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

${\displaystyle \mathrm {B} (z_{1},z_{2})=\int _{0}^{1}t^{z_{1}-1}(1-t)^{z_{2}-1}\,dt}$

for complex number inputs ${\displaystyle z_{1},z_{2}}$ such that ${\displaystyle \Re (z_{1}),\Re (z_{2})>0}$.

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.