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 Documentation / Mathematica / Built-in Functions / Mathematical Functions / Hypergeometric Related  /
Gamma

  • Gamma[ z ] is the Euler gamma function .
  • Gamma[ a , z ] is the incomplete gamma function .
  • Gamma[ a , , ] is the generalized incomplete gamma function .
  • Mathematical function (see Section A.3.10).
  • The gamma function satisfies .
  • The incomplete gamma function satisfies .
  • The generalized incomplete gamma function is given by the integral .
  • Note that the arguments in the incomplete form of Gamma are arranged differently from those in the incomplete form of Beta.
  • Gamma[ z ] has no branch cut discontinuities.
  • Gamma[ a , z ] has a branch cut discontinuity in the complex z plane running from to .
  • See the Mathematica book: Section 3.2.10.
  • See also Implementation NotesA.9.44.15MainBookLinkOldButtonDataA.9.44.15.
  • See also: Factorial, LogGamma, InverseGammaRegularized, PolyGamma, RiemannSiegelTheta.

    Further Examples

    Here are the first 10 integer values of the gamma function; these are the factorials offset by 1.

    In[1]:=

    Out[1]=

    The gamma function generalizes factorials to numbers beyond the positive integers.

    In[2]:=

    Out[2]=

    This is the derivative.

    In[3]:=

    Out[3]=

    This is a series expansion around .

    In[4]:=

    Out[4]=