The Poisson integral with ,where is a Bessel function of the first kind.
Gamma functions of argument can be expressed in terms of gamma functions of smaller arguments. From the definition of the beta function,(1)Now, let , then(2)and , so and(3)(4)(5)(6)Now, use the beta function identity(7)to write the above as(8)Solving for and using then gives(9)(10)
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)
An approximation for the gamma function with is given by(1)where is an arbitrary constant such that ,(2)where is a Pochhammer symbol and(3)and(4)(5)with (Lanczos 1964; Luke 1969, p. 30). satisfies(6)and if is a positive integer, then satisfies the identity(7)(Luke 1969, p. 30).A similar result is given by(8)(9)(10)where is a Pochhammer symbol, is a factorial, and(11)The first few values of are(12)(13)(14)(15)(16)(OEIS A054379 and A054380; Whittaker and Watson 1990, p. 253). Note that Whittaker and Watson incorrectly give as 227/60.Yet another related result gives(17)(Whittaker and Watson 1990, p. 261), where is a Hurwitz zeta function and is a polygamma function.
The symbol defined by(1)(2)(3)where is the Pochhammer symbol and is the gamma function. Note that the definition by Erdélyi et al. (1981, p. 52) incorrectly gives the prefactor of (3) as .
where is a binomial coefficient and is a gamma function.
Thomae's theorem, also called Thomae's transformation, is the generalizedhypergeometric function identity(1)where is the gamma function, is a generalized hypergeometric function,(2)and (Bailey 1935, p. 14). It is a generalization of Dixon's theorem (Slater 1966, p. 52).An equivalent formulation is given by(3)(Hardy 1999, p. 104). The symmetry of this form was used by Ramanujan in his proof of the identity, which is essentially the same as Thomae's. Interestingly, this is one of the few cases in which Ramanujan gives an explicit proof of one of his propositions (Hardy 1999, p. 104).A special case of the theorem is given by(4)(J. Sondow, pers. comm., May 25, 2003).
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..
The "complete" gamma function can be generalized to the incomplete gamma function such that . This "upper" incomplete gamma function is given by(1)For an integer (2)(3)where is the exponential sum function. It is implemented as Gamma[a, z] in the Wolfram Language.The special case of can be expressed in terms of the subfactorial as(4)The incomplete gamma function has continued fraction(5)(Wall 1948, p. 358).The lower incomplete gamma function is given by(6)(7)(8)where is the confluent hypergeometric function of the first kind. For an integer ,(9)(10)It is implemented as Gamma[a,0, z] in the Wolfram Language.By definition, the lower and upper incomplete gamma functions satisfy(11)The exponential integral is closely related to the incomplete gamma function by(12)Therefore, for real ,(13)..
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)
The regularized gamma functions are defined by(1)(2)where and are incomplete gamma functions and is a complete gamma function. The function is implemented in the Wolfram Language as GammaRegularized[a, 0, z], and is implemented as GammaRegularized[a, z]. and satisfy the identity(3)The derivatives of and are(4)(5)and the second derivatives are(6)(7)The integrals are(8)(9)
A generalization of the complete beta function definedby(1)sometimes also denoted . The so-called Chebyshev integral is given by(2)The incomplete beta function is implemented in the Wolfram Language as Beta[z, a, b].It is given in terms of hypergeometric functionsby(3)(4)It is also given by the series(5)where is a Pochhammer symbol.The incomplete beta function reduces to the usual beta function when ,(6)It has derivative(7)and indefinite integral(8)
For ,where is the gamma function.
The regularized beta function is defined bywhere is the incomplete beta function and is the (complete) beta function. The regularized beta function is sometimes also denoted and is implemented in the Wolfram Language as BetaRegularized[z, a, b]. The four-argument version BetaRegularized[z1, z2, a, b] is equivalent to .
The symbol defined by(1)(2)where is the gamma function. If is an integer, then this simplifies to(3)given incorrectly by Erdélyi et al. (1981, p. 52).
Letwhere is a Pochhammer symbol, and let be a negative integer. Thenwhere is the gamma function.
A special function mostly commonly denoted , , or which is given by the st derivative of the logarithm of the gamma function (or, depending on the definition, of the factorial ). This is equivalent to the th normal derivative of the logarithmic derivative of (or ) and, in the former case, to the th normal derivative of the digamma function . Because of this ambiguity in definition, two different notations are sometimes (but not always) used, namely(1)(2)(3)which, for can be written as(4)(5)where is the Hurwitz zeta function.The alternate notation(6)is sometimes used, with the two notations connected by(7)Unfortunately, Morse and Feshbach (1953) adopt a notation no longer in standard use in which Morse and Feshbach's "" is equal to in the usual notation. Also note that the function is equivalent to the digamma function and is sometimes known as the trigamma function. is implemented in the Wolfram Language as PolyGamma[n, z] for positive..
Binet's first formula for , where is a gamma function, is given byfor (Erdélyi et al. 1981, p. 21; Whittaker and Watson 1990, p. 249).Binet's second formula isfor (Erdélyi et al. 1981, p. 22; Whittaker and Watson 1990, pp. 250-251).
for , where is a (Gauss) hypergeometric function. If is a negative integer , this becomeswhich is known as the Chu-Vandermonde identity.
(1)(2)where is the gamma function (Erdélyi et al. 1981, p. 217; Prudnikov et al. 1990, p. 799; Gradshteyn and Ryzhik 2000, p. 1109).The notation is also sometimes used to denote the divisor function giving the number of divisors or the number of distinct prime factors of a positive integer .
There are two functions commonly denoted , each of which is defined in terms of integrals. Another unrelated mathematical function represented using the Greek letter is the Möbius function.The two-argument -function is defined by the definite integralwhere is the gamma function (Erdélyi et al. 1981a, p. 388; Prudnikov et al. 1990, p. 798; Gradshteyn and Ryzhik 2000, p. 1109), while the three-argument -function is defined by(Prudnikov et al. 1990, p. 798; Gradshteyn and Ryzhik 2000, p. 1109).
(1)(2)where is a Pochhammer symbol and is the gamma function. This is a special case of the identity(3)
If a contour in the complex plane is curved such that it separates the increasing and decreasing sequences of poles, thenwhere is the gamma function (Bailey 1935, p. 7).Barnes' second lemma states thatprovided that (Bailey 1935, pp. 42-43).
where is the digamma function and is the gamma function.
The integral representation of by(1)(2)where is the log gamma function and is the digamma function.
Let be the gamma function and denote a double factorial, thenWriting the sums explicitly, Bailey's theorem states
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..
For and ,(1)(2)(3)where is the Pochhammer symbol and is the beta function.
The beta function is the name used by Legendre and Whittaker and Watson (1990) for the beta integral (also called the Eulerian integral of the first kind). It is defined by(1)The beta function is implemented in the Wolfram Language as Beta[a, b].To derive the integral representation of the beta function, write the product oftwo factorials as(2)Now, let , , so(3)(4)Transforming to polar coordinates with , (5)(6)(7)(8)The beta function is then defined by(9)(10)Rewriting the arguments then gives the usual form for the beta function,(11)(12)By symmetry,(13)The general trigonometric form is(14)Equation (14) can be transformed to an integral over polynomials by letting ,(15)(16)(17)(18)For any with ,(19)(Krantz 1999, p. 158).To put it in a form which can be used to derive the Legendre duplication formula, let , so and , and(20)(21)To put it in a form which can be used to develop integral representations of the Bessel functions and hypergeometric..
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,..
The Pochhammer symbol(1)(2)(Abramowitz and Stegun 1972, p. 256; Spanier 1987; Koepf 1998, p. 5) for is an unfortunate notation used in the theory of special functions for the rising factorial, also known as the rising factorial power (Graham et al. 1994, p. 48) or ascending Factorial (Boros and Moll 2004, p. 16). The Pochhammer symbol is implemented in the Wolfram Language as Pochhammer[x, n].In combinatorics, the notation (Roman 1984, p. 5), (Comtet 1974, p. 6), or (Graham et al. 1994, p. 48) is used for the rising factorial, while or denotes the falling factorial (Graham et al. 1994, p. 48). Extreme caution is therefore needed in interpreting the notations and .The first few values of for nonnegative integers are(3)(4)(5)(6)(7)(OEIS A054654).In closed form, can be written(8)where is a Stirling number of the first kind.The Pochhammer symbol satisfies(9)the dimidiation formulas(10)(11)and..
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..
Stirling's approximation gives an approximate value for the factorial function or the gamma function for . The approximation can most simply be derived for an integer by approximating the sum over the terms of the factorial with an integral, so that(1)(2)(3)(4)(5)(6)The equation can also be derived using the integral definition of the factorial,(7)Note that the derivative of the logarithm of the integrandcan be written(8)The integrand is sharply peaked with the contribution important only near . Therefore, let where , and write(9)(10)Now,(11)(12)(13)so(14)(15)(16)Taking the exponential of each side thengives(17)(18)Plugging into the integral expression for then gives(19)(20)Evaluating the integral gives(21)(22)(Wells 1986, p. 45). Taking the logarithm of bothsides then gives(23)(24)This is Stirling's series with only the first term retained and, for large , it reduces to Stirling's approximation(25)Taking successive..
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.
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)...
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).