===== Euler beta function =====
==== Function ====
| @#FF9944: definition  | @#FF9944: $ {\mathrm B}: \{z\ |\ \mathfrak{R}(z) > 0 \}^2 \to \mathbb C$ |
| @#FF9944: definition  | @#FF9944: $ {\mathrm B}(p,q) := \int_0^1 \tau^{p-1}(1-\tau)^{q-1}\,\mathrm d\tau $ |

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=== Theorems ===

For natural numbers

  * ${\large{n \choose k}}=(n+1)\cdot\dfrac{1}{{\mathrm B}(n-k+1,k+1)}$

  * $\dfrac{1}{{\mathrm B}(x,y)} = \frac{x\,y}{x+y} \prod_{n=1}^\infty \left( 1 + \dfrac{x\,y}{n\,(x+y+n)}\right)$

=== Reference ===
Wikipedia: [[http://en.wikipedia.org/wiki/Beta_function|Beta function]]

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=== Subset of ===
[[ℂ valued function]]
=== Related ===
[[Gamma function]]