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

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15 views

Finding zeros of a function involving Gamma function.

I am looking for the zeros of following function ($a$ and $b$ are real): $$ F(a,b) = 4^{a+ib} \Gamma(a+ib) \Gamma(-a) \Gamma(-ib) + \Gamma(-a-ib) \Gamma(ib)\Gamma(a) $$ and I have no idea on ...
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1answer
42 views

Alternate formula for $Γ(t + 1)$

I believe that: $$\frac{d^{n}}{dx^{n}}[x^{n}] \equiv Γ(n + 1) \equiv n!$$ Would this have any application, if it has not already been discovered, which I am almost certain that it has?
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2answers
62 views

how can I show this integral diverges?

I want to show $E(T_a)=\infty$ $$E(T_a)=\int_0^{\infty}{{x|a|}\over\sqrt{2\pi}}x^{-3/2}e^{-a^2/x}dx$$ to show this I need to show this integral diverges. I know gamma function that $$\Gamma ...
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0answers
29 views

convergence and holomorphic function - explanations and proofs

I haven't had analysis for a long time and I've forgotten plenty. Could you please explain me and prove that the following converges: $$ \sum_{n \geq 1}e^{-n^2t\pi}, t>0 c $$ and explain why the ...
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0answers
24 views

Decomposing a series

When I insert the following function \begin{equation} F(X,Y)=-\frac{1}{Y^{2/3}}\sum _{m=0}^{\infty } \frac{\Gamma \left(\frac{m+2}{3}\right)}{m! \Gamma (m+1)}\left(-\frac{X^2}{2^2 ...
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1answer
47 views

Gamma and Beta function proof.

I'm trying to proof the equality $B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ when $x,y>0,$ without using calculus in many variables. I've investigated about the topic but all references make ...
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1answer
27 views

How to obtain the Laruent expansion of gamma function around $z=0$?

I want to prove, the laurent expansion of gamma function. \begin{align} \Gamma(z) = \frac1z-\gamma+\frac12\left(\gamma^2+\frac {\pi^2}6\right)z-\frac16\left(\gamma^3+\frac {\gamma\pi^2}2+2 ...
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1answer
21 views

Gamma and Beta Functions [closed]

\begin{equation*} \int \limits _0 ^\infty x^m \mathbb e ^{-x^n} \mathbb d x = \frac 1 n \Gamma (\frac {m+1} n), \space m>-1, \space n>0. \end{equation*} \begin{equation*} \int \limits _0 ^1 ...
2
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0answers
21 views

Derivation of Gamma distribution characteristic function reference?

I was wondering if there was a derivation of the Gamma distribution characteristic function without expanding the $e^{itx}$ into an infinite summation?
2
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1answer
64 views

Concerning Hurwitz Zeta function, how to prove the following identity?

It is claimed that $$\zeta'(0,s)=\ln\left(\frac{\Gamma(s)}{\sqrt{2\pi}}\right)$$ where the derivative is meant by the first argument (as usual with Hurwitz Zeta). How to prove this? Wolfram Alpha ...
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1answer
32 views

Express each of the following integrals in terms of the gamma or beta functions and simplify when possible

$$\begin{align} \\ &= \int_{0}^{1} \ ( {(1/x) - 1} ) ^ {1/4} dx \\ &= \end{align}$$ I try to solve but not sure if correct or not and need help in these also ,I try in first one ...
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1answer
45 views

Proving $\int_0^\infty e^{-ax}x^n\,dx = \frac{1}{a^{n+1}} \Gamma(n+1)$

Prove that $$ \int_{0}^{\infty} \ e^ {-ax} x^{n} dx = \frac{1}{a^{n+1}} \Gamma(n+1) \qquad (n>-1, \, a>0). $$ My try: Let $dv = e^{-ax}$ and $u = x^n$. Then $v = -\frac{1}{a}e^{-ax}$ ...
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0answers
20 views

Can I calculate the upper limit of the definite integral in this equation?

while working on an engineering thesis I came to this equation: $$\sum_{k=1}^N\int_0^Te_k(t)dt=s$$ The $e_k(t)$ is erlangian distribution with shape k, N is an integer (its value can be assumed to any ...
2
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1answer
39 views

Series of gamma function with fixed real part and increasing imaginary part

I'm trying to evaluate the series or to pursue a upper bound, theoretically or numerically: $$ \sum_{k \ge 1} \left| \Gamma(m+2\pi ik/\log q) \right| $$ I know this series is convergent because each ...
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3answers
72 views

Show that: $\int_{0}^{\infty} \frac{\sin{x^{q}}}{x^{q}} dx = \frac{\Gamma{\frac{1}{q}}}{q-1}\cos{\frac{\pi}{2q}} \mbox{, q > 1}$

How do you show that: $$ \int_{0}^{\infty} \frac{\sin{x^{q}}}{x^{q}} dx = \frac{\Gamma{\frac{1}{q}}}{q-1}\cos{\frac{\pi}{2q}} \mbox{, q > 1} $$ Without using Gamma function?
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1answer
38 views

prove an identity involving beta function and gamma function

We know that $B(p,q)=\Gamma(p)\Gamma(q)/\Gamma(p+q)$ where $p, q>0$, and $B(p,q)$ is related to binomial coefficients if one of $p,q$ is an integer. I want to prove the following identity. ...
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0answers
39 views

Inverse Gamma function for integers (Hankel)

So I want to prove that for all integers $n \in \mathbb{Z}$ it holds that $$F(n):= \frac{1}{2\pi i} \int\limits_{\gamma}^{} s^{-n}e^{s} ds = \frac{1}{\Gamma(n)},$$ with $\gamma$ the 'Hankel'-contour, ...
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1answer
26 views

Intuitive explanation for this Gamma function identity

Wolfram Alpha says that this result is true: $$\frac{\Gamma(n+1)}{\Gamma(\frac{n}{2}+1)} = \frac{\Gamma(\frac{n}{2} +\frac{1}{2})}{\Gamma(\frac{1}{2})} \times 2^n$$ This implies a curious result for ...
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0answers
23 views

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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0answers
41 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
43 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
148 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, ...
2
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1answer
23 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) = ...
3
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0answers
163 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 $a,b \in \mathbb{C}, |z| <1$, and $- \pi < \arg(z) < \pi$. Let $C$ be the right half ...
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1answer
40 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 ...
3
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0answers
38 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 ...
4
votes
2answers
338 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 ...
3
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2answers
51 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
88 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
79 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
50 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: $ ...
0
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1answer
29 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: ...
1
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1answer
60 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
42 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
20 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 ...
1
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1answer
46 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 ...
0
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1answer
32 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
89 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 ...
4
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3answers
119 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 ...
3
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2answers
66 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
46 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 ...
2
votes
2answers
93 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 ...
0
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0answers
29 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 ...
1
vote
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 ...
0
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1answer
52 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 ...
9
votes
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 ...
0
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0answers
43 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 $n\to \infty$?