Beta function

In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral





B

(
x
,
y
)
=

∫

0


1



t

x
−
1


(
1
−
t

)

y
−
1



d
t


{\displaystyle \mathrm {B} (x,y)=\int _{0}^{1}t^{x-1}(1-t)^{y-1}\,dt}
for complex number inputs x, y such that Re x > 0, Re y > 0.
The beta function was studied by Euler and Legendre and was given its name by Jacques Binet; its symbol Β is a Greek capital beta.

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