gamma( z )

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

gamma(z)=Γ(z)=0tz1et dt
gamma(z,x)=Γ(z,x)=xtz1et dt
gamma(z,0,x)=γ(z,0,x)=x0tz1et dt=Γ(z)Γ(z,x)
gamma(z,a,x)=γ(z,a,x)=xatz1et dt=γ(z,0,x)γ(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 :

-4 -3 -2 -1 1 2 3 4 5 -5.0 -2.5 2.5 5.0

Imaginary part on the real axis is zero.

Real part on the imaginary axis:

-5.0 -2.5 2.5 5.0 -0.5 -0.4 -0.3 -0.2 -0.1

Imaginary part on the imaginary axis:

-5.0 -2.5 2.5 5.0 -1.0 -0.5 0.5 1.0

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