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Gamma

Gamma[z]
is the Euler gamma function Gamma(z).
Gamma[a, z]
is the incomplete gamma function Gamma(a,z).
Gamma[a, z0, z1]
is the generalized incomplete gamma function Gamma(a,z_0)-Gamma(a,z_1).
  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • 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 -Infinity to 0.
  • For certain special arguments, Gamma automatically evaluates to exact values.
  • Gamma can be evaluated to arbitrary numerical precision.
  • Gamma automatically threads over lists.
EXAMPLESCLOSE ALL
Scope   (5)
Evaluate for large arguments:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
Series expansion:
Incomplete gamma function:
Generalizations & Extensions   (11)
Infinite arguments give symbolic results:
Gamma threads element-wise over lists:
Gamma can be applied to a power series:
Series expansion at poles:
Expansion at symbolically specified negative integers:
Series expansion at infinity (Stirling approximation):
Give the result for an arbitrary symbolic direction:
TraditionalForm formatting:
Evaluate symbolically at integer and half-integer orders:
Series expansion at a generic point:
Series expansion at infinity:
Evaluate symbolically at integer and half-integer orders:
Series expansion at a generic point:
Applications   (5)
Plot of the absolute value of Gamma in the complex plane:
Find the asymptotic expansion of ratios of gamma functions:
Volume of an n-dimensional unit hypersphere:
Low-dimensional cases:
Plot the volume of the unit hypersphere as a function of dimension:
Plot of the real part of the incomplete gamma function over the parameter plane:
CDF of the chi^2-distribution:
Calculate the PDF:
Plot the CDF for different numbers of degrees of freedom:
Properties & Relations   (6)
Use FullSimplify to simplify gamma functions:
Numerically find a root of a transcendental equation:
Sum expressions involving Gamma:
Generate from integrals, products, and limits:
Obtain Gamma as the solution of a differential equation:
Integrals:
Possible Issues   (2)
Large arguments can give results too large to be computed explicitly:
Machine-number inputs can give high-precision results:
Neat Examples   (2)
Nest Gamma over the complex plane:
Fractal from iterating Gamma:
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