 Beta function

For $\mathrm{Re}p,\mathrm{Re}q>0$, Euler's beta function is

The Beta function is a special function in mathematics that is closely related to the gamma function and is defined for positive real numbers. It is often denoted as \( B(x, y) \) and defined as follows: \[ B(x, y) = \int_0^1 t^{x-1} (1-t)^{y-1} \, dt \] for \( x > 0 \) and \( y > 0 \).