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Questions tagged [beta-function]

For questions about the Beta function, a special function closely related to the Gamma function. It is advisable to also use the [special-functions] tag in conjunction with this tag.

4
votes
1answer
311 views

Closed form for this integral with Beta function $\int_0^\infty x \mathrm B (x,x)~dx$

Is it possible to find the closed form of $$\int_0^\infty x \mathrm B (x,x)~dx=2.44333\dots$$ This integral converges because for small $x$ Beta function behaves like $2/x$. Using the integral ...
1
vote
2answers
176 views

An integral involving beta function

For $-1 \leq \theta \leq 1$ and $\nu > -1/2$, prove that the function $$ f(x; \theta,\nu) = \frac{(1-x^2)^{\nu -1/2}} {(1-2\theta x +\theta^2)^\nu B(\nu+1/2,1/2)} $$ is a valid probability density ...
1
vote
0answers
140 views

Is there any special property about incomplete beta function $B(x;1-x,x)$?

guys, I encountered a function like $$ f(x) = B(x;1-x,x) $$ where $B(\cdot)$ is the incomplete beta function and input $0 < x < 1$ is some positive small real value close to zero . I want to ...
0
votes
1answer
39 views

I like to know beta function`s property related with my question

Fact 1. ($B$ is beta-function) $$\int_{0}^{1}\frac{z^{-(n+1)}}{(1+\frac{1}{z})^{2n}}=\frac{1}{2}B(n,n)$$ I can find above fact by using MATLAB. But i like to show above fact using beta function ...
1
vote
0answers
126 views

Riemann's zeta function in a definite integral

I am trying to evaluate the value of the definite integral $ I= \int_0^1((1-\delta)\log[p]-\delta\log[1-p])^4dp$ Using Binomial expansion I get $I=(1-\delta)^4\int_0^1(\log[p])^4dp-4(1-\delta)^3\...
0
votes
0answers
33 views

Simplify $B(ix,2+iy,0)$ whre $B$ is the incomplete Beta function

Is there any way to re-write or simplify this function for $x,y\in\mathbb{R}$, in the limit $x\rightarrow\pm\infty$? Or any laws regarding symmetry with respect to $x\rightarrow -x$ or $y\rightarrow -...
1
vote
1answer
647 views

Compute $\int_0^1x^m(1-x^n)^pdx$

Compute $\int_0^1x^m(1-x^n)^pdx$ Hint in question says, express in terms of gamma function, which I don't see how. I can put $x=\frac{1}{e^t}$ and make limit from $0$ to infinity, as in gamma ...
2
votes
2answers
915 views

Evaluate$\int_0^1 \frac{x^{m-1} + x^{n-1}}{(1+x)^{m+n}}dx$ in terms of Beta function

I have no idea of Beta functions. Based on some properties of beta functions like, $B(m,n) = \int_{0}^\infty \frac{x^{m-1}}{(1+x)^{m+n}}dx \;\; ;B(m,n) = B(n,m)$ I arrived at $2B(m,n) = I+ \int_{1}^\...
1
vote
3answers
294 views

Asymptotics of incomplete Beta function $B_{1/2}(y+1,y)$ when $y\to\infty$

My question concerns the behavior of the incomplete Beta function $$B_{1/2}(y+1,y)=\int_0^{1/2}x^y (1-x)^{y-1}dx$$ in the large $y$ limit. I have been looking everywhere, but I can't find anything. ...
1
vote
1answer
122 views

Value of the series $\sum_{n=1}^{\infty}\big (\frac{H_n}{\binom{3n}{n}}\big)^2$

I want to find the value of following series $$ \sum_{n=1}^{\infty} \left(\frac{H_n}{\binom{3n}{n}}\right)^2\tag{1} $$ where $H_n = 1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n}$. I know that $$\sum_{...
1
vote
0answers
56 views

The integral $\int_0^{\frac{1}{2}}\frac{x^{k-1}}{1-x^{2^k}}dx$ and how simplify the Pochhmammer symbol in related series

Inspired in the shape of useful integrals to compute $\pi$ (see *), I've consider for each integer $k\geq 1$ $$\int_0^{\frac{1}{2}}\frac{x^{k-1}}{1-x^{2^k}}dx=\int_0^{\frac{1}{2}}x^{k-1}\sum_{n=0}^\...
1
vote
1answer
191 views

About the relation between $\int_{0}^{\infty}\frac{\cosh((\alpha-\beta)x)}{(\cosh(x))^{\alpha+\beta }} \,dx$ and $B(\alpha,\beta)$

How do I get from $$B(\alpha,\beta)=\int_{0}^{1}{ t^{ \alpha -1} (1-t)^{ \beta -1} \,dt}$$ to: $$2^{2- \alpha - \beta } \int_{0}^{ \infty}{ \frac{ \cosh (( \alpha - \beta )x)}{( \cosh(x))^{ \...
2
votes
1answer
221 views

Integral representation of the Beta function

How do I get from $ \int_{0}^{1}{t^{x-1} (1-t)^{y-1}\,\mathrm dt} $ to $ \displaystyle\int_{0}^{\infty}{ \frac{t^{x-1} + t^{y-1}}{(1+t)^{x+y}} \,\mathrm dt} $ ? I have tried different change of ...
7
votes
1answer
105 views

Anti-treta function in terms of standard special functions

Define treta$^*$ function as $$ \tau(\alpha_1,\alpha_2,\alpha_3) = \iint_{0< x_1< x_2<1} x_1^{\alpha_1-1}(x_2-x_1)^{\alpha_2-1}(1-x_2)^{\alpha_3-1}\, d(x_1,x_2).\tag{1} $$ Similarly to the ...
2
votes
1answer
53 views

How to compute $E[Xr / (Xr +1 - X)] $ where $X$ follows a Beta distibution?

I would like to compute $E[Xr / (Xr +1 - X)] $ where $X$ follows a Beta distribution $\operatorname{Beta}(\alpha, \beta)$ with $\alpha, \beta > 1$, $\alpha < \beta$ and $r \in (0,1)$. This is ...
1
vote
0answers
41 views

prove the following problem based on beta function [closed]

$\int_{0}^{1} \frac {x^{m-1}\cdot (1-x)^{n-1}}{(a+x)^{m+n}}dx = \frac{B(m,n)}{a^{n}\cdot (1+a)^{m}}$
3
votes
0answers
159 views

Solving equation involving factorials

I have this particular equation $\frac{(\alpha-1)!(\beta-1)!}{(\alpha+\beta-1)!} = \frac{\Gamma(p)(1+q)^{n+2p} 2^n}{q^{p}(2+q)^{n+p}}$. Now, given the values of $\alpha$ and $\beta$, I need to find ...
3
votes
1answer
104 views

How to compute $\int_0^1x^a(1-x)^be^{cx}dx$?

How to compute the integral $I(a,b,c) = \int_0^1x^a(1-x)^be^{cx}dx$ ? I know that, $\int_0^1{x^a(1-x)^b}dx = \frac{\Gamma(a+1)\Gamma(b+1)}{\Gamma(a+b+2)}$. Using this result, I tried integration by ...
0
votes
1answer
91 views

Approximation of product of Bernoulli with different proportions

I want to update a variable $Y$ with Beta (uniform for simplicity, $Y \sim U(0, 1)$) distribution, with Bernoulli information each period... But each period the proportion parameter of the Bernoulli ...
0
votes
0answers
43 views

Why is this a beta distribution?

I'm given a circle with point $A$ defined by $(x,y)$. Then $T=1-d[O,A]$, so $T=1-\sqrt{(x^2+y^2)}$. Asked to find: $P[T<=u]$ $E[T]$ $Var(T)$ Alright, so $d[O,A]$ has the CDF $u^2$. So, for the ...
3
votes
1answer
52 views

Mixed partials of the Beta function B$(a,b)$ at $(1,0^+)$

In this post M.N.C.E gave the equality below $$\frac{\partial ^{5}}{\partial a^{3}\partial b^{2}}\mathrm{B}\left ( 1,0^{+} \right )=\left [ \frac{1}{b}+O\left ( 1 \right ) \right ]\left [ \left ( 12\...
25
votes
2answers
1k views

Show that $\sum_{n=0}^{\infty}\frac{2^n(5n^5+5n^4+5n^3+5n^2-9n+9)}{(2n+1)(2n+2)(2n+3){2n\choose n}}=\frac{9\pi^2}{8}$

I don't how prove this series and I have try look through maths world and Wikipedia on sum for help but no use at all, so please help me to prove this series. How to show that $$\sum_{n=0}^{\infty}...
1
vote
1answer
905 views

Expected value of exponential function

Suppose two identical component are connected in a piece of factory equipment. The two lifetimes X1 and X2 are independent each having exponential distribution with beta =2. The value of the equipment ...
2
votes
0answers
144 views

When is the inequality $\beta(a_1+b_1, a_2+b_2)\ge \beta(a_1, a_2)\beta(b_1, b_2)$ true?

Let $\beta(a, b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$. Does there exist some general condition on $a_1, a_2, b_1, b_2\in \mathbb{N}^+$ such that $$\beta(a_1+b_1, a_2+b_2)\ge \beta(a_1, a_2)\beta(...
1
vote
1answer
45 views

Finite series that gives Beta function at integers

Playing around with Mathematica I have found that $$ \sum_{k=0}^n \binom{n}{k} \frac{(-1)^{n+k}}{2n+1-k} = \beta(n+1,n+1), $$ where $\beta(x,y)$ is the Beta function. Now, I'm not versed neither with ...
0
votes
1answer
75 views

Beta function. What is wrong? Misunderstanding.

I am only interested in real numbers, so here $x,y>0$ real numbers. The Beta function is defined as $$B(x,y)= \int_0^1 w^{x-1}(1-w)^{y-1}dw.$$ The above integral is defined for every $x,y>0$. ...
1
vote
1answer
78 views

2nd moment of the Beta Function / Beta function simplification

Let $B(x,y)$ denote the Beta function. I have $$\frac{B(x+2, y)}{B(x, y)}$$ Wikipedia says $$B(x+1, y) = B(x, y) \cdot \frac {x}{x+y}$$ Thus, $$\frac{B(x+2, y)}{B(x, y)}$$ should equal $$\frac {x^2}{(...
3
votes
1answer
103 views

Write area in terms of Gamma function

Let $p>0$. Then express the area of the region bounded by the coordinate axes and the curve $x^p+y^p=1$ in the first quadrant in terms of the gamma function. My thought is to consider polar ...
1
vote
2answers
45 views

Calculating $f(x), f(x\mid y), f(y\mid x)$ from $f(x,y)\propto \left(\begin{array}n n\\ x\end{array}\right)y^{x+\alpha-1}(1-y)^{n-x+\beta-1}$

I'm reading about Gibbs sampling from a paper by Casella and George and in an example I'm given the following joint distribution for random variables $X$ and $Y$: $$f(x,y)\propto \left(\begin{array}...
0
votes
2answers
159 views

Is it possible to integrate $\int_0^1 x^5 (2-x)^4 \text dx$ using Beta functions?

The integral $$\int_0^2 x^5 (2-x)^4 \text dx$$ Can be integrated using beta function by substituting $x=2\sin^{2}\theta$ and using the definition of the beta function $$\beta(x,y)=2\int_{0}^{\frac{...
0
votes
1answer
57 views

Check this value of $\int_{0}^{x}\frac{t^m}{(x-t)^\alpha}dt$

I want to prove that: $$\int_{0}^{x}\frac{t^m}{(x-t)^\alpha}dt=\frac{\Gamma(1-\alpha)\Gamma(m+1)}{\Gamma(m-\alpha+2)}x^{m-\alpha+1}$$ where $m$ is a positive integer and $\alpha \in [0,1]$. I try ...
1
vote
1answer
104 views

Evaluating an integral involving Beta function

What is a good starting point for approaching the following integral? $$\int_t^z \frac{\mathrm{d}x}{\left(z - x\right)^{1 - \alpha} (x - t)^\alpha} $$ I have tried several change of variables (with ...
4
votes
1answer
335 views

Limit involving the inverse beta regularized function

Let $0<p<\frac{1}{2}$. I am looking for the limit: $$\lim_{t \to \infty} \left(\frac{t}{\frac{t}{I_{2 p}^{-1}\left(\frac{t}{2},\frac{1}{2}\right)}-2 \sqrt{t} \sqrt{\frac{1}{I_{2 p}^{-1}\left(\...
3
votes
1answer
177 views

How to prove $\int_{0} ^{1}{\sqrt{1-x^4}dx}=\frac{1}{12}\sqrt {\frac{2}{\pi}}(\Gamma\left(\frac{1}{4}\right))^2$?

To prove , $$\int_{0} ^{1}{\sqrt{1-x^4}dx}=\frac{1}{12}\sqrt {\frac{2}{\pi}}(\Gamma\left(\frac{1}{4}\right))^2$$ When we substitute $x^4$ with t we get the equation $$\frac{1}{4}\int_{0}^1t^{\frac{-...
3
votes
2answers
583 views

Proving the multiplication formula of Gamma function

Evaluate this integral $$\int_{0}^{\infty} \frac{x^{2m}}{1+x^{2n}}dx$$ then use the result and the relationship between gamma and beta functions to prove that $$\Gamma({x})\Gamma(1-x)= \frac{\pi}{sin(\...
1
vote
1answer
88 views

Evaluating an integral by substitution and special functions [duplicate]

How can I evaluate this integral? $$\int_{0}^{1} \frac{dx}{\sqrt{{1+x^4} }}$$ I tried using the substitution $x=\mathrm{e}^{-u}$ but I got nowhere.
3
votes
1answer
674 views

Show that $\Gamma(z)\Gamma(1-z) \sin \pi z$ is bounded in the complex plane

Attempt I know that $\Gamma(z)=\int_0^\infty e^{-t}t^{z-1} \ dt$ so $$\lvert \Gamma(z) \rvert \leq \int_0^\infty e^{-t}|t^{z-1}| \ dt=\int_0^{\infty} \frac{e^{-t}}{t} t^{\Re(z)} dt .$$ After this, I'...
4
votes
1answer
481 views

To evaluate integral using Beta function - Which substitution should i use?

$$\int_{0}^{1} \frac{x^{m-1}(1-x)^{n-1}}{(a+bx)^{m+n}}dx = \frac{B(m,n)}{(a+b)^ma^n}$$ I have to use some kind of substitution but i do not understand what i use and why ? Thanks
6
votes
1answer
258 views

Another beta integral due to Cauchy.

I have the following identity which I want to prove: $$C(x,y):= \int_{-\infty}^{\infty} \frac{dt}{(1+it)^x(1-it)^y} = \frac{\pi \cdot 2^{2-x-y}\Gamma(x+y-1)}{\Gamma(x)\Gamma(y)}$$ where $\Re(x+y)>...
2
votes
0answers
109 views

Parameterizing monotonic functions

I am trying to design a function $f_{\alpha} : \left[0,1\right] \rightarrow [0,+\inf)$ that looks like the following: It should have the following properties: Its integral should be 1 for any value ...
7
votes
3answers
223 views

How could one solve $\int_{0}^{\infty} \frac{1}{1-t^4}dt$ with special functions?

How could one solve $$\int_0^\infty \frac{1}{1-t^4} \, dt\,?$$ I have to apply special functions, so I thought that I have to use the change variable $$u=t^4,$$ but $$du=4t^3\,dt$$ and when $$t\...
-1
votes
1answer
369 views

How do I renormalize these probability distributions?

So I have two random variables, $X_1$ and $X_2$, both uniformly distributed on $[0, 1]$. If $Y = (X_1 + X_2) / 2$, it will also be distributed between 0 and 1, but it won't be uniformly distributed ...
-1
votes
2answers
276 views

Alpha and Beta question [Addmath, quadratic equations]

Please help. Question: Addmath (Quadratic Equations) Given $\alpha$ and $\beta$ are the roots of the quadratic equation $2x^2 - 6x + 5 = 0$, form an quadratic equation with the roots $\alpha + 1$ ...
2
votes
2answers
3k views

Proof that Beta-function $B(m,n)$ = $\frac{n-1}{m}B(m+1,n-1)$?

When m and n are positive integers. It probably has to do with the incomplete Beta-function $B_{sin^2(x)}(m,n)$.
0
votes
1answer
174 views

How do I calculate the Beta-function $B(m,n) = 2\int_0^{\frac{\pi}{2}}\sin ^{2 m-1}(t) \cos ^{2 n-1}(t)\, dt$

The Beta-Function $$B(m,n) =2\int_0^{\frac{\pi}{2}}\sin ^{2 m-1}(t) \cos ^{2 n-1}(t)\, dt \tag{a}$$ is equal to $$\frac{n-1}{m}B(m-1,n+1) \tag{b}.$$ How do I go from (a) to (b)? (I tried with ...
2
votes
1answer
220 views

Show $\int_{0}^{\infty} \frac{x^{-z}}{(1 + x)^{2}} ~ \mathrm{d}{x} = \frac{\pi z}{sin(\pi z)}$

I need to solve the following integral: $$ I = \int_{0}^{\infty} \frac{x^{-z}}{(1 + x)^{2}} ~ \mathrm{d}{x}. $$ Wolfram Alpha gives the answer as $ \frac{\pi z}{sin(\pi z)}$, or equivalently, $\pi z ...
1
vote
1answer
690 views

Integral of the incomplete beta function

Does anyone know how to evaluate $$I(a,b) = \int_0^1 B_t(a,b) dt$$? Here $B_t(a,b)$ is the incomplete Beta function, defined as $$B_t(a,b) = \int_0^t x^{a-1}(1-x)^{b-1} dx $$ Of course $I(a,b)$ ...
2
votes
2answers
222 views

Using the Eulerian integrals evaluate $\int_0^\infty \frac{\ln^2{x}}{1+x^4} \mathrm{d}x$

The question asks to evaluate the integral: $$\int_0^\infty \frac{\ln^2{x}}{1+x^4} \mathrm{d}x.$$ I have tried a few substitutions but am not getting anywhere. Thanks in advance!
4
votes
1answer
564 views

Double integral involving beta functions

I want to prove the following result: $$\int_0^1 \int_0^1 {f(xy)(1-x)^{p-1}y^p(1-y)^{q-1}} \mathrm{d}x \, \mathrm{d}y=\frac{\Gamma(p) \Gamma(q)}{\Gamma(p+q)} \int_0^1 {f(t)(1-t)^{p+q-1}} \mathrm{d}t.$$...
3
votes
0answers
104 views

Evaluate $\int_0^1\frac{\ln{\left(1-kt^2(1-t)+\frac{t^4(1-t)^2}{4}\right)}}{t}dt$ , where $k=\cos1$

I want to find the value of $\displaystyle\sum_{n=1}^\infty \dfrac{\cos n}{n^22^n\binom{3n}{n}}.$ Since $\displaystyle\dfrac{1}{\binom{3n}{n}}=2n\beta(2n,n+1)=2n\int_0^1t^{2n-1}(1-t)^ndt$, $\...