Beta function

Remarks

Example: $B(\frac12, \frac12) = \int_0^1 \frac{dt}{\sqrt{t(1-t)}}$. The substitution $t = \sin^2 \theta$ gives $dt = 2 \sin\theta \cos\theta\, d\theta$ and $\sqrt{t(1-t)} = \sin\theta\cos\theta$, so $B(\frac12, \frac12) = \int_0^{\pi/2} 2\, d\theta = \pi$.

Non-example of a valid argument: for $x = 0$ and $y = 1$ the integral is $\int_0^1 t^{-1}\, dt = \infty$, so $(0, 1)$ is outside the domain.

The beta function is symmetric, $B(x, y) = B(y, x)$, and it satisfies $B(x, y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x + y)}$, where $\Gamma$ is the gamma function. For positive integers $m, n$ this gives $B(m, n) = \frac{(m-1)!\,(n-1)!}{(m + n - 1)!}$. The same integral defines $B(x, y)$ for complex $x, y$ with positive real parts.