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Questions tagged [gamma-function]

Questions on the gamma function $\Gamma(z)$ of Euler extending the usual factorial $n!$ for arbitrary argument, and related functions.

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How to find CDF of Gamma distribution for the Time (t) by integration?

I know the CDF of Gamma Distribution for the Time ($T \sim gamma(\alpha, \lambda)$) and shape $\alpha>0$ , rate $\lambda>0$ and $t>0$, is $$F(t)= \frac{\Gamma_t(\alpha)}{\Gamma(\alpha)}, $$ ...
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92 views

Evaluate $\int_{0}^{1} \log^2\left(\frac{\Gamma(x)}{\Gamma(1-x)}\right)\log^2\left(\frac{\Gamma(x)}{\Gamma(1+x)}\right) dx$

Evaluate : $$\int_{0}^{1} \log^2\left(\frac{\Gamma(x)}{\Gamma(1-x)}\right)\log^2\left(\frac{\Gamma(x)}{\Gamma(1+x)}\right) dx$$ This is my attempt in below ... but what I want is simplify more ...
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19 views

Computation of a sum involving gamma functions

Let $l$ denote a positive integer and $m$ be an integer $-l \leq m \leq l$. I would like to prove the following identity: $$\sum_{0 \leq j \leq \left\lfloor\frac{l - m}{2}\right\rfloor}\sum_{0 \leq k ...
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1answer
27 views

Confusing closed form for $\int_{-\infty}^{0}e^{(x^{2n+1})}dx$ for $n\in\Bbb N$

Given that $n\in\Bbb N$, $$f(n)=\int_{-\infty}^{0}e^{(x^{2n+1})}dx$$ How can one get to a closed form of $f(n)$? According to the online integral calculator, $$\int e^{x^{2n+1}}dx=\frac{-\Gamma(\frac{...
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1answer
33 views

Unique extensions of the factorial

There are many ways to extend the factorial beyond the naturals. However, some extensions are particularly notable for being unique under certain conditions, namely the Gamma function. And so I'm ...
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19 views

Conditional probability with Gamma distribution.

I am stuck on a vital step in this process, and that is about the conditional probability aspect of it. Once I get that step, the integration and using the gamma function should go fine. So my ...
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1answer
48 views

Prove that $e^{\frac12i\,\text{gd}(\pi t)}\cos^\frac12(\text{gd}(\pi t))=\frac{1+i}{e^{\pi z}+i}e^{\frac{\pi z}2}$

This question was derived from this post about Gamma function. Juan said: $$ \Gamma\left(\frac12+it\right)=\sqrt{\frac{\pi}{\cosh\pi t}}\exp\left\{i\left(2\vartheta(t)+t\log(2\pi)+\arctan\tanh\...
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Gamma function integral [closed]

How does the derivation of this integral from the Gamma function is equivalent to (n-1)!?$$\Gamma(n)=\displaystyle\int_0^1\Bigg(\ln\frac1x\Bigg)^{n-1}dx$$
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Evaluating integral (comes from a bigger problem in statistics)

Let $\alpha, \beta>0$ be parameters. I wish to compute $$\int_0^\infty \frac{x^\alpha}{x-1} e^{-\beta x} dx.$$ I managed to reduce this problem when $\alpha$ is integer by using $$\frac{x^\alpha}{...
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The right way to find $\frac{d}{ds}\Gamma (s)$

I thought up this method to find $\frac{d}{ds}\Gamma (s)$, and I want to know if it is valid. I know that there are other questions about this on MSE but they don't really help me understand whether ...
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51 views

An integration related to incomplete gamma function

I have no clues about the following equation, expecting some help from anyone. $\int_\beta^\infty e^{-x \theta}\frac{1}{\theta}(\frac{\theta}{\beta}-1)^{-\alpha}d \theta=\Gamma(1-\alpha)\Gamma(\alpha,\...
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Show that $\frac{1}{\Gamma(a)}\int_{0}^{\infty}[1-e^{-z(u^{-1/a}-1)}][1-e^{-z(v^{-1/a}-1)}]z^{a-1}e^{-z}dz=[max(u^{-a}+v^{-a}-1,0)]^{-1/a}$

Assume $X, Y$ two independent exponential random variables with $\lambda = 1$, and $Z$ to be a gamma variate with $\alpha = \theta > 0, \beta = 1$, independent from $X$ and $Y$. Next, assume $...
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$\phi(x)$ for negative integers

During the the proof of the following formula I faced with $\phi(x)$ for negative integers. That is in order to finish proof of the mentioned formula (which is not proved in the book), after a long ...
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1answer
16 views

Median of beta distribution for alpha = beta

I've seen that the median of the Beta Distribution cannot be defined by a closed form analytic expression but in most sites I see people give that, when the parameteres alpha and beta are the same, ...
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1answer
36 views

A query while showing that the Gamma function $\Gamma$ is logarithmically convex for $x \gt 0.$

We are using the general definition of gamma function defined on $\Bbb C \setminus \{\text{non-positive integers}\}$ i.e. $\Gamma(Z)=\frac {e^{-\gamma z}}{z} \prod (1+ \frac zn)^{-1} e^{\frac zn}$. ...
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Evaluating $K\big(\frac{3-\sqrt{7}}{4\sqrt{2}}\big)$

On MSE, I have seen derivations of the elliptic integral special values $$K(1/\sqrt{2})=\frac{\Gamma^2(1/4)}{4\sqrt{\pi}}$$ $$K(\tan(\pi/8))=\frac{\sqrt{\sqrt{2} +1} \Gamma (1/8)\Gamma (3/8)}{2^{13/4}\...
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1answer
44 views

What is the Laplace transform of gamma function?

What is $$\mathfrak L (\Gamma(z))$$? And how can it be derived? $$\Gamma(z)=\int_0^\infty t^{z-1}e^t dt$$ $$\mathfrak L(\Gamma(z)) = \int_0^\infty \int_0^\infty t^{z-1}e^t dt e^{-sz} dz$$ Yes, of ...
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the gamma function being undefined

Could someone help me with part C of this question, relating to the Gamma Function. I am aware it is something to do with convergence but I am not sure how to show this (a) Use integration by parts ...
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1answer
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Need help isolating variables (average translational energy equation)

I need to rewrite equation so that i can use gamma function. Below are assignment text and my steps and reasoning so far: The probability of finding a translational energy in the range $$E_{tr}, ...
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1answer
26 views

Gamma and incomplete gamma functions- when are they equal?

I do not know anything about these gamma functions. In a series sum I got two terms as follows: $\Gamma(b/c) - \Gamma(b/c,a/c)$. The first is a gamma function, the second is an incomplete gamma ...
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Is there a term for extensions of functions on discrete sets to continuous sets?

The pi function, $\Pi(z)$ – defined as $\Pi(z) = \Gamma(z+1)$, where $\Gamma(z)$ is the gamma function – extends the factorial in that $$\Pi(n) = (n)!$$ for all positive integers $n$. In other words,...
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1answer
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Is it legal to use factorials of negative integers when summing Taylor series of different derivatives?

My question is about the use of the factorial function appearing in the Taylor series, in the case that expansions of the derivatives of several order, of the same function, around the same point, are ...
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Proof of equivalence gamma function definitions (Integral & Euler) allows a limit: Why?

I am researching the equivalence between $\Gamma_{Integral}$ and $\Gamma_{Euler}$. I have found 3 sources all using the proof as found on https://proofwiki.org/wiki/...
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2answers
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Integral for $\Gamma(z) \Gamma(1-z)$

In gamma function How do you evaluatate $\Gamma(z)\Gamma(1-z) = \int_{0}^\infty \frac{v^{z-1}}{v+1} dv$ which method could be fit to solve the integral? The result should be $ \Gamma(z)\Gamma(1-z) ...
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1answer
28 views

Reflection Formula on Gamma Function, integral substitution

In proving $ \Gamma(z) \Gamma(1-z)= \Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin \pi z} $ appears a $\int_{0}^{\infty}\int_{0}^{\infty}s^{-z}t^{z-1}e^{-(s+t)}\,ds\, dt \qquad (0<\Re z <1)$ where we ...
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Solve an equation involving gamma functions

I have a question involving gamma functions and the related functions. Let me first give you a simple problem: Find $y$ which satisfies the equation: $x = Q (a,y)$, $a > 0$ and $y > 0$, Eq. (...
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Prove $\lim\limits_{r \to \infty} r-\frac{\sqrt{r}}{e^{r}}\sum_{n=0}^{\infty}\frac{\Gamma{(n+3/2)}r^n}{(n!)^2}=1/4$

This is a follow-up question for the one described here. Evaluate $$\lim\limits_{r \to \infty} r-\frac{\sqrt{r}}{e^{r}}\sum_{n=0}^{\infty}\frac{\Gamma{(n+3/2)}r^n}{(n!)^2}=1/4$$ There is a ...
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Is there any $a, b \in \mathbb R, b \ne 0$ such that $\Gamma(a+bi)$ can be evaluated manually?

Is there any $a, b \in \mathbb R, b \ne 0$ such that $\Gamma(a+bi)$ can be evaluated manually? (Like $\Gamma(\frac 12)$) If there is/are, could you show me how to calculate it? I found that $\Gamma(...
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Sum of series $\frac{ 4}{10}+\frac{4\cdot 7}{10\cdot 20}+\frac{4\cdot 7 \cdot 10}{10\cdot 20 \cdot 30}+\cdots \cdots$

Sum of the series $$\frac {4}{10}+\frac {4\cdot7}{10\cdot20}+ \frac {4\cdot7\cdot10}{10\cdot20\cdot30}+\cdots $$ Sum of series $\frac {4}{10}+\frac {4\cdot7}{10\cdot20}+ \frac {4\cdot7\cdot10}{...
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Sum of $1+\frac{1\cdot 3}{6}+\frac{1\cdot 3 \cdot 5}{6 \cdot 8}+\cdots \cdots$

Finding sum of $\displaystyle 1+\frac{1\cdot 3}{6}+\frac{1\cdot 3 \cdot 5}{6\cdot 8}+\frac{1\cdot 3 \cdot 5 \cdot 7}{6 \cdot 8 \cdot 10}+\cdots \cdots$ Try: We can write sum as $$ \mathcal{S} ...
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58 views

Set the mode and median of a gamma distribution equal to each other

I am trying to generate a set of random positive steps that will result in a final location that is close to what I would have gotten from taking a similar number of fixed steps$^1$. I would like the ...
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How to approximate a fraction of gamma functions evaluated at huge values

For sufficiently large $m$, one can approximate the function $$f:m\mapsto\frac{\Gamma \left(\frac{m+1}{2}\right)^2}{\Gamma \left(\frac{m}{2}\right) \Gamma \left(\frac{m}{2}+1\right)}$$ using the ...
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49 views

exponential integral with arbitrary power of variable

I have been trying to solve a simple mathematical integral. I know that the solution exists but it is not being verified by Matlab. Here is the integral that I am solving along with its answer which I ...
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The series $\sum_{n=0}^\infty \frac{(-\beta)^n}{n!}\Gamma\left(\frac{n+1}{2},\alpha\right)$

I've worked out the solution to an integral which involves the following expression: $$\sum_{n=0}^\infty \frac{(-\beta)^n}{n!}\Gamma\left(\frac{n+1}{2},\alpha\right)$$ Where $\Gamma(s,z)$ is the ...
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Gamma function: relating infinite product and integral definitions. Stuck on a step in the proof.

This has been asked and answered before, but at a level far above anything I could understand. When proving that: $\Gamma(z+1)=\int_0^\infty x^{z}e^{-x} dx$ is the same as: $\Gamma(z+1)=N^z \frac{1}...
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1answer
46 views

How to switch to polar coordinates with Gaussian Integral?

I'm trying to do the Gamma function of 3/2, so $$\int_0^{\infty } e^{-x} \sqrt{x} \, dx$$ So far I have this u substitution $$u=\sqrt{x}$$ $$du=\frac{dx}{2 \sqrt{x}}$$ $$\int_0^{\infty } e^{-u^2} u2u \...
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156 views

Limit of sum of sequences at infinity

Given two sequences $$a_n=\int_0^1 (1-x^2)^n dx$$ and $$b_n=\int_0^1 (1-x^3)^n dx$$ ,($n\in N$) then find the value of $$L=\lim_{n\to \infty} (10 \sqrt [n]{a_n} +5\sqrt [n]{b_n})$$ My try: $$a_n=\...
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Show that $\int_{0}^{1}(1-u)^p u^{p+1} \text{d}u = \frac{\Gamma(p+1)\Gamma(p+2)}{\Gamma(2p+3)}$

Assume $p > 1$. I want to show that $$ \int_{0}^{1}(1-u)^p u^{p+1} \text{d}u = \frac{\Gamma(p+1)\Gamma(p+2)}{\Gamma(2p+3)},$$ here the result is from WolframAlpha. However, not having any ...
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85 views

Looking for an intuitive explanation of the gamma function which can be comprehended with high school maths. [closed]

I am looking for an "intuition pump" (Daniel Dennett's phrase) for the gamma function, showing why $\Gamma(n + 1)$ is the same as $\int_0^\infty x^n e^{-x} \, dx$. For instance, is the fact that $e^{...
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1answer
67 views

Beta distribution with parameters $\alpha = \beta \to 0$ is Bernoulli distribution

In the article https://en.wikipedia.org/wiki/Beta_distribution#Symmetric_(α_=_β) it is said that a Beta distribution with parameters $\alpha = \beta \to 0$ has a Bernoulli distribution with ...
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187 views

Definite Integral over the Gamma Function

Do we have any methods for evaluating $$\int_1^{\infty} \frac{1}{\Gamma(s)} \,ds$$? I thought about perhaps rewriting as $$\int_1^{\infty} \frac{\Gamma(1-s)}{\Gamma(1-s) \Gamma(s)} \,ds$$ $$=\frac{1}{...
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Solutions for $\Gamma(n+1)$ = $\alpha$

There's a general method of solving the equation $n!$ = $\alpha$, where $n$ and $\alpha$ can be any real numbers? Here's why my question may be important: There's no solution for the equation $n! = ...
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contour integration path in the complex s-plane running from $R−iW$ to $R+iW$

I would like to run the following function : "The generalized upper incomplete Fox’s H function" given by $$ \large{H}_{m,n}^{p,q}\left( z \left| \begin{array}{cc} (a_1,\alpha_1,A_1)\cdots (a_p,\...
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1answer
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Can any one prove for me $\ln(1+x) = \large{G}_{2,2}^{1,2}\left( x \left| \begin{array}{cc} 1,1 \\ 1,0 \end{array} \right. \right).$

I am a PhD student in Wireless Communications and recently I found a paper about the use of "The generalized upper incomplete Fox’s H function". I think that in order to understand this function, I ...
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1answer
72 views

simplifying triple summations from series expansion

I have the following equation I would like to extract $x^k$ out of $$ \sum^\infty_{k=0}x^k \Gamma \bigg(l,\frac{ax-1}{a^2}\bigg) $$ I start by expanding the incomplete gamma function, and then the ...
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1answer
73 views

Gamma function, a roadblock: $\int_0^\infty e^{-t}t^{x-1}\,dt = \frac{1}{x}\int_0^\infty e^{-u^{1/x}}\,du$?

In Spivak's Calculus 19 - 44, I'm being asked to prove that the previously encountered gamma function, defined as: $$\Gamma(x)=\int_0^\infty e^{-t}t^{x-1}\,dt$$ is equivalent to $$\frac{1}{x}\int_0^\...
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1answer
34 views

integrating modified gamma functions

I have an integral that looks like an incomplete gamma function, with another factor in front \begin{equation} I = \int_x^y \frac{1}{a^{c+1}}[1-t]^bt^ce^{-\frac{t}{a^2}}dt \end{equation} Is there a ...
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1answer
29 views

Find E(X) for a certain function using the gamma function

The question I have to do is essentially this: A distribution, X, is modelled by $\displaystyle f(x)= \frac{x}{\sigma^2}e^{-x^2/2\sigma^2},\ x\ge0. $ Show that $\displaystyle E(X)=\sigma \sqrt{\...
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57 views

Gamma function proof

I came across this theorem somewhere and it looked really interesting. I don't know how one would go about proving it though? Can anyone give me some pointers? I'd like to understand this statement ...
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66 views

Close form for a sequence of integrals involving the Gamma function

I'm trying to find a close form for these integrals (with $n\in\mathbb{N}$) : $$\int_{0}^{1}\ln(\Gamma(x+1))\cdot x^n \, \text{d}x$$ Wolfram is able to give a close form for every specific value of n,...