and AiryBi[z]. Their derivatives are implemented as AiryAiPrime[z] and AiryBiPrime[z].For the special case , the functions can be written as(5)(6)(7)where is a modified Bessel function of the first kind and..

Complex exponentiation

A complex number may be taken to the power of another complex number. In particular, complex exponentiation satisfies(1)where is the complex argument. Written explicitly in terms of real and imaginary parts,(2)An explicit example of complex exponentiation is given by(3)A complex number taken to a complex number can be real. In fact, the famous example(4)shows that the power of the purely imaginary to itself is real. Min Max Re Im In fact, there is a family of values such that is real, as can be seen by writing(5)This will be real when , i.e., for(6)for an integer. For positive , this gives roots or(7)where is the Lambert W-function. For , this simplifies to(8)For , 2, ..., these give the numeric values 1, 2.92606 (OEIS A088928), 4.30453, 5.51798, 6.63865, 7.6969, ......

Complex argument

Min Max Re Im A complex number may be represented as(1)where is a positive real number called the complex modulus of , and (sometimes also denoted ) is a real number called the argument. The argument is sometimes also known as the phase or, more rarely and more confusingly, the amplitude (Derbyshire 2004, pp. 180-181 and 376).The complex argument of a number is implemented in the Wolfram Language as Arg[z].The complex argument can be computed as(2)Here, , sometimes also denoted , corresponds to the counterclockwise angle from the positive real axis, i.e., the value of such that and . The special kind of inverse tangent used here takes into account the quadrant in which lies and is returned by the FORTRAN command ATAN2(y, x) and the Wolfram Language function ArcTan[x, y], and is often (including by the Wolfram Language function Arg) restricted to the range . In the degenerate case when ,(3)Special values of the complex argument include(4)(5)(6)(7)(8)From..

Power tower

The power tower of order is defined as(1)where is Knuth up-arrow notation (Knuth 1976), which in turn is defined by(2)together with(3)(4)Rucker (1995, p. 74) uses the notation(5)and refers to this operation as "tetration."A power tower can be implemented in the WolframLanguage as PowerTower[a_, k_Integer] := Nest[Power[a, #]&, 1, k]or PowerTower[a_, k_Integer] := Power @@ Table[a, {k}]The following table gives values of for , 2, ... for small .OEIS1A0000271, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...2A0003121, 4, 27, 256, 3125, 46656, ...3A0024881, 16, 7625597484987, ...41, 65536, ...The following table gives for , 2, ... for small .OEIS1A0000121, 1, 1, 1, 1, 1, ...2A0142212, 4, 16, 65536, , ...3A0142223, 27, 7625597484987, ...44, 256, , ...Consider and let be defined as(6)(Galidakis 2004). Then for , is entire with series expansion:(7)Similarly, for , is analytic for in the domain of the principal branch of , with series expansion:(8)For..

Factorial

The factorial is defined for a positive integer as(1)So, for example, . An older notation for the factorial was written (Mellin 1909; Lewin 1958, p. 19; Dudeney 1970; Gardner 1978; Conway and Guy 1996).The special case is defined to have value , consistent with the combinatorial interpretation of there being exactly one way to arrange zero objects (i.e., there is a single permutation of zero elements, namely the empty set ).The factorial is implemented in the Wolfram Language as Factorial[n] or n!.The triangular number can be regarded as the additive analog of the factorial . Another relationship between factorials and triangular numbers is given by the identity(2)(K. MacMillan, pers. comm., Jan. 21, 2008).The factorial gives the number of ways in which objects can be permuted. For example, , since the six possible permutations of are , , , , , . The first few factorials for , 1, 2, ... are 1, 1, 2, 6, 24, 120, ... (OEIS A000142).The..

Tangent

The tangent function is defined by(1)where is the sine function and is the cosine function. The notation is sometimes also used (Gradshteyn and Ryzhik 2000, p. xxix).The common schoolbook definition of the tangent of an angle in a right triangle (which is equivalent to the definition just given) is as the ratio of the side lengths opposite to the angle and adjacent the angle, i.e.,(2)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).The word "tangent" also has an important related meaning as a line or plane which touches a given curve or solid at a single point. These geometrical objects are then called a tangent line or tangent plane, respectively. Min Max Re Im The definition of the tangent function can be extended to complex arguments using the definition(3)(4)(5)(6)where..

Probability integral

Min Max Min Max Re Im (1)(2)(3)(4)where is the normal distribution function and erf is the error function.

Dirichlet lambda function

Min Max Re Im The Dirichlet lambda function is the Dirichlet L-series defined by(1)(2)where is the Riemann zeta function. The function is undefined at . It can be computed in closed form where can, that is for even positive .The Dirichlet lambda function is implemented in the WolframLanguage as DirichletLambda[x].It is related to the Riemann zeta functionand Dirichlet eta function by(3)and(4)(Spanier and Oldham 1987). Special values of include(5)(6)

Riemann zeta function

The Riemann zeta function is an extremely important special function of mathematics and physics that arises in definite integration and is intimately related with very deep results surrounding the prime number theorem. While many of the properties of this function have been investigated, there remain important fundamental conjectures (most notably the Riemann hypothesis) that remain unproved to this day. The Riemann zeta function is denoted and is plotted above (using two different scales) along the real axis. Min Max Re Im In general, is defined over the complex plane for one complex variable, which is conventionally denoted (instead of the usual ) in deference to the notation used by Riemann in his 1859 paper that founded the study of this function (Riemann 1859). is implemented in the Wolfram Language as Zeta[s].The plot above shows the "ridges" of for and . The fact that the ridges appear to decrease monotonically for is not..

Absolute square

Min Max Min Max Re Im The absolute square of a complex number , also known as the squared norm, is defined as(1)where denotes the complex conjugate of and is the complex modulus.If the complex number is written , with and real, then the absolute square can be written(2)If is a real number, then (1) simplifies to(3)An absolute square can be computed in terms of and using the Wolfram Language command ComplexExpand[Abs[z]^2, TargetFunctions -> Conjugate].An important identity involving the absolute square is given by(4)(5)(6)If , then (6) becomes(7)(8)If , and , then(9)Finally,(10)(11)(12)

Erfi

Min Max Min Max Re Im The "imaginary error function" is an entire function defined by(1)where is the erf function. It is implemented in the Wolfram Language as Erfi[z]. has derivative(2)and integral(3)It has series about given by(4)(where the terms are OEIS A084253), and seriesabout infinity given by(5)(OEIS A001147 and A000079).

Erfc

Erfc is the complementary error function, commonly denoted , is an entire function defined by(1)(2)It is implemented in the Wolfram Languageas Erfc[z].Note that some authors (e.g., Whittaker and Watson 1990, p. 341) define without the leading factor of .For ,(3)where is the incomplete gamma function.The derivative is given by(4)and the indefinite integral by(5)It has the special values(6)(7)(8)It satisfies the identity(9)It has definite integrals(10)(11)(12)For , is bounded by(13) Min Max Re Im Erfc can also be extended to the complex plane, as illustrated above.A generalization is obtained from the erfcdifferential equation(14)(Abramowitz and Stegun 1972, p. 299; Zwillinger 1997, p. 122). The general solution is then(15)where is the repeated erfc integral. For integer ,(16)(17)(18)(19)(Abramowitz and Stegun 1972, p. 299), where is a confluent hypergeometric function of the first kind and is a..

Erf

is the "error function" encountered in integrating the normal distribution (which is a normalized form of the Gaussian function). It is an entire function defined by(1)Note that some authors (e.g., Whittaker and Watson 1990, p. 341) define without the leading factor of .Erf is implemented in the Wolfram Language as Erf[z]. A two-argument form giving is also implemented as Erf[z0, z1].Erf satisfies the identities(2)(3)(4)where is erfc, the complementary error function, and is a confluent hypergeometric function of the first kind. For ,(5)where is the incomplete gamma function.Erf can also be defined as a Maclaurin series(6)(7)(OEIS A007680). Similarly,(8)(OEIS A103979 and A103980).For , may be computed from(9)(10)(OEIS A000079 and A001147;Acton 1990).For ,(11)(12)Using integration by parts gives(13)(14)(15)(16)so(17)and continuing the procedure gives the asymptoticseries(18)(19)(20)(OEIS A001147..

Dirichlet eta function

Min Max Min Max Re Im The Dirichlet eta function is the function defined by(1)(2)where is the Riemann zeta function. Note that Borwein and Borwein (1987, p. 289) use the notation instead of . The function is also known as the alternating zeta function and denoted (Sondow 2003, 2005). is defined by setting in the right-hand side of (2), while (sometimes called the alternating harmonic series) is defined using the left-hand side. The function vanishes at each zero of except (Sondow 2003).The eta function is related to the Riemann zeta function and Dirichlet lambda function by(3)and(4)(Spanier and Oldham 1987). The eta function is also a special case of the polylogarithmfunction,(5)The value may be computed by noting that the Maclaurin series for for is(6)Therefore, the natural logarithm of 2 is(7)(8)(9)(10)The derivative of the eta function is given by(11)or in the special case , by(12)(13)(14)(15)This latter fact provides a remarkable..

Dirichlet beta function

Min Max Min Max Re Im The Dirichlet beta function is defined by the sum(1)(2)where is the Lerch transcendent. The beta function can be written in terms of the Hurwitz zeta function by(3)The beta function can be defined over the whole complexplane using analytic continuation,(4)where is the gamma function.The Dirichlet beta function is implemented in the WolframLanguage as DirichletBeta[x].The beta function can be evaluated directly special forms of arguments as(5)(6)(7)where is an Euler number.Particular values for are(8)(9)(10)(11)where is Catalan's constant and is the polygamma function. For , 3, 5, ..., , where the multiples are 1/4, 1/32, 5/1536, 61/184320, ... (OEIS A046976 and A053005).It is involved in the integral(12)(Guillera and Sondow 2005).Rivoal and Zudilin (2003) proved that at least one of the seven numbers , , , , , , and is irrational.The derivative can also be computed analytically at a number of integer values of including(13)(14)(15)(16)(17)(18)(19)(OEIS..

Cube root

Min Max Min Max Re Im Given a number , the cube root of , denoted or ( to the 1/3 power), is a number such that . The cube root is therefore an nth root with . Every real number has a unique real cube root, and every nonzero complex number has three distinct cube roots.The schoolbook definition of the cube root of a negative number is . However, extension of the cube root into the complex plane gives a branch cut along the negative real axis for the principal value of the cube root as illustrated above. By convention, "the" (principal) cube root is therefore a complex number with positive imaginary part. As a result, the Wolfram Language and other symbolic algebra languages and programs that return results valid over the entire complex plane therefore return complex results for . For example, in the Wolfram Language, ComplexExpand[(-1)^(1/3)] gives the result .When considering a positive real number , the Wolfram Language function CubeRoot[x],..

Integer part

The function gives the integer part of . In many computer languages, the function is denoted int(x). It is related to the floor and ceiling functions and by(1)The integer part function satisfies(2)and is implemented in the Wolfram Language as IntegerPart[x]. This definition is chosen so that , where is the fractional part. Although Spanier and Oldham (1987) use the same definition as in the Wolfram Language, they mention the formula only very briefly and then say it will not be used further. Graham et al. (1994), and perhaps most other mathematicians, use the term "integer" part interchangeably with the floor function . Min Max Re Im The integer part function can also be extended to the complexplane, as illustrated above.Since usage concerning fractional part/value and integer part/value can be confusing, the following table gives a summary of names and notations used. Here, S&O indicates Spanier and Oldham (1987).notationnameS&OGraham..

Riemann zeta function zeros

Zeros of the Riemann zeta function come in two different types. So-called "trivial zeros" occur at all negative even integers , , , ..., and "nontrivial zeros" occur at certain values of satisfying(1)for in the "critical strip" . In general, a nontrivial zero of is denoted , and the th nontrivial zero with is commonly denoted (Brent 1979; Edwards 2001, p. 43), with the corresponding value of being called .Wiener (1951) showed that the prime number theorem is literally equivalent to the assertion that has no zeros on (Hardy 1999, p. 34; Havil 2003, p. 195). The Riemann hypothesis asserts that the nontrivial zeros of all have real part , a line called the "critical line." This is known to be true for the first zeros.An attractive poster plotting zeros of the Riemann zeta function on the critical line together with annotations for relevant historical information, illustrated above,..

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