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

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Proof: Derivative of the Gamma-Function

Let $0 < a < b$ be real Numbers and for $x>0$ let $\Gamma_{a,b}(x):=\int_a^b e^{-t}t^{x-1}\mathrm{d}t$. I've to show: a) Show that $\Gamma_{a,b}$ is a continuous Function from ...
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32 views

Help with an Integral with Substitution

I cam across this generalization, but could not understand it completely:$$$$ We will use $$\Gamma(x)=\int_0^{\infty}t^{x-1}e^{-t}\,dt$$ $$\Gamma(\frac{1}{2}+n)=\frac{(2n-1)!!}{2^n}\sqrt{\pi}$$ & ...
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1answer
41 views

Understand Substitution used in Integral

This is a solution I had come across for a general case.$$$$ We will use $$\Gamma(x)=\int_0^{\infty}t^{x-1}e^{-t}\,dt$$ $$\Gamma(\frac{1}{2}+n)=\frac{(2n-1)!!}{2^n}\sqrt{\pi}$$ & we will first ...
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3answers
129 views

Evaluating $\lim\limits_{n\rightarrow \infty} \frac1{n^2}\ln \left( \frac{(n!)^n}{(0!1!2!…n!)^2} \right)$

Evaluating $$\lim\limits_{n\rightarrow \infty} \frac1{n^2}\ln \left( \frac{(n!)^n}{(0!1!2!...n!)^2} \right)$$ I'm not quite sure where to start in evaluating this. Some pointers, or a solution, ...
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1answer
21 views

How can I prove this Bessel function relation

Prove $$J_{s \pm 1}(z) = \frac{s}{z}J_s(z) \mp J_s'(z)$$ from $$J_s(z) = \sum^\infty_{j=0} \frac{(-1)^j}{\Gamma(j+1)\Gamma(j+s+1)}\Big(\frac{z}{2}\Big)^{2j+s}$$ I proved $J_{s - 1}(z) = ...
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61 views

Arguing that a complex function tends to zero fast enough to ensure that $\lim_{N \to \infty} \int_{C} f(s) \, ds = 0$

Consider the complex function $$ f(s) = \frac{\Gamma(a+s)}{\Gamma(b+s)}\frac{z^{s}}{\sin (\pi s)}$$ where $|z| <1$ and $- \pi < \arg(z) < \pi$. Let $C$ be the right half of the circle $|z|= ...
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1answer
34 views

How to calculate a limit with gamma function?

$$\lim _{n\rightarrow +\infty }\dfrac {n^{\left( \dfrac {b-a}{c}\right) }\prod ^{n}_{i=0}\left( a+ic\right) }{\prod ^{n}_{i=0}\left( b+ic\right) }$$ where $a,b,c$are positive integers. How to ...
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36 views

Expansion of gamma function

The lecturer wrote down $\Gamma(x-2)=-\frac{1}{2x}+\cdots$ , but I can't figure out where this comes from? It needs to be in this form so that I can cancel the $x$ with the expansion of another ...
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2answers
326 views

Is the gamma function expressible as a proper integral?

Is the gamma function expressible as a proper integral of elementary functions? You're also allowed to compose it with however many elementary functions. But strictly no limits. [edit] So far the ...
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2answers
42 views

Derivative of the Gamma function

How do you prove that $$ \Gamma'(1)=-\gamma, $$ where $\gamma$ is the Euler-Mascheroni constant?
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3answers
81 views

Evaluating an integral using Gamma function [closed]

For $r \in (0,2)$, I would like to evaluate the integral $$\frac{2}{r} \int_0^{\infty} \frac{\sin(u)}{u^r} du.$$ The answer should be $$\frac{\pi \cdot \mathrm{cosec}{\frac{r\pi}{2}} ...
3
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2answers
75 views

Solution of Integral $\int_0^\infty \frac{x^{r/\beta}e^{-\alpha x}}{\left(1-(1-\beta)e^{-x}\right)^{\alpha+1}} \, dx$

I need to solve the following integral $$\int_0^\infty \frac{x^{r/\beta}e^{-\alpha x}}{\left(1-(1-\beta)e^{-x}\right)^{\alpha+1}} \, dx$$ where $r,\alpha,\beta>0$. kindly help me to solve this ...
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0answers
44 views

Hardy Littlewood Circle Method

I'm working through Vaughan's book on the Hardy Littlewood circle method, which uses the following lemma: Suppose that $\alpha \geq \beta$ are positive real numbers, and that $\beta \leq 1$. Then: $ ...
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1answer
26 views

Recurrance relation with non-integer part

Apologies if this has already been asked, but I'm stuck. I have a recurrance relation that I want resolve to the form $a_n=f(n,\gamma)a_0$ but I don't know what to do. The relation is: ...
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1answer
59 views

Proving that the gamma function is a certain limit

This time I want to prove that $\displaystyle \Gamma(x) = \lim_{\varepsilon\rightarrow 0} \int_\varepsilon ^{1/\varepsilon} t^{x-1} e^{-t}$, I know this is true because we have defined $\displaystyle ...
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0answers
38 views

The inverse of the Gamma function at $-\infty$

Let $\Gamma$ be the analytic continuation of the Gamma function $$\Gamma:z\mapsto \int_0^{+\infty} x^{z-1}e^{-x}dx$$ on the complex plane except non-positive integers. We know that $\Gamma$ has no ...
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0answers
17 views

Deeply confused while trying to understand the derivation of infinite product representation of gamma function

I am trying to understand how in the world Euler figured out the infinite product representation of Gamma function. $$\Gamma (z)=\lim_{n\rightarrow \infty }\frac{n!n^{z}}{z(z+1)\cdots(z+n)}$$ Of ...
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1answer
45 views

Product with $\Gamma$ function

Evaluate the product: $$\Gamma \left ( \frac{1}{n} \right )\Gamma \left ( \frac{2}{n} \right )\cdots \Gamma \left ( \frac{n-1}{n} \right )$$ A path to a solution: $$\begin{aligned}\Gamma\left ...
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1answer
30 views

Solve for X in the given Equation(Gamma Curve)

I'm having a set of points in the form: ${(\frac A{255})}^{\frac 1x}=\frac B{255}$ I need to Find $x$ in the Equation.Where $A$ & $B$ are set of constants ranging from $0$ to $255$. Please ...
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4answers
86 views

Evaluating the limit of a gamma function

I have the following to start: $$F(x)=x^{b-a}\frac{\Gamma(x+a+1)}{\Gamma(x+b+1)}$$ And I'm trying to evaluate: $$\lim_{x\rightarrow\infty}F(x)$$ I have simplified this to yield the same outcome as ...
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1answer
28 views

Evaluating integral $\displaystyle \int_0^\infty x^n e^\frac{-x}{n} dx$ with Gamma-function?

Consider the following integral: $\int_0^\infty x^n e^\frac{-x}{n} dx $. One can find a recursion formula for $R_k = \int_0^\infty x^k e^\frac{-x}{n} dx $: $R_k = n k R_{k-1}$. This yields $R_n ...
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3answers
116 views

How to integrate $ \int_0^\infty \sin x \cdot x ^{-1/3} dx$ (using Gamma function)

How can I calculate the following integral: $$\int_0^\infty x ^{-\frac{1}{3}}\sin x \, dx$$ WolframAlpha gives me $$ \frac{\pi}{\Gamma\Big(\frac{1}{3}\Big)}$$ How does WolframAlpha get this? I ...
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2answers
65 views

$\frac{1}{2}!$ aka $\Gamma(\frac{3}{2})$ [duplicate]

I know it's $\frac{\sqrt{\pi}}{2}$ but how can this be evaluated by hand? (Or can it not?) For quick reference: $$n!=\Gamma(n+1)$$ $$\Gamma(n)=(n-1)!$$ $$\Gamma(n)=\int_0^\infty x^{n-1} e^{-x}\,dx$$ ...
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3answers
44 views

Finding the moment generating function with parameter θ.

I've been stuck on this question for a while now and my exam is coming up so,any hints/comments etc. would be greatly appreciated. Question: Find the moment generating function of the probability ...
2
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1answer
48 views

How do I find the closed form of this integral?

I was trying to calculate the expected value of $\log(Y)$ where $Y$ has gamma distribution and I got to something like this: $$\int_0^\infty \log(z)z^{\phi-1}e^{-z} \, dz,$$ wich based on other ...
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2answers
88 views

Closed form for an infinite sum over Gamma functions?

I am having quite a bit of trouble trying to find a closed form (or a really fast way to compute) for the infinite sum $$\sum_{n=1}^{\infty} a^n \dfrac{\gamma(n+1,b)}{\Gamma(n+1)\Gamma(n)}$$ where ...
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0answers
23 views

Mean and variance of Gamma distribution

How do I calculate the mean and the variance of a Gamma distribution? I was told to prove the variance was sigma/lambda(^2), I don't know how to find the variance much less the variance.
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1answer
21 views

A question about the digamma function and a recursion

The recursion is $T_n=\frac{b}{n+b}+T_{n-1}$ and $T_1=1$. When expanding the recursion we have $T_n=\frac{b}{b+n-1}+\frac{b}{b+n-2}+...+\frac{b}{b+1}+1$. However, the solution shows $T_n=b({\psi _0}(b ...
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2answers
50 views

Equations involving factorial/Gamma function

Are there any known methods to formally solve equations like: 1)$x^3!+(2x^2)!-x!+3=0$ 2)$x!=e^x$ ($0$ is trivial but there must be another one) 3)$(2x!)^2+x!-1=0$ 4)$x!!+x!=7$ I don't need ...
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1answer
50 views

How rapidly does $\Gamma(x)$ diverge as $x$ approaches $0$?

Notoriously $$\lim\limits_{x\to0^{\pm}}\Gamma(x)=\pm\infty,$$ but can we be more precise (tightly) bounding from above $\left\lvert \Gamma(x) \right\rvert$ when $x$ is close to $0$? I could not find ...
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1answer
73 views

$ \frac{\Gamma(r)\Gamma(s)\Gamma(k)}{\Gamma(r+s+k)} $ as a nice integral?

I saw the beta function: $$ \frac{\Gamma(r)\Gamma(s)}{\Gamma(r+s)}= \int_0^1 t^{(r-1)}(1-t)^{(s-1)} dt $$ and got me wondering if I could do something similar the product of 3 or more gamma ...
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42 views

Taylor coefficients for $\zeta(x)\Gamma(x)$

Could you show that in a right neighbourhood of $x=0$ $$\frac{1}{n!}\frac{d^n}{dx^n}\Gamma(x)\zeta(x)$$ behaves like $(-1)^n \zeta (-1)+1$ when $n\to \infty$?
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1answer
18 views

Zero/First order Bessel function (of first kind) identity proof

Here's my attempt to establish that $\dfrac{d}{dx}J_0(x)=-J_1(x)$: $$\begin{align*} J_0(x)&=\sum_{k=0}^{\infty}\frac{(-1)^k}{k! \Gamma(k+1)}\left(\frac{x}{2}\right)^{2k}\\\\ ...
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1answer
34 views

Proof for 1/k! using n choose k as n approaches infinity and its relation to the gamma function

Prove that $\lim_{ n \to \infty }\binom{n}{k}(1/n)^k =\frac{1}{k!}$ How is this related to the gamma function?
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2answers
13 views

Gamma Family Density Function of Y

I have tried to think of how to prove this but at a loss. I keep getting it squared so not sure what I'm doing wrong.
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24 views

Sums of infinite series containing terms involving Gamma function

I am looking for relevant literature on sums of series involving the Gamma function. In particular, my interest is in series of the following form. \begin{equation} \sum_{j=0}^\infty ...
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1answer
42 views

Asymptotic behavior of the zeros of the digamma function

The gamma function has just one extremum on each interval $(k,k+1)$, where $k$ is a negative integer. These extrema occur at the zeros of the derivative of the gamma function. Let $z_n$ denote the ...
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1answer
57 views

Monte Carlo gamma function

This question was asked before but I'd like to ask something more precise given the answer that was given. [ Estimate gamma function using monte carlo ] What is the criterion for a random point to ...
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1answer
59 views

Proving that $~\lim_{x\to1^-}~\bigg(\sqrt[a]{1-x}\cdot\sum_{n=0}^\infty~x^{n^a}\bigg)~=~\Gamma\bigg(1+\frac1a\bigg)$

How could we prove that $$\lim_{x\to1^-}~\bigg(\sqrt[a]{1-x}\cdot\sum_{n=0}^\infty~x^{n^a}\bigg)~=~\Gamma\bigg(1+\frac1a\bigg)$$ for $a>0$ ? The inspiration came to me while trying find a ...
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1answer
39 views

Bound on the Beta function

For positive integers x and y, we have that $$ B(x,y) = \frac{\Gamma(x) \Gamma(y)}{\Gamma(x+y)} = \frac{1}{x} \left( \begin{array}{c} x+y-1 \\ x \end{array} \right)^{-1} . $$ However, $$ \left( ...
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2answers
69 views

Estimate gamma function using monte carlo

Let $\Gamma(\beta) = \int_0^\infty x^{\beta - 1} e^{-x} dx$ how to estimate the above gamma function using monte carlo? Any idea?
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1answer
42 views

Is the Gamma Function multivalued??

Consider the definition of the Gamma function $$ \Gamma(s) = \int_{0}^{\infty}\left[x^{s-1}e^{-x} \right] dx $$ Clearly: $x^{s-1}$ may have multiple defined values for $s$ if $s-1$ is rational or ...
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1answer
53 views

How to show that this limit exists?

How to show that $$S =\lim_{n \to \infty} \frac{\Gamma(\frac{n+1}2)}{\Gamma(\frac n2)\sqrt{n}}$$ exists? If it exists I can show that is equal to $\displaystyle \frac 1{\sqrt 2}$ [0], but I don't ...
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1answer
43 views

Gamma function and logarithms question

I'm trying to find $$ \int_0^\infty\ln(x)\,x^2e^{-x}\,\mathrm{d}x $$ Could anyone help explain this to me? I'm also interested in changing the $e^{-x}$ to an $e^{-ax}$. Thank you.
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23 views

Approximation using Stirling

In the article "Series evaluation of Tweedie exponential dispersion model densities" by Peter Dunn and Gordon Smyth it is stated that $$-\log\Gamma(1+j)-\log\Gamma(-\alpha j)\approx ...
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1answer
55 views

Evaluating $\frac{1}{2\pi j}\int_{c-j\infty}^{c+j\infty}x^{-s}\sigma ^{ms-m} [ \frac{\Gamma ( s )}{\Gamma ( s+2)}]^{m}ds$

I have been trying to solve the problem for $m=3$: $$f(x)=\frac{1}{2\pi j}\int_{c-j\infty}^{c+j\infty}x^{-s}\sigma ^{ms-m}\left [ \frac{\Gamma \left ( s \right )}{\Gamma \left ( s+2 \right )} \right ...
3
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3answers
103 views

Residue of $\Gamma^{2}$ and $\Gamma^{3}$

Based on wiki, the residues of $\Gamma$ at non positive integers are given by: $$\text{Res}\left ( \Gamma(z),z=-n \right )=\frac{(-1)^{n}}{n!}.$$ I have been trying to find residue for $\Gamma^{2}$ ...
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0answers
43 views

Proof of Ramanujan's Identities of Euler's Function

Consider Euler's Function defined as (and not to be confused with the totient!) $$ \phi(x) = (1-x)(1 - x^2) (1 - x^3 ) .... = \prod_{i=1}^{\infty} \left[(1 - x)^i \right] = (1;x)_{\infty}$$ ...
2
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1answer
24 views

Is there any way to extend the domain of this function through analytic continuation?

$\prod_{k=2}^x \log k=F(x)$ It looks a lot like the gamma function (a sort of logarithmic factorial), and I wonder if it can be similarly expressed as an integral or something. Any ideas?
3
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0answers
44 views

Closed form expression for a sum

I want to calculate a sum of the form $$\sum_{k=0}^m \frac{\Gamma[m+1+\alpha-k]^2}{\Gamma[m+1-k]^2}\frac{\Gamma[x+k]}{\Gamma[x]k!}$$ where $m>0$ and belongs to integers and $\alpha$ takes half ...