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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.
as the only positive function f , with domain on the interval x > 0, that simultaneously has the following three properties: f (1) = 1, and f (x + 1) = x f (x) for x > 0 and f is logarithmically convex. A treatment of this theorem is in Artin's book The Gamma Function, [4] which has been reprinted by the AMS in a collection of Artin's writings.
The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic except at zero and the negative integers, where it has simple poles. The gamma function has no zeros, so the reciprocal gamma function 1 / Γ(z) is an entire function.
Inverse gamma distribution is a special case of type 5 Pearson distribution; A multivariate generalization of the inverse-gamma distribution is the inverse-Wishart distribution. For the distribution of a sum of independent inverted Gamma variables see Witkovsky (2001)
Interpolated approximations and bounds are all of the form ~ () + (~ ()) where ~ is an interpolating function running monotonially from 0 at low α to 1 at high α, approximating an ideal, or exact, interpolator (): = () () For the simplest interpolating function considered, a first-order rational function ~ = + the tightest lower bound has ...
Thus computing the gamma function becomes a matter of evaluating only a small number of elementary functions and multiplying by stored constants. The Lanczos approximation was popularized by Numerical Recipes , according to which computing the gamma function becomes "not much more difficult than other built-in functions that we take for granted ...
One of the advantage of defining this type incomplete-version of Bessel function (,) is that even for example the associated Anger–Weber function defined in Digital Library of Mathematical Functions [6] can related:
A detailed analysis of this spectrum is typically used to determine the identity and quantity of gamma emitters present in a gamma source, and is a vital tool in radiometric assay. The gamma spectrum is characteristic of the gamma-emitting nuclides contained in the source, just like in an optical spectrometer , the optical spectrum is ...