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Graphs of the Beta Function

position

opacity

2D plot

density

contour

axes

Real parts of the special function

Β(x,y)

, the beta function, which was studied by Euler, Legendre, Binet, Whittaker, Watson, and others.

Here are two equivalent definitions of the beta function:

Β(x,y)

Γ(x)Γ(y)

Γ(x+y)

, where

Γ(z)

is Euler's gamma function;

Β(x,y)

∫

y-1

x-1

(1-t)

t,Re(x)>0,Re(y)>0

Daniel de Souza Carvalho

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