page_title: Gamma function
section_title: General
hierarchical_section_title: Gamma function / Properties / General
caption_attribution_description: English: Derivative of the function &#x0393; ( z ) {\displaystyle \Gamma (z)}
context_page_description: In mathematics, the gamma function is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For any positive integer
  Derived by Daniel Bernoulli, for complex numbers with a positive real part the gamma function is defined via a convergent improper integral:
  The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except the non-positive integers, where the function has simple poles.
  The gamma function has no zeroes, so the reciprocal gamma function is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential function:
  Other extensions of the factorial function do exist, but the gamma function is the most popular and useful. It is a component in various probability-distribution functions, and as such it is applicable in the fields of probability and statistics, as well as combinatorics.
context_section_description: Other important functional equations for the gamma function are Euler's reflection formula
  which implies
  and the Legendre duplication formula
  The duplication formula is a special case of the multiplication theorem (See, Eq. 5.5.6)
  A simple but useful property, which can be seen from the limit definition, is:
  In particular, with z = a + bi, this product is
  If the real part is an integer or a half-integer, this can be finitely expressed in closed form:
  Perhaps the best-known value of the gamma function at a non-integer argument is
  which can be found by setting  in the reflection or duplication formulas, by using the relation to the beta function given below with , or simply by making the substitution  in the integral definition of the gamma function, resulting in a Gaussian integral. In general, for non-negative integer values of  we have:
  where  denotes the double factorial of n and, when , . See Particular values of the gamma function for calculated values.
  It might be tempting to generalize the result that  by looking for a formula for other individual values  where  is rational, especially because according to Gauss's digamma theorem, it is possible to do so for the closely related digamma function at every rational value. However, these numbers  are not known to be expressible by themselves in terms of elementary functions. It has been proved that  is a transcendental number and algebraically independent of  for any integer  and each of the fractions . In general, when computing values of the gamma function, we must settle for numerical approximations.
  Another useful limit for asymptotic approximations is:
  The derivatives of the gamma function are described in terms of the polygamma function. For example:
  For a positive integer m the derivative of the gamma function can be calculated as follows (here  is the Euler–Mascheroni constant):
  For  the th derivative of the gamma function is:
  (This can be derived by differentiating the integral form of the gamma function with respect to , and using the technique of differentiation under the integral sign.)
  Using the identity
  where  is the Riemann zeta function, and  is a partition of  given by
  we have in particular
caption: Derivative of the function Γ(z)
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