In mathematics, the reciprocal gamma function is the function
where denotes the gamma function. Since the gamma function is meromorphic and nonzero everywhere in the complex plane, its reciprocal is an entire function. As an entire function, it is of order 1 (meaning that grows no faster than ), but of infinite type (meaning that grows faster than any multiple of , since its growth is approximately proportional to in the left-half plane).
The reciprocal is sometimes used as a starting point for numerical computation of the gamma function, and a few software libraries provide it separately from the regular gamma function.
Karl Weierstrass called the reciprocal gamma function the "factorielle" and used it in his development of the Weierstrass factorization theorem.
Infinite product expansion Following from the infinite product definitions for the gamma function, due to Euler and Weierstrass respectively, we get the following infinite product expansion for the reciprocal gamma function:
where is the Euler–Mascheroni constant. These expansions are valid for all complex numbers .
Taylor series Taylor series expansion around gives:
where is the Euler–Mascheroni constant. For , the coefficient for the term can be computed recursively as and
where is the Riemann zeta function. An integral representation for these coefficients was recently found by Fekih-Ahmed (2014):
For small values, these give the following values: 1 +1.0000000000000000000000000000000000000000 2 +0.5772156649015328606065120900824024310422 3 −0.6558780715202538810770195151453904812798 4 −0.0420026350340952355290039348754298187114 5 +0.1665386113822914895017007951021052357178 6 −0.0421977345555443367482083012891873913017 7 −0.0096219715278769735621149216723481989754 8 +0.0072189432466630995423950103404465727099 9 −0.0011651675918590651121139710840183886668 10 −0.0002152416741149509728157299630536478065 11 +0.0001280502823881161861531986263281643234 12 −0.0000201348547807882386556893914210218184 13 −0.0000012504934821426706573453594738330922 14 +0.0000011330272319816958823741296203307449 15 −0.0000002056338416977607103450154130020573 16 +0.0000000061160951044814158178624986828553 17 +0.0000000050020076444692229300556650480600 18 −0.0000000011812745704870201445881265654365 19 +0.0000000001043426711691100510491540332312 20 +0.0000000000077822634399050712540499373114 21 −0.0000000000036968056186422057081878158781 22 +0.0000000000005100370287454475979015481323 23 −0.0000000000000205832605356650678322242954 24 −0.0000000000000053481225394230179823700173 25 +0.0000000000000012267786282382607901588938 26 −0.0000000000000001181259301697458769513765 27 +0.0000000000000000011866922547516003325798 28 +0.0000000000000000014123806553180317815558 29 −0.0000000000000000002298745684435370206592 30 +0.0000000000000000000171440632192733743338
Fekih-Ahmed (2014) also gives an approximation for :
where , and is the negative-first branch of the Lambert function.
The Taylor expansion around has the same (but shifted) coefficients, i.e.:
(the reciprocal of Gauss's pi function).
Asymptotic expansion As goes to infinity at a constant we have:
Contour integral representation An integral representation due to Hermann Hankel is
where is the Hankel contour, that is, the path encircling 0 in the positive direction, beginning at and returning to positive infinity with respect for the branch cut along the positive real axis. According to Schmelzer & Trefethen,; ; numerical evaluation of Hankel's integral is the basis of some of the best methods for computing the gamma function.
Integral representations at the positive integers
For positive integers , there is an integral for the reciprocal factorial function given by
Similarly, for any real and such that we have the next integral for the reciprocal gamma function along the real axis in the form of:
where the particular case when provides a corresponding relation for the reciprocal double factorial function,
Integral along the real axis
Integration of the reciprocal gamma function along the positive real axis gives the value
which is known as the Fransén–Robinson constant.
We have the following formula
See also Bessel–Clifford function Inverse-gamma distribution Nu function
References
Mette Lund, An integral for the reciprocal Gamma function Milton Abramowitz & Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables Eric W. Weisstein, Gamma Function, MathWorld