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

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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
24 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
57 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
32 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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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
43 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
29 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
26 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
115 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
61 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
42 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 ...
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1answer
46 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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1answer
52 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
19 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
47 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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71 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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0answers
40 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
16 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
32 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
11 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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20 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
40 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
53 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
36 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
65 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
41 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
52 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
42 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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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
53 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 ...
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3answers
101 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!) $$ \phi(x) = (1-x)(1 - x^2) (1 - x^3 ) .... = \prod_{i=1}^{\infty} \left[(1 - x)^i \right] = (1;x)_{\infty}$$ ...
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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?
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0answers
43 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 ...
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2answers
36 views

Composite Gamma function simplification

I'm running some python code using the gamma function and it involves dividing one gamma function with another. Unfortunately because both the numerator and denominator are so large, python outputs a ...
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23 views

How do I compute the limit as $N \to \infty$ of $N^{-q} \sum_{k=0}^{N-1} \frac{\Gamma(k+q)}{\Gamma(k+1)}$?

How do I compute $$\lim_{N \to \infty} N^{-q} \sum_{k=0}^{N-1} \frac{\Gamma(k+q)}{\Gamma(k+1)}$$ for $q > 0$ real?
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2answers
49 views

Calculate improper integral using Euler's integral

I have to evaluate the following integral $$\int_0^2 \frac{dx}{\sqrt[5]{x^3(2-x)^2}}$$ Thanks in advance.
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1answer
46 views

Prove equality with product of improper integrals

I have to prove the following equality: $$ \int\limits_0^{+\infty} \frac{dx}{\sqrt{\cosh x}} \cdot \int\limits_0^{+\infty} \frac{dx}{\sqrt{\cosh^3 x}} = \pi. $$ The first integral Wolfram Mathematica ...
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1answer
25 views

Convergence of $\int_0 ^{\infty}\frac{\cos t}{t^{\alpha}} dt$ related to $\Gamma$ function

I would like to show that the integral $$\int_0 ^\infty \frac{\cos t}{t^{\alpha}} dt$$ converges for $0<\alpha <1$. I already showed that it does not converge for $\alpha\leq 0$ or $\alpha \geq ...
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2answers
72 views

When is it not okay to substitute $n! = \Gamma(n + 1)$?

Let's say you just derived some formula that included integer parameters $m$ and $n$, and you wound up with something that had $m!$ or $n!$ or something similar in it, like the following example: $$ ...
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How can I get the value of k?

I have to get the value of k in this equation: $\frac{(\lambda T)^k [ln(\lambda T)-\psi(k+1)]}{\Gamma(k+1)}=0$, where $\psi$ is the digamma function. Since $\Gamma(k+1)$ is in the denominator and the ...
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3answers
129 views

Gamma function proof of gamma $\;Γ(1/2) = \sqrt \pi\;$

So our teacher doesnt use the same demonstration as most other sites use for proving that gamma of a half is the square root of pi. I dont understand the demonstration from the first step because he ...
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1answer
41 views

Calculus over integers for derivative/integral of factorial?

One usually introduces the Gamma function to define a derivative of the factorial. However couldn't one define a derivative over integers like $$ f'(n) = \frac{f(n+1) - f(n)}{1} ? $$ Such a discrete ...
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4answers
80 views

How to calculate $\int(5x)^4e^{-2x}dx$ without using IBP?

How do I integrate this? (UPDATE: Sorry, I should have clarified that I've been told to use the gamma function and not just integration by parts, also that this is a definite integral.) ...
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1answer
36 views

gamma function manipulation

For $t = 0$, let $$f_t(x) = e^{-tx}(1+\frac{x}{t})^{t^2} ; (x>-t)$$ and $0$ otherwise. a) For $s > -1$, proof that $$ \Gamma(s+1) = s^{s+0.5}e^{-s}\int^{\infty}_{-\sqrt{s}} f_\sqrt{s}(x) dx$$ ...
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1answer
14 views

Problem with proof of Poisson distribution as the difference of two gamma distributions

I found the following equations related to the Erlang distribution, ie the difference of two gamma distributions, which is a Poisson process with rate $\lambda t$: $ P(N(t)=t)=\frac { { \lambda }^{ ...