﻿ gamma — Math

gamma( z )

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

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

plot : gamma(x)

complex3D : gamma(x)

usage : gamma

handbook : gamma

source : gamma

gamma(n,x) - upper incomplete gamma function
gamma(z,0) = gamma(z) = Gamma(z)
plot : gamma(1,x) = exp(-x)
complex3D : gamma(1,x) = exp(-x)
plot 2D : gamma(n,x) in plot 2D for dy/dx, y' = f(x) and polar graph.
plot 2D : im(gamma(n,x)) imaginary part in plot 2D for dy/dx, y' = f(x) and polar graph.
plot 3D : gamma(n,x) in plot 3D graph.
plot 3D : re(gamma(n,x)) real part in plot 3D graph.
plot 3D : im(gamma(n,x)) imaginary part in plot 3D graph.
plot 3D : gamma(x,y) in plot 3D graph.
plot 3D : re(gamma(x,y)) real part in plot 3D graph.
plot 3D : im(gamma(x,y)) imaginary part in plot 3D graph.

gamma(n,0,x) - lower incomplete gamma function
gamma(n,0,x) = gamma(n) - gamma(n,x)
complex2D : gamma(1,0,x) = 1-exp(-x)
complex3D : gamma(1,0,x) = 1-exp(-x)
plot 2D : gamma(n,0,x) in plot 2D for dy/dx, y' = f(x) and polar graph.
plot 2D : im(gamma(n,0,x)) imaginary part in plot 2D for dy/dx, y' = f(x) and polar graph.
plot 3D : gamma(n,0,x) in plot 3D graph.
plot 3D : re(gamma(n,0,x)) real part in plot 3D graph.
plot 3D : im(gamma(n,0,x)) imaginary part in plot 3D graph.

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

Real part on the real axis :

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