Questions tagged [beta-function]

Question 1

Since I have to teach a course involving some statistical hypothesis tests, while not being a statistician myself, I am looking for some simple, but possibly accurate, bounds for cumulative ...

Question 2

Solving a problem I found several integrals that look like $$I(z) = \int \frac{f(x,y)}{(x y)^{1 + z}} dx dy$$ where the integral is over the triangle $\left\{(x,y) | x>0, y>0, x+y<1\right\}$. ...

Question 3

Using the integral, $$\frac{1}{\binom{n}{r}} = (n+1)\int_0^1 x^r(1-x)^{n-r} \mathrm dx,$$ I want to solve some binomial questions. For instance, it seems, I might be unclear as to what I require. I am ...

Question 4

Where can I find the derivation of this formula? https://functions.wolfram.com/GammaBetaErf/Beta/06/03/0001/ $$B(a, b) = \sum^\infty_{k=0}\frac{(1 - b)_k}{(a + k) k!} ;\quad \Re(a) > 0 \;\text{ and ...

Question 5

Considering the incomplete beta function $$B_x(a, b) = \int_0^x y^{a-1}(1-y)^{b-1} \:dy,$$ I am interested in deriving the following asymptotic expansion that I determined using Mathematica $$B_{1-x}(...

Question 6

Working on a problem I came across this series, \begin{equation} \frac{1}{2}+(\frac{n-1}{n})\sum_{k=1}^{\infty}\frac{x^{-\frac{k(n-1)}{n}}}{k(k+1)B\left(k,-\frac{k}{n}\right)} \end{equation} Removing ...

Question 7

I am dealing with integrals in the form $\mathcal{I} = \int_0^{1} dx \, x^\alpha \, (1-x)^\beta \, log(x)$ And $\mathcal{J} = \int_0^{1} dx \, x^\alpha \, (1-x)^\beta \, Li_n(x)$ . I have written the ...

Question 8

Context Some numerical work suggests that the following integral is true: $$\begin{align*} I&=\int_{0}^{1}\arcsin{(x^2)}\left(\frac{\log{(\sqrt{1-x^4}+1)}}{x^2}-\frac{2x(x(1-x^4)^{1/4}-1)}{(1-x^4)^...

Question 9

I'm trying to tackle integrals of the form $$ \int_{1}^{\infty} \frac{1}{x^{2k}+1} \ dx $$ for large $k \in \mathbb{Z}^{+}$. An easily computable general solution is desired, rather than, say, a ...

Question 10

Prove the given identity: $$ \sum_{k=0}^{2n} (-1)^k \beta\left(\frac{x+k}{2n+1}\right) = (2n+1) \beta(x) $$ Where $$ \beta(x) = \sum_{j=0}^{\infty} \frac{(-1)^j}{x+j} $$ The proof is more ...

Question 11

I'm stuck on this problem and would appreciate any help. I'm trying to compute the integral $$ \int_0^1 \frac{\ln(x)}{\sqrt{x - x^2}}\, dx. $$ To approach it, I define a parametric function $$ F(a) = \...

Question 12

I want to compute the sum: $$\sum_{0}^{n}k^{i}(n-k)^{j}$$ If I simply expand the polynomial, I get: $=\sum_{k=0}^{R}\sum_{l=0}^{j}k^{i}\binom{j}{l}R^{l}(-1)^{j-l}k^{j-l}\\ =\sum_{l=0}^{j}\binom{j}{l}...

Question 13

I found this beta function representation integral very pleasing, so I tried to work it out—but something went horribly wrong. I'm wondering if I am wrong, or if the book is. I've been working on this ...

Question 14

Hmm, that is what we have. A "close form" using Beta function: $$\begin{align}\int_0^\pi \left(\ln(\sin x)\right)^k\text dx=\frac{1}{2^k} \frac{\partial^k}{\partial a^k} \text B\left(a, \...

Question 15

For finding when the median of a beta distribution is $\frac{1}{2}$, this answer says: If a $\mathrm{Beta}(a,b)$ distribution has $a>b$ then $\mathbb P(X \le \frac12) \lt \frac12$ and its median ...

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Questions tagged [beta-function]

Good known bounds for cumulative distribution functions

How to derive relations between integrals over a triangle without evaluating them explicitly?

Using the integral $\frac{1}{\binom{n}{r}} = (n+1)\int_0^1 x^r(1-x)^{n-r}\mathrm dx$

Representation of the beta Euler function by an infinite sum

Asymptotic expansion of incomplete Beta function

Closed form or representation as a hypergeometric series of $\sum _{k=1}^{\infty } \frac{x^{k-1}}{B\left(k,-\frac{k}{n}\right)}$

Definite integrals involving logarithm and polylog [closed]

How to prove $I=\Gamma(\frac{1}{4})^2\frac{\sqrt{2}(\pi+4)}{8\sqrt{\pi}}-\pi-4$.

General solution to $\int_{1}^{\infty} \frac{1}{x^{2k}+1} \ dx$ via beta function

Prove $ \sum_{k=0}^{2n} (-1)^k \beta\left(\frac{x+k}{2n+1}\right) = (2n+1) \beta(x) $

Calculating $\int_0^1 \frac{\ln(x)}{\sqrt{x - x^2}}\, dx$

Evaluating the sum $\sum_{0}^{n}k^{i}(n-k)^{j}$ and its connection to the beta function

Confusion on Beta Function Integral Representation

How can I evaluate the close form of $\int_0^{\pi}\left(\ln\left(\sin x\right)\right)^k\text dx$

Finding values that make the Beta function symmetric

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