gamma( z )

The gamma function of z in Math, is special function. Defined by

\[ \Gamma(z) = \int_0^\infty \; t^{z-1} e^{-t} \ dt \] \[ \Gamma(z,x) = \int_x ^\infty \; t^{z-1} e^{-t} \ dt \] \[ \gamma(z,a,x) = \int_a ^x \; t^{z-1} e^{-t} \ dt = \gamma(z,0,x) - \gamma(z,0,a) \] gamma(n) = (n-1)! when n is positive integer.

gamma(n,x) - upper incomplete gamma function \[ gamma(n,x) = Gamma(n,x) = \Gamma(z,x) \]
gamma(1,x) = exp(-x)

gamma(n,0,x) - lower incomplete gamma function
gamma(n,0,x) = gamma(n) - gamma(n,x)
gamma(1,0,x) = 1-exp(-x)

gamma( x, y, z ) — generalized incomplete gamma function
gamma(x,y,z) = gamma(x,0,z) - gamma(x,0,y)

Real part on the real axis:

gamma(x,y)

Imaginary part on the real axis is zero.

Real part on the imaginary axis:

Imaginary part on the imaginary axis:

Real part on the complex plane:

Imaginary part on the complex plane:

Absolute value on the complex plane:

Reference

  1. Related functions:   logGamma   beta
  2. Function category: gamma functions
  3. special function