Newton's iteration is an algorithm for computing the square root of a number via the recurrence equation(1)where . This recurrence converges quadratically as .Newton's iteration is simply an application of Newton'smethod for solving the equation(2)For example, when applied numerically, the first few iterations to Pythagoras's constant are 1, 1.5, 1.41667, 1.41422, 1.41421, ....The first few approximants , , ... to are given by(3)These can be given by the analytic formula(4)(5)These can be derived by noting that the recurrence can be written as(6)which has the clever closed-form solution(7)Solving for then gives the solution derived above.The following table summarizes the first few convergents for small positive integer OEIS, , ...11, 1, 1, 1, 1, 1, 1, 1, ...2A001601/A0510091, 3/2, 17/12, 577/408, 665857/470832, ...3A002812/A0715791, 2, 7/4, 97/56, 18817/10864, 708158977/408855776, .....
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..
Let(1)(2)(3)(OEIS A104457), where is the golden ratio, and(4)(5)(OEIS A002390).Define the Fibonacci hyperbolic sine by(6)(7)(8)The function satisfies(9)and for ,(10)where is a Fibonacci number. For , 2, ..., the values are therefore 1, 3, 8, 21, 55, ... (OEIS A001906).Define the Fibonacci hyperbolic cosine by(11)(12)(13)This function satisfies(14)and for ,(15)where is a Fibonacci number. For , 2, ..., the values are therefore 2, 5, 13, 34, 89, ... (OEIS A001519).Similarly, the Fibonacci hyperbolic tangent is defined by(16)and for ,(17)For , 2, ..., the values are therefore 1/2, 3/5, 8/13, 21/34, 55/89, ... (OEIS A001906 and A001519).
Schur's partition theorem lets denote the number of partitions of into parts congruent to (mod 6), denote the number of partitions of into distinct parts congruent to (mod 3), and the number of partitions of into parts that differ by at least 3, with the added constraint that the difference between multiples of three is at least 6. Then (Schur 1926; Bressoud 1980; Andrews 1986, p. 53).The values of for , 2, ... are 1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, ... (OEIS A003105). For example, for , there are nine partitions satisfying these conditions, as summarized in the following table (Andrews 1986, p. 54).15The identity can be established using the identity(1)(2)(3)(4)(5)(Andrews 1986, p. 54). The identity is significantly trickier.
The Wolstenholme numbers are defined as the numerators of the generalized harmonic number appearing in Wolstenholme's theorem. The first few are 1, 5, 49, 205, 5269, 5369, 266681, 1077749, ... (OEIS A007406).By Wolstenholme's theorem, for prime , where is the th Wolstenholme number. In addition, for prime .The first few prime Wolstenholme numbers are 5, 266681, 40799043101, 86364397717734821, ... (OEIS A123751), which occur at indices , 7, 13, 19, 121, 188, 252, 368, 605, 745, ... (OEIS A111354).
The term "Euler function" may be used to refer to any of several functions in number theory and the theory of special functions, including 1. the totient function , defined as the number of positive integers that are relatively prime to , where 1 is counted as being relatively prime to all numbers; 2. the function(1)(2)(3)where and are q-Pochhammer symbols; 3. the Euler L-function , which is a special case of the Artin L-function for the polynomial and is defined by(4)where(5)(6)with a Legendre symbol.
Based on a problem in particle physics, Dyson (1962abc) conjectured that the constantterm in the Laurent seriesis the multinomial coefficientThe theorem was proved by Wilson (1962) and independently by Gunson (1962). A definitive proof was subsequently published by Good (1970).
The logarithmic derivative of a function is defined as the derivative of the logarithm of a function. For example, the digamma function is defined as the logarithmic derivative of the gamma function,
Polynomials which form the Sheffer sequence for(1)(2)which have generating function(3)The first few are(4)(5)(6)
Polynomials which form the Sheffer sequence for(1)and have exponential generating function(2)The first few are(3)(4)(5)(6)(7)(8)
Polynomials which form a Sheffer sequence with(1)(2)and have generating function(3)The first few are(4)(5)(6)Jordan (1965) considers the related polynomials which form a Sheffer sequence with(7)(8)These polynomials have generating function(9)The first few are(10)(11)(12)(13)The Peters polynomials are a generalizationof the Boole polynomials.
The polynomials which form the Sheffer sequence for(1)(2)which have generating function(3)The first few are(4)(5)(6)
Polynomials which form the Sheffer sequence for(1)(2)where is the inverse function of , and have generating function(3)The first few polynomials are(4)(5)(6)(7)The Stirling polynomials are related to the Stirling numbers of the first kind by(8)where is a binomial coefficient and is an integer with (Roman 1984, p. 129).
Polynomials which form the Sheffer sequence for(1)(2)and have generating function(3)The are given in terms of the hypergeometricseries by(4)where is the Pochhammer symbol (Koepf 1998, p. 115). The first few are(5)(6)(7)Koekoek and Swarttouw (1998) defined the Meixner polynomials without the Pochhammersymbol as(8)The Krawtchouk polynomials are a specialcase of the Meixner polynomials of the first kind.
Polynomials which form the Sheffer sequence for(1)where is the inverse function of , and have generating function(2)The first few are(3)(4)(5)(6)(7)(8)
Polynomials which form a Sheffer sequence with(1)(2)giving generating function(3)Roman (1984) defines Bernoulli numbers of the second kind as . They are related to the Stirling numbers of the first kind by(4)(Roman 1984, p. 115), and obey the reflection formula(5)(Roman 1984, p. 119).The first few Bernoulli polynomials of the second kind are(6)(7)(8)(9)(10)
The polynomials given by the associated Sheffer sequence with(1)where . The inverse function (and therefore generating function) cannot be computed algebraically, but the generating function(2)can be given in terms of the sum(3)This results in(4)where is a falling factorial. The first few are(5)(6)(7)(8)(9)The binomial identity obtained from the Sheffer sequencegives the generalized Chu-Vandermonde identity(10)(Roman 1984, p. 69; typo corrected).In the special case , the function simplifies to(11)which gives the generating function(12)giving the polynomials(13)(14)(15)(16)(17)
A symmetric function on variables , ..., is a function that is unchanged by any permutation of its variables. In most contexts, the term "symmetric function" refers to a polynomial on variables with this feature (more properly called a "symmetric polynomial"). Another type of symmetric functions is symmetric rational functions, which are the rational functions that are unchanged by permutation of variables.The symmetric polynomials (respectively, symmetric rational functions) can be expressed as polynomials (respectively, rational functions) in the elementary symmetric polynomials. This is called the fundamental theorem of symmetric functions.A function is sometimes said to be symmetric about the y-axis if . Examples of such functions include (the absolute value) and (the parabola)...
A function built up of a finite combination of constant functions, field operations (addition, multiplication, division, and root extractions--the elementary operations)--and algebraic, exponential, and logarithmic functions and their inverses under repeated compositions (Shanks 1993, p. 145; Chow 1999). Among the simplest elementary functions are the logarithm, exponential function (including the hyperbolic functions), power function, and trigonometric functions.Following Liouville (1837, 1838, 1839), Watson (1966, p. 111) defines the elementarytranscendental functions as(1)(2)(3)and lets , etc.Not all functions are elementary. For example, the normaldistribution function(4)(5)is a notorious example of a nonelementary function, where is erf (sometimes known as the error function). The elliptic integral(6)is another, where is an elliptic integral of the first kind...
The spherical harmonics can be generalized to vector spherical harmonics by looking for a scalar function and a constant vector such that(1)(2)(3)(4)so(5)Now interchange the order of differentiation and use the fact that multiplicative constants can be moving inside and outside the derivatives to obtain(6)(7)(8)and(9)(10)Putting these together gives(11)so satisfies the vector Helmholtz differential equation if satisfies the scalar Helmholtz differential equation(12)Construct another vector function(13)which also satisfies the vector Helmholtzdifferential equation since(14)(15)(16)(17)(18)which gives(19)We have the additional identity(20)(21)(22)(23)(24)In this formalism, is called the generating function and is called the pilot vector. The choice of generating function is determined by the symmetry of the scalar equation, i.e., it is chosen to solve the desired scalar differential equation. If is taken as(25)where..
The Poisson integral with ,where is a Bessel function of the first kind.
The reciprocal of the arithmetic-geometric mean of 1 and ,(1)(2)(3)(4)(5)(6)(7)(OEIS A014549), where is the complete elliptic integral of the first kind, is a Jacobi theta function, and is the gamma function. This correspondence was first noticed by Gauss, and was the basis for his exploration of the lemniscate function (Borwein and Bailey 2003, pp. 13-15).Two rapidly converging series for are given by(8)(9)(Finch 2003, p. 421).Gauss's constant has continued fraction [0,1, 5, 21, 3, 4, 14, 1, 1, 1, 1, 1, 3, 1, 15, ...] (OEIS A053002).The inverse of Gauss's constant is given by(10)(OEIS A053004; Finch 2003, p. 420; Borwein and Bailey 2003, p. 13), which has [1, 5, 21, 3, 4, 14, 1, 1, 1, 1, 1, 3, 1, 15, 1, ...] (OEIS A053003).The value(11)(OEIS A097057) is sometimes called the ubiquitous constant (Spanier and Oldham 1987; Schroeder 1994; Finch 2003, p. 421), and(12)(OEIS A076390) is sometimes called the secondlemniscate..
The entire functionwhere and , named after the British mathematician E. M. Wright.
Dawson's integral (Abramowitz and Stegun 1972, pp. 295 and 319), also sometimes called Dawson's function, is the entire function given by the integral(1)(2)where is erfi, that arises in computation of the Voigt lineshape (Harris 1948, Hummer 1963, Sajo 1993, Lether 1997), in heat conduction, and in the theory of electrical oscillations in certain special vacuum tubes (McCabe 1974). It is commonly denoted (McCabe 1974; Coleman 1987; Milone and Milone 1988; Sajo 1993; Lether 1997; Press et al. 2007, p. 302), although Spanier and Oldham (1987) denote it by .Dawson's integral is implemented in the WolframLanguage as DawsonF[z].It is an odd function, so(3)Its derivative is(4)and its indefinite integral is(5)where is a generalized hypergeometric function.It is the particular solution to the differential equation(6)(McCabe 1974).Its Maclaurin series is given by(7)(8)(OEIS A122803 and A001147).If has the asymptotic series(9)It..
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...
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)..
Fermat's sandwich theorem states that 26 is the only number sandwiched between a perfect square number ( and a perfect cubic number (). According to Singh (1997), after challenging other mathematicians to establish this result while not revealing his own proof, Fermat took particular delight in taunting the English mathematicians Wallis and Digby with their inability to prove the result.
The conjecture made by Belgian mathematician Eugène Charles Catalan in 1844 that 8 and 9 ( and ) are the only consecutive powers (excluding 0 and 1). In other words,(1)is the only nontrivial solution to Catalan'sDiophantine problem(2)The special case and is the case of a Mordell curve.Interestingly, more than 500 years before Catalan formulated his conjecture, Levi ben Gerson (1288-1344) had already noted that the only powers of 2 and 3 that apparently differed by 1 were and (Peterson 2000).This conjecture had defied all attempts to prove it for more than 150 years, although Hyyrő and Makowski proved that no three consecutive powers exist (Ribenboim 1996), and it was also known that 8 and 9 are the only consecutive cubic and square numbers (in either order). Finally, on April 18, 2002, Mihăilescu sent a manuscript proving the entire conjecture to several mathematicians (van der Poorten 2002). The proof has now appeared in..
The function defined by Minkowski for the purpose of mapping the quadratic surds in the open interval into the rational numbers of in a continuous, order-preserving manner. takes a number having continued fraction to the number(1)The function satisfies the following properties (Salem 1943). 1. is strictly increasing. 2. If is rational, then is of the form , with and integers. 3. If is a quadratic surd, then the continued fraction is periodic, and hence is rational. 4. The function is purely singular (Denjoy 1938). can also be constructed as(2)where and are two consecutive irreducible fractions from the Farey sequence. At the th stage of this definition, is defined for values of , and the ordinates corresponding to these values are for , 1, ..., (Salem 1943).The function satisfies the identity(3)A few special values include(4)(5)(6)(7)(8)(9)(10)(11)where is the golden ratio...
Gauss's continued fraction is given by the continuedfractionwhere is a hypergeometric function. Many analytic expressions for continued fractions of functions can be derived from this formula.
Euler's continued fraction is the name given by Borwein et al. (2004, p. 30)to Euler's formula for the inverse tangent,An even more famous continued fraction related to Euler which is perhaps a more appropriate recipient of the appellation "Euler's continued fraction" is the simple continued fraction for e, namely
"The" Jacobi identity is a relationship(1)between three elements , , and , where is the commutator. The elements of a Lie algebra satisfy this identity.Relationships between the Q-functions are also known as Jacobi identities:(2)equivalent to the Jacobi triple product (Borweinand Borwein 1987, p. 65) and(3)where(4) is the complete elliptic integral of the first kind, and . Using Weber functions(5)(6)(7)(5) and (6) become(8)(9)(Borwein and Borwein 1987, p. 69).
For even ,(1)(Nagell 1951, p. 176). Writing out symbolically,(2)which gives(3)where is a q-Pochhammer symbol.For example, for ,(4)and for ,(5)
A numberwhere is an integer or rational number, is the inverse tangent, and is the inverse cotangent. Gregory numbers arise in the determination of Machin-like formulas. Every Gregory number can be expressed uniquely as a linear combination of s where the s are Størmer numbers.
Lehmer (1938) showed that every positive irrational number has a unique infinite continued cotangent representation of the form(1)where the s are nonnegative and(2)Note that this growth condition on coefficients is essential for the uniqueness of Lehmer expansion.The following table summarizes the coefficients for various special constants.OEISeA0026682, 8, 75, 8949, 119646723, 15849841722437093, ...Euler-Mascheroni constant A0817820, 1, 3, 16, 389, 479403, 590817544217, ...golden ratio A0062671, 4, 76, 439204, 84722519070079276, ...Lehmer's constant A0020650, 1, 3, 13, 183, 33673, ...A0026673, 73, 8599, 400091364,371853741549033970, ...Pythagoras's constant A0026661, 5, 36, 3406, 14694817,727050997716715, ...The expansion for the golden ratio has the beautiful closed form(3)where is a Lucas number.An illustration of a different cotangent expansion for that is not a Lehmer expansion because its coefficients..
The Lehmer cotangent expansion for whichthe convergence is slowest occurs when the inequality in the recurrence equation(1)for(2)is replaced by equality, giving and(3)for .This recurrences gives values of corresponding to 0, 1, 3, 13, 183, 33673, ... (OEIS A002065), and defines the constant known as Lehmer's constant as(4)(5)(6)(OEIS A030125). is not an algebraic number of degree less than 4, but Lehmer's approach cannot show whether is transcendental.
For ,where is the gamma function.
(1)(2)(3)where is a gamma function and is a double factorial.
Hadjicostas's formula is a generalization of the unitsquare double integral(1)(Sondow 2003, 2005; Borwein et al. 2004, p. 49), where is the Euler-Mascheroni constant. It states(2)for , where is the gamma function and is the Riemann zeta function (although care must be taken at because of the removable singularity present there). It was conjectured by Hadjicostas (2004) and almost immediately proved by Chapman (2004). The special case gives Beukers's integral for ,(3)(Beukers 1979). At , the formula is related to Beukers's integral for Apéry's constant , which is how interest in this class of integrals originally arose.There is an analogous formula(4)for , due to Sondow (2005), where is the Dirichlet eta function. This includes the special cases(5)(6)(7)(OEIS A094640; Sondow 2005) and(8)(9)(OEIS A103130), where is the Glaisher-Kinkelin constant (Sondow 2005)...
The logarithmic integral is defined as theCauchy principal value(1)(2)Soldner's constant, denoted (or sometimes ) is the root of the logarithmic integral,(3)so that(4)for (Soldner 1812; Nielsen 1965, p. 88). Ramanujan calculated (Hardy 1999, Le Lionnais 1983, Berndt 1994), while the correct value is 1.45136923488... (OEIS A070769; Derbyshire 2004, p. 114).
A plot in the complex plane of the points(1)where and are the Fresnel integrals (von Seggern 2007, p. 210; Gray 1997, p. 65). The Cornu spiral is also known as the clothoid or Euler's spiral. It was probably first studied by Johann Bernoulli around 1696 (Bernoulli 1967, pp. 1084-1086). A Cornu spiral describes diffraction from the edge of a half-plane.The quantities and are plotted above. The slope of the curve's tangentvector (above right figure) is(2)plotted below. The Cesàro equation for a Cornu spiral is , where is the radius of curvature and the arc length. The torsion is .Gray (1997) defines a generalization of the Cornu spiral given by parametricequations(3)(4)(5)(6)where is a generalized hypergeometric function.The arc length, curvature,and tangential angle of this curve are(7)(8)(9)The Cesàro equation is(10)Dillen (1990) describes a class of "polynomial spirals" for which the..
The decimal expansion of the natural logarithm of 10 is given by(1)(OEIS A002392).It is also given by the BBP-type formulas(2)(3)(4)(5)(6)(7)(E. W. Weisstein, Aug. 28, 2008).
The catacaustic of the natural logarithm specified parametrically as(1)(2)is a complicated expression for an arbitrary radiantpoint.However, for a point , the catacaustic becomes(3)(4)Making the substitution then gives the equivalent parametrization(5)(6)which is the equation of a catenary.
The Mercator series, also called the Newton-Mercator series (Havil 2003, p. 33), is the Taylor series for the natural logarithm(1)(2)for , which was found by Newton, but independently discovered and first published by Mercator in 1668.Plugging in gives a beautiful series for the natural logarithm of 2,(3)also known as the alternating harmonic series and equal to , where is the Dirichlet eta function.
A Fourier series-like expansion of a twice continuouslydifferentiable function(1)for , where is a zeroth order Bessel function of the first kind. The coefficients are then given by(2)(3)(Gradshteyn and Ryzhik 2000, p. 926), where and care should be taken to avoid the two typos of Iyanaga and Kawada (1980) and Itô (1986).As an example, consider , which has and therefore(4)(5)(6)(7)(8)so(9)(Whittaker and Watson 1990, p. 378; Gradshteyn and Ryzhik 2000, p. 926). This is illustrated above with 1 (red), 2 (green), 3 (blue), and 4 terms (violet) included.Similarly, for ,(10)
The series for the inversetangent,Plugging in gives Gregory's formulaThis series is intimately connected with the number of representations of by squares , and also with Gauss's circle problem (Hilbert and Cohn-Vossen 1999, pp. 27-39).
The series(1)is called the harmonic series. It can be shown to diverge using the integral test by comparison with the function . The divergence, however, is very slow. Divergence of the harmonic series was first demonstrated by Nicole d'Oresme (ca. 1323-1382), but was mislaid for several centuries (Havil 2003, p. 23; Derbyshire 2004, pp. 9-10). The result was proved again by Pietro Mengoli in 1647, by Johann Bernoulli in 1687, and by Jakob Bernoulli shortly thereafter (Derbyshire 2004, pp. 9-10).Progressions of the form(2)are also sometimes called harmonic series (Beyer 1987).Oresme's proof groups the harmonic terms by taking 2, 4, 8, 16, ... terms (after the first two) and noting that each such block has a sum larger than 1/2,(3)(4)and since an infinite sum of 1/2's diverges, so does the harmonic series.The generalization of the harmonic series(5)is known as the Riemann zeta function.The sum of the first few terms of..
How far can a stack of books protrude over the edge of a table without the stack falling over? It turns out that the maximum overhang possible for books (in terms of book lengths) is half the th partial sum of the harmonic series.This is given explicitly by(1)where is a harmonic number. The first few values are(2)(3)(4)(5)(OEIS A001008 and A002805).When considering the stacking of a deck of 52 cards so that maximum overhang occurs, the total amount of overhang achieved after sliding over 51 cards leaving the bottom one fixed is(6)(7)(8)(Derbyshire 2004, p. 6).In order to find the number of stacked books required to obtain book-lengths of overhang, solve the equation for , and take the ceiling function. For , 2, ... book-lengths of overhang, 4, 31, 227, 1674, 12367, 91380, 675214, 4989191, 36865412, 272400600, ... (OEIS A014537) books are needed.When more than one book or card can be used per level, the problem becomes much more complex. For..
There are (at least) two mathematical constants associated with Theodorus. The first Theodorus's constant is the elementary algebraic number , i.e., the square root of 3. It has decimal expansion(1)(OEIS A002194) and is named after Theodorus, who proved that the square roots of the integers from 3 to 17 (excluding squares 4, 9,and 16) are irrational (Wells 1986, p. 34). The space diagonal of a unit cube has length . has continued fraction [1, 1, 2, 1, 2, 1, 2, ...] (OEIS A040001). In binary, it is represented by(2)(OEIS A004547).Another constant sometimes known as the constant of Theodorus is the slope of a continuous analog of the discrete Theodorus spiral due to Davis (1993) at the point , given by(3)(4)(5)(6)(OEIS A226317; Finch 2009), where is the Riemann zeta function.
In this work, the name Pythagoras's constant will be given to the squareroot of 2,(1)(OEIS A002193), which the Pythagoreans provedto be irrational.In particular, is the length of the hypotenuse of an isosceles right triangle with legs of length one, and the statement that it is irrational means that it cannot be expressed as a ratio of integers and . Legend has it that the Pythagorean philosopher Hippasus used geometric methods to demonstrate the irrationality of while at sea and, upon notifying his comrades of his great discovery, was immediately thrown overboard by the fanatic Pythagoreans. A slight generalization is sometimes known as Pythagoras's theorem.Theodorus subsequently proved that the square roots of the numbers from 3 to 17 (excluding 4, 9,and 16) are also irrational (Wells 1986, p. 34).It is not known if Pythagoras's constant is normalto any base (Stoneham 1970, Bailey and Crandall 2003).The continued fraction for..
If is a prime , then the numerator of the harmonic number(1)is divisible by and the numerator of the generalized harmonic number(2)is divisible by . The numerators of are sometimes known as Wolstenholme numbers.These imply that if is prime, then(3)
There are at least two statements which go by the name of Artin's conjecture.If is any complex finite-dimensional representation of the absolute Galois group of a number field, then Artin showed how to associate an -series with it. These -series directly generalize zeta functions and Dirichlet -series, and as a result of work by Richard Brauer, is known to extend to a meromorphic function on the complex plane. Artin's conjecture predicts that it is in fact holomorphic, i.e., has no poles, with the possible exception of a pole at (Artin 1923/1924). Compare with the generalized Riemann hypothesis, which deals with the locations of the zeros of certain -series.The second conjecture states that every integer not equal to or a square number is a primitive root modulo for infinitely many and proposes a density for the set of such which are always rational multiples of a constant known as Artin's constant. There is an analogous theorem for functions instead..
The sum-of-factorial powers function is defined by(1)For ,(2)(3)(4)where is the exponential integral, (OEIS A091725), is the En-function, is the real part of , and i is the imaginary number. The first few values are 1, 3, 9, 33, 153, 873, 5913, 46233, 409113, ... (OEIS A007489). cannot be written as a hypergeometric term plus a constant (Petkovšek et al. 1996). The only prime of this form is , since(5)(6)(7)is always a multiple of 3 for .In fact, is divisible by 3 for and , 5, 7, ... (since the Cunningham number given by the sum of the first two terms is always divisible by 3--as are all factorial powers in subsequent terms ) and so contains no primes, meaning sequences with even are the only prime contenders.The sum(8)does not appear to have a simple closed form, but its values for , 2, ... are 1, 5, 41, 617, 15017, 533417, 25935017, ... (OEIS A104344). It is prime for indices 2, 3, 4, 5, 7, 8, 10, 18, 21, 42, 51, 91, 133, 177, 182, 310, 3175, 9566, 32841,..
The alternating factorial is defined as the sum of consecutive factorialswith alternating signs,(1)They can be given in closed form as(2)where is the exponential integral, is the En-function, and is the gamma function.The alternating factorial will is implemented in the WolframLanguage as AlternatingFactorial[n].A simple recurrence equation for is given by(3)where .For , 2, ..., the first few values are 1, 1, 5, 19, 101, 619, 4421, 35899, ... (OEIS A005165).The first few values for which are (probable) primes are , 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160, 661, 2653, 3069, 3943, 4053, 4998, 8275, 9158, 11164, 43592, 59961, ... (OEIS A001272; extending Guy 1994, p. 100). ivković (1999) has shown that the number of such primes is finite. was verified to be prime in Jul. 2000 by team of G. La Barbera and others using the Certifix program developed by Marcel Martin.The following table summarizes the largest..
A series of the form(1)where is a real and is a Bessel function of the first kind. Special cases are(2)where is the gamma function, and(3)where(4)and is the floor function.
The exponential sum function , sometimes also denoted , is defined by(1)(2)where is the upper incomplete gamma function and is the (complete) gamma function.
A Kapteyn series is a series of the form(1)where is a Bessel function of the first kind. Examples include Kapteyn's original series(2)and(3)
A function is said to be piecewise constant if it is locally constant in connected regions separated by a possibly infinite number of lower-dimensional boundaries. The Heaviside step function, rectangle function, and square wave are examples of one-dimensional piecewise constant functions. Examples in two dimensions include and (illustrated above) for a complex number, the real part, and the floor function. The nearest integer function is also piecewise constant.
A -variate multivariate normal distribution (also called a multinormal distribution) is a generalization of the bivariate normal distribution. The -multivariate distribution with mean vector and covariance matrix is denoted . The multivariate normal distribution is implemented as MultinormalDistribution[mu1, mu2, ..., sigma11, sigma12, ..., sigma12, sigma22, ..., ..., x1, x2, ...] in the Wolfram Language package MultivariateStatistics` (where the matrix must be symmetric since ).In the case of nonzero correlations, there is in general no closed-form solution for the distribution function of a multivariate normal distribution. As a result, such computations must be done numerically.
The bivariate normal distribution is the statistical distribution with probabilitydensity function(1)where(2)and(3)is the correlation of and (Kenney and Keeping 1951, pp. 92 and 202-205; Whittaker and Robinson 1967, p. 329) and is the covariance.The probability density function of the bivariate normal distribution is implemented as MultinormalDistribution[mu1, mu2, sigma11, sigma12, sigma12, sigma22] in the Wolfram Language package MultivariateStatistics` .The marginal probabilities are then(4)(5)and(6)(7)(Kenney and Keeping 1951, p. 202).Let and be two independent normal variates with means and for , 2. Then the variables and defined below are normal bivariates with unit variance and correlation coefficient :(8)(9)To derive the bivariate normal probability function, let and be normally and independently distributed variates with mean 0 and variance 1, then define(10)(11)(Kenney and Keeping..
The Gauss map is a function from an oriented surface in Euclidean space to the unit sphere in . It associates to every point on the surface its oriented unit normal vector. Since the tangent space at a point on is parallel to the tangent space at its image point on the sphere, the differential can be considered as a map of the tangent space at into itself. The determinant of this map is the Gaussian curvature, and negative one-half of the trace is the mean curvature.Another meaning of the Gauss map is the function(Trott 2004, p. 44), where is the floor function, plotted above on the real line and in the complex plane.The related function is plotted above, where is the fractional part.The plots above show blowups of the absolute values of these functions (a version of the left figure appears in Trott 2004, p. 44)...
A piecewise function is a function that is defined on a sequence of intervals. Acommon example is the absolute value,(1)Piecewise functions are implemented in the Wolfram Language as Piecewise[val1, cond1, val2, cond2, ...].Additional piecewise functions include the Heaviside step function, rectangle function, and triangle function.Semicolons and commas are sometimes used at the end of either the left or the right column, with particular usage apparently depending on the author. The words "if" and "for" are sometimes used in the right column, as is "otherwise" for the final (default) case.For example, Knuth (1996, pp. 175 and 180) uses the notations(2)(3)(4)both with and without the left-column commas. Similarly, Arfken (1985, pp. 488-489) uses(5)which lacks semicolons but only sometimes lacks right-column commas.In this work, commas and semicolons are not used...
The natural logarithm of 2 is a transcendental quantity that arises often in decay problems, especially when half-lives are being converted to decay constants. has numerical value(1)(OEIS A002162).The irrationality measure of is known to be less than 3.8913998 (Rukhadze 1987, Hata 1990).It is not known if is normal (Bailey and Crandall 2002).The alternating series and BBP-typeformula(2)converges to the natural logarithm of 2, where is the Dirichlet eta function. This identity follows immediately from setting in the Mercator series, yielding(3)It is also a special case of the identity(4)where is the Lerch transcendent.This is the simplest in an infinite class of such identities, the first few of which are(5)(6)(E. W. Weisstein, Oct. 7, 2007).There are many other classes of BBP-type formulas for , including(7)(8)(9)(10)(11)Plouffe (2006) found the beautiful sum(12)A rapidly converging Zeilberger-type sum..
The Dottie number is the name given by Kaplan (2007) to the unique real root of (namely, the unique real fixed point of the cosine function), which is 0.739085... (OEIS A003957). The name "Dottie" is of no fundamental mathematical significance since it refers to a particular French professor who--no doubt like many other calculator uses before and after her--noticed that whenever she typed a number into her calculator and hit the cosine button repeatedly, the result always converged to this value.The number is well-known, having appeared in numerous elementary works on algebra already by the late 1880s (e.g., Bertrand 1865, p. 285; Heis 1886, p. 468; Briot 1881, pp. 341-343), and probably much earlier as well. It is also known simply as the cosine constant, cosine superposition constant, iterated cosine constant, or cosine fixed point constant. Arakelian (1981, pp. 135-136; 1995) has used the Armenian..
The Heaviside step function is a mathematical function denoted , or sometimes or (Abramowitz and Stegun 1972, p. 1020), and also known as the "unit step function." The term "Heaviside step function" and its symbol can represent either a piecewise constant function or a generalized function.When defined as a piecewise constant function, the Heaviside step function is given by(1)(Abramowitz and Stegun 1972, p. 1020; Bracewell 2000, p. 61). The plot above shows this function (left figure), and how it would appear if displayed on an oscilloscope (right figure).When defined as a generalized function, it can be defined as a function such that(2)for the derivative of a sufficiently smooth function that decays sufficiently quickly (Kanwal 1998).The Wolfram Language represents the Heaviside generalized function as HeavisideTheta, while using UnitStep to represent the piecewise function Piecewise[1,..
The rectangle function is a function that is 0 outside the interval and unity inside it. It is also called the gate function, pulse function, or window function, and is defined by(1)The left figure above plots the function as defined, while the right figure shows how it would appear if traced on an oscilloscope. The generalized function has height , center , and full-width .As noted by Bracewell (1965, p. 53), "It is almost never important to specify the values at , that is at the points of discontinuity. Likewise, it is not necessary or desirable to emphasize the values in graphs; it is preferable to show graphs which are reminiscent of high-quality oscillograms (which, of course, would never show extra brightening halfway up the discontinuity)."The piecewise version of the rectangle function is implemented in the Wolfram Language as UnitBox[x] (which takes the value 1 at ), while the generalized function version is implemented..
The fractal-like two-dimensional functionThe function is named for the appearance of a butterfly-like pattern centered around the origin (left figure). In the above illustration, the left plot runs from to 5 and the right plot runs from to 20.
There are two kinds of Bell polynomials.A Bell polynomial , also called an exponential polynomial and denoted (Bell 1934, Roman 1984, pp. 63-67) is a polynomial that generalizes the Bell number and complementary Bell number such that(1)(2)These Bell polynomial generalize the exponentialfunction.Bell polynomials should not be confused with Bernoulli polynomials, which are also commonly denoted .Bell polynomials are implemented in the Wolfram Language as BellB[n, x].The first few Bell polynomials are(3)(4)(5)(6)(7)(8)(9)(OEIS A106800). forms the associated Sheffer sequence for(10)so the polynomials have that exponentialgenerating function(11)Additional generating functions for are given by(12)or(13)with , where is a binomial coefficient.The Bell polynomials have the explicit formula(14)where is a Stirling number of the second kind.A beautiful binomial sum is given by(15)where is a binomial coefficient.The..
For , the Riemann zeta function is given by(1)(2)where is the th prime. This is Euler's product (Whittaker and Watson 1990), called by Havil (2003, p. 61) the "all-important formula" and by Derbyshire (2004, pp. 104-106) the "golden key."This can be proved by expanding the product, writing each term as a geometricseries, expanding, multiplying, and rearranging terms,(3)Here, the rearrangement leading to equation (1) follows from the fundamental theorem of arithmetic, since each product of prime powers appears in exactly one denominator and each positive integer equals exactly one product of prime powers.This product is related to the Möbius function via(4)which can be seen by expanding the product to obtain(5)(6)(7)(8)(9), but the finite product exists, giving(10)For upper limits , 1, 2, ..., the products are given by 1, 2, 3, 15/4, 35/8, 77/16, 1001/192, 17017/3072, ... (OEIS A060753 and..
Landau (1911) proved that for any fixed ,as , where the sum runs over the nontrivial Riemann zeta function zeros and is the Mangoldt function. Here, "fixed " means that the constant implicit in depends on and, in particular, as approaches a prime or a prime power, the constant becomes large.Landau's formula is roughly the derivative of the explicitformula.Landau's formula is quite extraordinary. If is not a prime or a prime power, then and the sum grows as a constant times . But if is a prime or a prime power, then and the sum grows much faster, like a constant times . This exhibits an amazing connection between the primes and the s; somehow the zeros "recognize" when is a prime and cause large contributions to the sum.
There are two types of singular values, one in the context of elliptic integrals, and the other in linear algebra.For a square matrix , the square roots of the eigenvalues of , where is the conjugate transpose, are called singular values (Marcus and Minc 1992, p. 69). The so-called singular value decomposition of a complex matrix is given by(1)where and are unitary matrices and is a diagonal matrix whose elements are the singular values of (Golub and Van Loan 1996, pp. 70 and 73). Singular values are returned by the command SingularValueList[m].If(2)where is a unitary matrix and is a Hermitian matrix, then the eigenvalues of are the singular values of .For elliptic integrals, a elliptic modulus such that(3)where is a complete elliptic integral of the first kind, and . The elliptic lambda function gives the value of . Abel (quoted in Whittaker and Watson 1990, p. 525) proved that if is an integer, or more generally whenever(4)where..
The asymptotic series for the gammafunction is given by(1)(OEIS A001163 and A001164).The coefficient of can given explicitly by(2)where is the number of permutations of with permutation cycles all of which are (Comtet 1974, p. 267). Another formula for the s is given by the recurrence relation(3)with , then(4)where is the double factorial (Borwein and Corless 1999).The series for is obtained by adding an additional factor of ,(5)(6)The expansion of is what is usually called Stirling's series. It is given by the simple analytic expression(7)(8)(OEIS A046968 and A046969), where is a Bernoulli number. Interestingly, while the numerators in this expansion are the same as those of for the first several hundred terms, they differ at , 1185, 1240, 1269, 1376, 1906, 1910, ... (OEIS A090495), with the corresponding ratios being 37, 103, 37, 59, 131, 37, 67, ... (OEIS A090496)...
The th subfactorial (also called the derangement number; Goulden and Jackson 1983, p. 48; Graham et al. 2003, p. 1050) is the number of permutations of objects in which no object appears in its natural place (i.e., "derangements").The term "subfactorial "was introduced by Whitworth (1867 or 1878; Cajori 1993, p. 77). Euler (1809) calculated the first ten terms.The first few values of for , 2, ... are 0, 1, 2, 9, 44, 265, 1854, 14833, ... (OEIS A000166). For example, the only derangements of are and , so . Similarly, the derangements of are , , , , , , , , and , so .Sums and formulas for include(1)(2)(3)(4)where is a factorial, is a binomial coefficient, and is the incomplete gamma function.Subfactorials are implemented in the WolframLanguage as Subfactorial[n].A plot the real and imaginary parts of the subfactorial generalized to any real argument is illustrated above, with the usual integer-valued subfactorial..
A harmonic number is a number of the form(1)arising from truncation of the harmonic series.A harmonic number can be expressed analytically as(2)where is the Euler-Mascheroni constant and is the digamma function.The first few harmonic numbers are 1, , , , , ... (OEIS A001008 and A002805). The numbers of digits in the numerator of for , 1, ... are 1, 4, 41, 434, 4346, 43451, 434111, 4342303, 43428680, ... (OEIS A114467), with the corresponding number of digits in the denominator given by 1, 4, 40, 433, 4345, 43450, 434110, 4342302, 43428678, ... (OEIS A114468). These digits converge to what appears to be the decimal digits of (OEIS A002285).The first few indices such that the numerator of is prime are given by 2, 3, 5, 8, 9, 21, 26, 41, 56, 62, 69, ... (OEIS A056903). The search for prime numerators has been completed up to by E. W. Weisstein (May 13, 2009), and the following table summarizes the largest known values.decimal digitsdiscoverer6394227795E. W. Weisstein..
Exponentiation is the process of taking a quantity (the base) to the power of another quantity (the exponent). This operation most commonly denoted . In TeX, the Wolfram Language, and many other computer languages, exponentiation is denoted with a caret, i.e., as b^e. However, in FORTRAN, it is denoted b**e (Calderbank 1989, p. 29).
Polynomials which form the Sheffer sequence for(1)(2)and have generating function(3)The first few are(4)(5)(6)(7)The Pidduck polynomials are related to the Mittag-Leffler polynomials by(8)(Roman 1984, p. 127).
Polynomials which are a generalization of the Boole polynomials, form the Sheffer sequence for(1)(2)and have generating function(3)The first few are(4)(5)and(6)
Let be a set of positive numbers. Then(which is given incorrectly in Gradshteyn and Ryzhik 2000). Here, the constant e is the best possible, in the sense that counterexamples can be constructed for any stricter inequality which uses a smaller constant. The theorem is suggested by writing in Hardy's inequalityand letting .
The Griewank function is a function widely used to test the convergence of optimization functions. The Griewank function of order is defined byfor (Griewank 1981), plotted above for . It has a global minimum of 0 at the point .The function has 191 minima, with global minimum at and local minima at for (OEIS A177889), 12.5601, 18.8401, 25.1202, .... Restricting the domain of the function to , the numbers of local minima for for , 2, ... are therefore given by 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 7, ... (OEIS A178832).
Let any finite or infinite set of points having no finite limit point be prescribed, and associate with each of its points a definite positive integer as its order. Then there exists an entire function which has zeros to the prescribed orders at precisely the prescribed points, and is otherwise different from zero. Moreover, this function can be represented as a product from which one can read off again the positions and orders of the zeros. Furthermore, if is one such function, thenis the most general function satisfying the conditions of the problem, where denotes an arbitrary entire function.This theorem is also sometimes simply known as Weierstrass's theorem. A spectacularexample is given by the Hadamard product.
There are (at least) two mathematical objects known as Weierstrass forms. The first is a general form into which an elliptic curve over any field can be transformed, given bywhere , , , , and are elements of .The second is the definition of the gamma functionaswhere is the Euler-Mascheroni constant (Krantz 1999, p. 157).
A symmetric polynomial on variables , ..., (also called a totally symmetric polynomial) is a function that is unchanged by any permutation of its variables. In other words, the symmetric polynomials satisfy(1)where and being an arbitrary permutation of the indices 1, 2, ..., .For fixed , the set of all symmetric polynomials in variables forms an algebra of dimension . The coefficients of a univariate polynomial of degree are algebraically independent symmetric polynomials in the roots of , and thus form a basis for the set of all such symmetric polynomials.There are four common homogeneous bases for the symmetric polynomials, each of which is indexed by a partition (Dumitriu et al. 2004). Letting be the length of , the elementary functions , complete homogeneous functions , and power-sum functions are defined for by(2)(3)(4)and for by(5)where is one of , or . In addition, the monomial functions are defined as(6)where is the set of permutations..
Orthogonal polynomials are classes of polynomials defined over a range that obey an orthogonality relation(1)where is a weighting function and is the Kronecker delta. If , then the polynomials are not only orthogonal, but orthonormal.Orthogonal polynomials have very useful properties in the solution of mathematical and physical problems. Just as Fourier series provide a convenient method of expanding a periodic function in a series of linearly independent terms, orthogonal polynomials provide a natural way to solve, expand, and interpret solutions to many types of important differential equations. Orthogonal polynomials are especially easy to generate using Gram-Schmidt orthonormalization.A table of common orthogonal polynomials is given below, where is the weighting function and(2)(Abramowitz and Stegun 1972, pp. 774-775).polynomialintervalChebyshev polynomial of the first kindChebyshev polynomial of the..