Equivalence of Definitions of Beta Function

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Theorem

The following definitions of the concept of Beta Function are equivalent:

For $\map \Re x, \map \Re y > 0$:

Definition 1

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

Definition 2

$\displaystyle \Beta \left({x, y}\right) := 2 \int_0^{\pi / 2} \left({\sin \theta}\right)^{2 x - 1} \left({\cos \theta}\right)^{2 y - 1} \rd \theta$

Definition 3

$\map \Beta {x, y} := \dfrac {\map \Gamma x \, \map \Gamma y} {\map \Gamma {x + y} }$

where $\Gamma$ is the Gamma function.


Proof

Definition 1 is equivalent to Definition 2

\(\displaystyle \map \Beta {x, y}\) \(=\) \(\displaystyle \int_0^1 t^{x - 1} \paren {1 - t}^{y - 1} \rd t\) Definition 1 of Beta Function
\(\displaystyle \) \(=\) \(\displaystyle \int_0^{\pi / 2} \paren {\sin \theta}^{2 x - 2} \paren {\cos \theta}^{2 y - 2} 2 \sin \theta \cos \theta \rd \theta\) Substitute $t = \sin^2 \theta$
\(\displaystyle \) \(=\) \(\displaystyle 2 \int_0^{\pi / 2} \paren {\sin \theta}^{2 x - 1} \paren {\cos \theta}^{2 y - 1} \rd \theta\) Definition 2 of Beta Function

$\Box$


Definition 2 is equivalent to Definition 3

By definition of Gamma function:

$\displaystyle \map \Gamma x \, \map \Gamma y = \int_0^\infty t^{x - 1} e^{-t} \rd t \int_0^\infty s^{y - 1} e^{-s} \rd s$


Substitute $t = u^2$ and $s = v^2$:

\(\displaystyle \map \Gamma x \, \map \Gamma y\) \(=\) \(\displaystyle 4 \int_0^\infty u^{2 x - 1} e^{-u^2} \rd u \int_0^\infty v^{2 y - 1} e^{-v^2} \rd v\)
\(\displaystyle \) \(=\) \(\displaystyle 4 \int_0^\infty \int_0^\infty u^{2 x - 1} v^{2 y - 1} e^{-\paren {u^2 + v^2} } \rd u \rd v\) Fubini's Theorem
\(\displaystyle \) \(=\) \(\displaystyle 4 \int_0^\infty \int_0^{\pi / 2} r^{2 x - 1} \paren {\sin \theta}^{2 x - 1} r^{2 y - 1} \paren {\cos \theta}^{2 y - 1} e^{-r^2} r \rd \theta \rd r\) Substitute $v = r \cos \theta$ and $u = r \sin \theta$
\(\displaystyle \) \(=\) \(\displaystyle 4 \int_0^\infty r^{2 x + 2 y - 2} e^{-r^2} r \rd r \int_0^{\pi / 2} \paren {\sin \theta}^{2 x - 1} \paren {\cos \theta}^{2 y - 1} \rd \theta\) Fubini's Theorem
\(\displaystyle \) \(=\) \(\displaystyle 2 \int_0^\infty u^{x + y - 1} e^{-u} \rd u \int_0^{\pi / 2} \paren {\sin \theta}^{2 x - 1} \paren {\cos \theta}^{2 y - 1} \rd \theta\) Substitute $u = r^2$
\(\displaystyle \) \(=\) \(\displaystyle 2 \, \map \Gamma {x + y} \int_0^{\pi / 2} \paren {\sin \theta}^{2 x - 1} \paren {\cos \theta}^{2 y - 1} \rd \theta\) Definition of Gamma Function
\(\displaystyle \leadsto \ \ \) \(\displaystyle \) \(\) \(\displaystyle \frac {\map \Gamma x \, \map \Gamma y} {\map \Gamma {x + y} }\) Definition 3 of Beta Function
\(\displaystyle \) \(=\) \(\displaystyle 2 \int_0^{\pi / 2} \paren {\sin \theta}^{2 x - 1} \paren {\cos \theta}^{2 y - 1} \rd \theta\) Definition 2 of Beta Function

$\blacksquare$


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