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euler_beta_function [2015/01/13 13:17] nikolaj |
euler_beta_function [2015/11/13 17:07] nikolaj |
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| ===== Euler beta function ===== | ===== Euler beta function ===== | ||
| ==== Function ==== | ==== Function ==== | ||
| - | | @#FFBB00: definiendum | @#FFBB00: $ {\mathrm B}: \{z\ |\ \mathfrak{R}(z) > 0 \}^2 \to \mathbb C$ | | + | | @#FF9944: definition | @#FF9944: $ {\mathrm B}: \{z\ |\ \mathfrak{R}(z) > 0 \}^2 \to \mathbb C$ | |
| - | | @#FFBB00: definiendum | @#FFBB00: $ {\mathrm B}(p,q) := \int_0^1 \tau^{p-1}(1-\tau)^{q-1}\,\mathrm d\tau $ | | + | | @#FF9944: definition | @#FF9944: $ {\mathrm B}(p,q) := \int_0^1 \tau^{p-1}(1-\tau)^{q-1}\,\mathrm d\tau $ | |
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| For natural numbers | For natural numbers | ||
| - | * $\dfrac{1}{{\mathrm B}(x,y)} = \frac{x\,y}{x+y} \prod_{n=1}^\infty \left( 1 + \dfrac{x\,y}{n\,(x+y+n)}\right)$ | ||
| * ${\large{n \choose k}}=(n+1)\cdot\dfrac{1}{{\mathrm B}(n-k+1,k+1)}$ | * ${\large{n \choose k}}=(n+1)\cdot\dfrac{1}{{\mathrm B}(n-k+1,k+1)}$ | ||
| + | |||
| + | * $\dfrac{1}{{\mathrm B}(x,y)} = \frac{x\,y}{x+y} \prod_{n=1}^\infty \left( 1 + \dfrac{x\,y}{n\,(x+y+n)}\right)$ | ||
| === Reference === | === Reference === | ||
