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Computing Beta Function for Polya's Urn Scheme

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#beta-function #polya-urn-model
404
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Prove that $\int_0^{\pi/2}\sqrt{\cos(\theta)}d\theta=\frac{(2.\pi)^{3/2}}{16.[\Gamma(5/4)]^2}$

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#gamma-function #beta-function
372
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$\int_{0}^{1} \sin x \cdot \ln x dx$

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#integration #beta-function
211
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Another bizarre sum involving a binomial coefficient and inverse powers of integers.

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#summation #harmonic-numbers #beta-function #polygamma
46
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Show that $\beta_{x,y}:(0,1) \to \mathbb{C}, \beta_{x,y}(t)=(1-t)^{x-1}t^{y-1}$ is integrable

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#integration #beta-function
78
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$B(x,y)\Gamma(x+y)=\Gamma(x) \Gamma(y)$ for all $x,y \in \mathbb{C}_{Re>0}$

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#gamma-function #beta-function
349
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Simplifying to a expression with a double factorial

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#factorial #gamma-function #beta-function
382
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Computing $\int_{-1}^1(1-x^2)^n\ dx$

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#definite-integrals #beta-function
34
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Calculate some function involves complex arguments by the results of real arguments

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#complex-analysis #gamma-function #beta-function
125
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Evaluate the Beta(3,1) and Beta(91,11) probabilities

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#probability #probability-distributions #beta-function
72
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How to evaluate this integral in terms of a beta function?

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#integration #beta-function
287
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Using prior conjugates to find posterior distribution

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#probability #probability-distributions #density-function #beta-function
42
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The special function $\beta(1/2,u,1-u).$

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#ordinary-differential-equations #hypergeometric-function #beta-function #sturm-liouville
111
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Euler's Beta function for positive integers derivation

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#calculus #integration #definite-integrals #beta-function
375
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Evaluating B(3/4,5/4) without the Legendre duplication formula?

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#alternative-proof #gamma-function #beta-function

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