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

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2
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1answer
41 views

Show that $\int_{0}^{\infty}x^ne^{-x}\,dx=n!$ by differentiating the equality $\int_{0}^{\infty}e^{-tx}\,dx=\frac{1}{t}$

I'm learning about measure theory, specifically Lebesgue integral, and need help with the following problem: Show that $\int_{0}^{\infty}x^ne^{-x}\,dx=n!$ by differentiating the equality ...
1
vote
1answer
22 views

Is $\Gamma (0)^{\Gamma (0)^{-1}}$ defined at negative integers / $0$?

Is $x$ here undefined? $$x=Γ(0)^{(1/Γ(0))}$$ I'm aware that for the gamma function: $$\lim_{n\to^{+}0} (Γ(n)) = +\infty $$ $$\lim_{n\to^{-}0} (Γ(n)) = -\infty $$ But for $x$, the limits seem to be: ...
0
votes
0answers
9 views

Infinite Sum of Incomplete Gamma (with “z” varying)

I am trying to see whether the following series can be "reduced" or re-written in terms of well-known functions. This seems different from other questions on here in that the summation is going ...
2
votes
1answer
15 views

How are Gauss sums the analogue of the gamma function for finite fields?

I've seen this statement on the internet in a few places and I don't really get the connection. Would anyone mind fleshing out the details? Thanks.
1
vote
1answer
40 views

Does $\int f(u) \, du \int g(v) \, dv = \int \int f(u) g(v) \, du \, dv $?

I saw a proof of $\Gamma (\frac{1}{2})=\sqrt{\pi}$ that had the step $$\left[ \Gamma\left(\frac{1}{2}\right) \right]^2 = \left\{2 \int_0^\infty e^{-u^2} \, du\right\} \left\{ 2 \int_0^\infty ...
0
votes
0answers
17 views

Finding the survival and distribution function of a system.

We have a random variable $X\sim Gamma(3,c)$, so that means $f(x)=\frac{c^3}{\Gamma(3)}x^2e^{-cx} ; \ x>0$, with $c$ being appropriately selected scale parameter. We also have $P(U_2 \leq x)=x^2$ ...
3
votes
3answers
90 views

Prove that $2\int_0^\infty \frac{e^x-x-1}{x(e^{2x}-1)}dx =\ln(\pi)-\gamma $

I have to prove that given $\gamma=0.577216\ldots$, the Euler-Mascheroni constant, and $\pi=3.14159\ldots$, we have: $$2\int_0^\infty \frac{e^x-x-1}{x(e^{2x}-1)}dx =\ln(\pi)-\gamma $$
0
votes
0answers
7 views

Deformation of Gamma function integral contour

Terence Tao has described the gamma function as the inner product of a multiplicative and an additive character with respect to the Haar measure on $\Bbb R^+$. The gamma function is defined as ...
0
votes
0answers
28 views

Eigenvalue of Integral Operator and Gamma Function

$''$ Prove that the following integral operator $ Ku(x) = \int_{0}^{ \infty } \ e ^{-xy} u(y) dy $ has as eigenfunction the $ φ_α(x) = \sqrt {Γ(α)} x^{-α} + \sqrt {Γ(1-α)} x^{α-1} $ for $ ...
1
vote
1answer
42 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 ...
0
votes
2answers
15 views

How can I calculate a future random date/time with a probability distribution like a normal distribution - gamma distribution or similar?

I need to write some code that calculates a future random date/time (i.e. essentially a period of time), and I'm looking for an appropriate probability distribution and function I can use to transform ...
2
votes
0answers
53 views

Sum of Gamma Function Residues

I was exploring Cauchy's residue theorem with the gamma function and came across an interesting identity. Consider $$\int_{C_R} \Gamma(z) \, dz $$ Over the complex plane where $C_R$ is the curve ...
0
votes
1answer
52 views

Taylor series has zero convergence radius?

Let $$f(x):=\sum_{n=0}^{\infty} \frac{f^{n}(0)x^n}{n!}$$ where the $$|f^{n}(0)| \le C\frac{\Gamma(\frac{n+1}{\alpha})}{\alpha^{\frac{n+1}{\alpha}+1}}$$ for a constant $C>0$ and $\alpha>0$. Does ...
1
vote
0answers
42 views

What's about $\sum_{n=1}^\infty e^{-p_n u}$, where $p_n$ is the nth-prime number?

I am assuming that the following function, for which I am asking as reference request, should be known in the literature, since Glaisher studied the Prime Zeta Function, and my computation is the ...
0
votes
1answer
32 views

Evaluating sum over partial gamma function

I am trying to calculate the expectation value of some quantity and it eventually boils down to the following sum (and another similar one): $$ \sum_{n=0}^{\infty} \frac{\beta^n ...
2
votes
0answers
49 views

Regularizing the sum of all factorials

Consider the series $$\sum_{n=0}^\infty n! = 0! + 1! + 2! + 3! + 4! + \ldots = 1 + 1 + 2 + 6 + 24 + \ldots$$ This series clearly diverges. Now, given that the Gamma function is defined by $$n! = ...
2
votes
1answer
23 views

A change of variables in Riemann's proof of the functional equation of $\zeta(s)$

In Riemann's functional equation proof it says: From $$\Gamma(s)=2\int_0^\infty e^{-x^{2}}x^{2s-1}dx,$$ after variable substitution, we get $$\Gamma\left(\frac{s}2\right)=n^{s} ...
4
votes
2answers
106 views

The Gamma function has no zeros

How can I prove the Gamma function has no zeros in its holomorphy domain $\Bbb C\setminus\Bbb Z_{\le0}$ using only its integral definition $\Gamma(z)=\int_0^{+\infty}t^{z-1}e^{-t}\,dt$ valid when $\Re ...
1
vote
0answers
36 views

Is the gamma function a solution to this problem?

A real differentiable function on $\mathbb{R}^+$ satisfying $f(x) = x!$ for $x\in\mathbb{N}$ and having minimal derivative $\left|\frac{\partial f}{\partial x}\right|$ everywhere, say in the sense ...
0
votes
0answers
15 views

Show $\lim\limits_{p\rightarrow\infty}\frac{1}{p}\beta\left(\frac{1}{p},\frac{1}{p}+1 \right)=1$

Show $\lim\limits_{p\rightarrow\infty}\frac{1}{p}\beta\left(\frac{1}{p},\frac{1}{p}+1 \right)=1$ So $\frac{1}{p}\beta\left(\frac{1}{p},\frac{1}{p}+1 ...
2
votes
1answer
107 views

Prove that $\lim_{\ r\ \to \ \infty} \dfrac{r! r^x}{x(x+1)(x+2) \dots (x+r)} = \int_{0}^{\infty} t^{x-1} e^{-t} dt $

From Havil & Dyson, "Gamma: Exploring Euler's Constant", section 6.1 I can't prove the following Euler's theorem : ... on 13 October 1729, Euler had already proposed to Goldbach the ...
5
votes
1answer
54 views

How to prove that $\frac{\Gamma(3/2)}{3\Gamma(11/6)}=\frac{\sqrt{3}\,\Gamma(1/3)^2}{10\pi\sqrt[3]{2}}$?

My task is to show that: $$\frac{\Gamma(3/2)}{3\Gamma(11/6)}=\frac{\sqrt{3}\,\Gamma(1/3)^2}{10\pi\sqrt[3]{2}}.$$ I'm trying to show this using properties of the gamma function, but it seams every ...
3
votes
1answer
37 views

Is $\Gamma(\alpha, 0, x)$ log-concave as a function of $x$ (for fixed $\alpha$)?

The incomplete Gamma function is defined as: $$\Gamma(\alpha, 0, x) = \int_0^x t^{\alpha - 1} e^{-t} dt$$ Let $\alpha \ge 1$ be a fixed number. Is $\Gamma(\alpha, 0, x)$ log-concave, as a function ...
0
votes
0answers
57 views

conjectured general continued fraction for the quotient of gamma functions

Given complex numbers $a=x+iy$, $b=m+in$ and a gamma function $\Gamma(z)$ with $x\gt0$ and $m\gt0$, it is conjectured that the following general continued fraction which is symmetric on $a$ and $b$ is ...
0
votes
0answers
42 views

How to prove that arc segment vanishes

I have this integral: $$iNx^a\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{x^{Ne^{i\theta}}e^{i\theta}}{\Gamma(Ne^{i\theta}+a-m)\zeta(2Ne^{i\theta}+2a+n)\sin (\pi \left(a+Ne^{i\theta}\right))} ...
0
votes
0answers
15 views

Gamma function expansion

In one of the most popular regularization schemes in QFT---dimensional regularization---the following formula is often used: $\Gamma(−k + \varepsilon) = \frac{(−1)^k}{k!} \left[\frac{1}{\varepsilon} + ...
0
votes
1answer
48 views

Show $\int\limits_0^1\sqrt{1-t^3}\,dt = \frac{\sqrt{3}\cdot\Gamma(\frac{1}{3} )^3}{ 10\pi\sqrt[3]{2}}$

Show $\int\limits_0^1\sqrt{1-t^3}\,dt = \frac{\sqrt{3}\cdot\Gamma(\frac{1}{3} )^3}{ 10\pi\sqrt[3]{2}}$ I know that $\Gamma(1/6)=\frac{\sqrt{3}}{\sqrt[3]{2}\sqrt{\pi}}\cdot\Gamma(\frac{1}{3} ) ^2$ so ...
1
vote
1answer
41 views

$V_0=\frac{\rho_0}{4\pi \epsilon_o}\iiint_0^{\infty}\frac{e^{-(x^2+2y^2+2z^2)}}{\sqrt{x^2+y^2+z^2}} dxdydz$

Here is what i'm given: "The so called Schwinger parametrization is based on identities like the following: $$\frac{1}{\sqrt{R}}=\frac{2}{\sqrt{\pi}}\int_0^{\infty}e^{-Ru^2}du$$ The first octant of ...
2
votes
1answer
42 views

$\int_{-\infty}^{\infty} e^{-(x^2-2)^2-(\frac{1}{x^2}-2)^2}dx$

Assuming that the integrals below exist, $$\int_{-\infty}^{\infty} f(x-\frac{1}{x})dx=\int_{-\infty}^{\infty}f(x)dx$$ Use the above relation to evaluate the following integral: ...
4
votes
1answer
65 views

Asymptotic behavior of an integral involving the gamma function

I'm trying to obtain an asymptotic large-$k$ approximation for the integral $$I(k) := e^{-k^2}\int_0^1 \frac{(1 + \xi^2)\Gamma(0, \xi^2 k^2) - 2\Gamma(0, k^2)}{1 - \xi} d\xi$$ where $\Gamma$ is the ...
2
votes
2answers
62 views

Upper bound on ratio of incomplete Gamma function and Gamma function $\frac{ \Gamma \left( x; a\right)}{\Gamma(x)}$

I am trying to find a tight upper bound the following expression \begin{align} \frac{ \Gamma \left( x; a\right)}{\Gamma(x)} \end{align} where $\Gamma \left( x; a\right)$ is incomplet Gamma function ...
2
votes
2answers
31 views

Gamma representation of certain sequence

I'm trying to find a gamma rep for $ 15 \cdot 13 \cdot 11 \cdot 9 \cdot 7 \cdot ... $ Steps so far: It's a simple sequence of $ n \cdot (n-2) \cdot (n-4) \cdot (n-6) \cdot (n-8)... $ and so on. ...
1
vote
2answers
37 views

Show that the second derivative $\Gamma''(x)$ is positive when $x>0$

Let $\Gamma(x)=\int_0^{\infty}t^{z-1}e^{-t}dt$. I know that the first derivative is positive, since $\Gamma(x)$ is increasing when $x>0$, but I don't know how to show that the second derivative is ...
3
votes
1answer
29 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 ...
0
votes
1answer
25 views

Infinite sum over Gamma functions?

I am having quite a bit of trouble understanding this sum. Can someone explain to me exactly how to this from 1 to 3,very easily way? Question its from this webpage Thanks.
0
votes
0answers
18 views

Gamma duplication formula via Hadamard

The Gamma duplication formula reads $$ F(s) := \frac{\Gamma(s)\Gamma(s + 1/2)}{\Gamma(2s)} = \sqrt{\pi} 2^{1 - 2s} $$ This is an entire function of order 1. Also, it does not vanish anywhere. Hence ...
0
votes
0answers
22 views

How to prove $\sum_{n\geq 0}{\frac{\Gamma(n+2+\alpha)}{n!\Gamma(2+\alpha)}z^n}=\frac{1}{(1-z)^{2+\alpha}}$

Let $z\in \{z\in\mathbb{C}:|z|<1\}, \alpha>-1,\Gamma(s)$ is the gamma function. How to prove $\sum_{n\geq 0}{\frac{\Gamma(n+2+\alpha)}{n!\Gamma(2+\alpha)}z^n}=\frac{1}{(1-z)^{2+\alpha}}$ ? If ...
0
votes
1answer
10 views

How to prove the product representation of Baenes G-function

How to prove this formula of Barnes G-dunction $$G(z+1)=(2\pi)^{z/2}\text{exp}\left(-\frac{z(z+1)}{2}- \frac{\gamma z^{2}}{2}\right)\, ...
0
votes
1answer
47 views

How to calculate $\int_{0}^{\infty}\frac{dx}{\sqrt{x^{7}}e^{x}}$?

I am trying to calculate the integral $\int_{0}^{\infty}\frac{dx}{\sqrt{x^{7}}e^{x}}$. I worked out the answer to be $\Gamma[-\frac{5}{2}]$, However when I used mathematica to check my answer it says ...
1
vote
0answers
12 views

Sum of Gamma functions (product pairs)

This is my first time asking a question on stackexchange. Is there an analytical expression for the following summation of Gamma functions? $\sum_{t=0}^m \Gamma (A + t) \Gamma (B + m -t) = ?$ for ...
2
votes
1answer
71 views

Evaluating the integral $\int_{-\infty}^{\infty} \frac{(1-ix)^{n-s_1}}{(1+ix)^{n+s_2}} dx$.

How can one show that for $s_1,s_2 \in \mathbb{C}$ $$ \begin{aligned} \int_{-\infty}^{\infty} \frac{(1-ix)^{n-s_1}}{(1+ix)^{n+s_2}} dx = & 2\sin(\pi(n+s_2))2^{1-s_1-s_2} ...
4
votes
2answers
88 views

Show that $\frac{\Gamma(n+k)}{\Gamma(n)}\sim n^k$ for large values of n

In order to prove the above result I proceeded as follows: We know that: $\Gamma(n)=(n-1)\Gamma(n-1)$ Using this fact, we have: $\Gamma(n+k)=(n+k-1)\Gamma(n+k-1)\\=(n+k-1)(n+k-2)\Gamma(n+k-2)\\ ...
2
votes
2answers
84 views

How to obtain a close form/fast expression for this integral?

I have the following integral: $$ I_p = \int_0^1 x^{p-1}(ax + b) e ^{-(ax + b)^2/N}dx$$ where $$ \{x,a,b\} \in (-\infty,\infty), \; N \in (0,\infty), \; p \in \mathbb Z, p \geq 1 . $$ I want to ...
0
votes
0answers
29 views

Gamma functions and Binomial Sums

I have a question regarding the properties of gamma functions. If I am given $$\frac{\Gamma(n+1)}{\Gamma(x+1)\Gamma(n-x)}$$ would that be equal to $$\frac{n!}{x!(n-x-1)!}\space $$
1
vote
1answer
54 views

Computing integral $\int_{-\infty}^{\infty} \frac{dx}{(ax^2 + b)^k}$

I know this is a difficult integral to compute, and I know that the answer is: $\int_{-\infty}^{\infty} \frac{dx}{(ax^2 + b)^k} = \frac{\sqrt\pi \Gamma(k-\frac{1}{2})}{\Gamma(k)}\frac{1}{\sqrt a ...
2
votes
1answer
32 views

Verification of the proof that Gamma functions has no zero

https://proofwiki.org/wiki/Zeroes_of_Gamma_Function Note that $\Gamma(z)= \frac{1}{z} \prod_{n=1}^\infty \frac{(1+1/n)^z}{1+z/n}$. Suppose $\Gamma(z)=0$ for some $z$ in the domain. The proof in the ...
3
votes
1answer
51 views

Deriving Stirling's approximation formula via the definition of the Gamma function

In my asymptotic analysis and combinatorics class I was asked this question: We first remember the definition f the Gamma function $ \Gamma(n+1) = n! = \int_{0}^{\infty} t^{n} e^{-t} dt $ and ...
1
vote
1answer
52 views

Is Binomial:Gamma ever an integer?

We consider: $$\dfrac{\Gamma(n)}{\Gamma(k)\Gamma(n-k)}\quad\quad[1]$$ for $n,k\in\mathbb{R}$. Is $[1]$ ever an integer, except for the obvious?
1
vote
1answer
185 views

Proof that these two expressions are equivalent

I'm looking for algebraic proof that $$ y=\lim_{N \to \infty}\frac{\prod\limits_{n=0}^{N}x-n}{\prod\limits_{n=0}^{N-\lfloor x\rfloor}{x-\lfloor ...
2
votes
1answer
49 views

Valid Induction Argument involving Closed Form for $\Gamma(1/2-n)$

Given $\Gamma\left(\frac1{2}\right)=\sqrt{\pi}$, $$\sqrt{\pi}=\Gamma\left(\frac1{2}\right)=\Gamma\left(-\frac1{2}+1\right)=-\frac{1}{2}\Gamma\left(-\frac1{2}\right)$$ and so ...