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} where z 1 , z 2 {\displaystyle z_{1},z_{2}} are complex numbers such that ℜ ( z 1 ) > 0 {\displaystyle \Re (z_{1})>0} and ℜ ( z 2 ) > 0 {\displaystyle \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 B {\displaystyle \mathrm {B} } is a Greek capital beta.