Beta Function is Defined for Positive Reals

Theorem

Let $x, y \in \R$ be real numbers.

Let $\map \Beta {x, y}$ be the Beta function:

$\ds \map \Beta {x, y} := \int_{\mathop \to 0}^{\mathop \to 1} t^{x - 1} \paren {1 - t}^{y - 1} \rd t$

Then $\map \Beta {x, y}$ exists provided that $x, y > 0$.

Consider the following inequalities, valid for $0 < t < 1$:

$\ds t^{x - 1} \paren {1 - t}^{y - 1}$

$<$

$\ds t^{x - 1}$

$\ds \leadsto \ \ $

$\ds t^{x - 1} \paren {1 - t}^{y - 1}$

$<$

$\ds \paren {1 - t}^{y - 1}$

Then:

$\ds \int_{\mathop \to 0}^{\mathop \to 1} t^{x - 1} \rd t$

$=$

$\ds \intlimits {\frac {t^x} x} {\mathop \to 0} {\mathop \to 1}$

$\ds $

$=$

$\ds x - 0$

and similarly:

$\ds \int_{\mathop \to 0}^{\mathop \to 1} \paren {1 - t}^{y - 1} \rd t$

$=$

$\ds -\intlimits {\frac {\paren {1 - t}^y} y} {\mathop \to 0} {\mathop \to 1}$

$\ds $

$=$

$\ds y - 0$

$\blacksquare$