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
(
z
1
,
z
2
)
=
∫
0
1
t
z
1
−
1
(
1
−
t
)
z
2
−
1
d
t
{\displaystyle \mathrm {B} (z_{1},z_{2})=\int _{0}^{1}t^{z_{1}-1}(1-t)^{z_{2}-1}\,dt}
for complex number inputs
z
1
,
z
2
{\displaystyle z_{1},z_{2}}
such that
Re
(
z
1
)
,
Re
(
z
2
)
>
0
{\displaystyle \operatorname {Re} (z_{1}),\operatorname {Re} (z_{2})>0}
.
The beta function was studied by Leonhard Euler and Adrien-Marie Legendre and was given its name by Jacques Binet; its symbol Β is a Greek capital beta.
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