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

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Analytic continuation of Euler's reflection formula with the Gamma function

Let $\widetilde\Gamma$ be an analytic continuation of $\Gamma$ on $\mathbb C\setminus(-\mathbb N_0)$. Show that the function $$\widetilde\Gamma(z)\widetilde\Gamma(1-z)-\frac{\pi}{\sin(\pi z)}$$ ...
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1answer
15 views

Derivative of Incomplete Gamma Function

For the following incomplete Gamma function: \begin{equation} Γ(1+d,A-c \ln x)=\int_{A-c\ln x}^{\infty}t^{(1+d)-1}e^{-t}dt \end{equation} I am trying to calculate the derivative of $Γ$ with respect ...
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0answers
42 views

Xi Function on Critical Strip - Mellin Transform

Story I'm trying to prove the following identity $$\int_0^\infty \frac{\Xi(t)}{t^2 + \frac{1}{4}} \cos(xt) dt = \frac{1}{2} \pi (e^{\frac{1}{2}x} - 2e^{-\frac{1}{2}x} \psi(e^{-2x}))$$ where $$\psi(...
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0answers
13 views

Can Euler's generalization of factorial be done for double factorial?

Can Euler's generalization of factorial be done for double factorial? Euler's generalization of factorial to non-integer values is $t! =\lim_{n \to \infty} \dfrac{n!n^t}{\prod_{k=1}^n(t+k)} $. I ...
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2answers
54 views

Prove that $\Gamma(-k+\frac{1}{2})=\frac{(-1)^k 2^k}{1\cdot 3\cdot 5\cdots(2k-1)}\sqrt{\pi}$.

I was able to prove that $$ \Gamma\left (k+\frac{1}{2} \right )=\frac{1\cdot 3\cdot 5\cdots(2k-1)}{2^k}\sqrt{\pi}.\tag{$k\geq 1$}$$ using the Legendre's duplication formula. But I can't do the same to ...
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1answer
65 views

Calculate the series $\sum_{n=1}^{\infty} \frac{x^n}{(3n)!}$ [duplicate]

I can't find the series $$\sum_{n=1}^{\infty} \frac{x^n}{(3n)!}$$ But I have no idea how to find. Thanks for any hints or solutions.
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1answer
17 views

Integral involving gamma function (finding the MGF of gamma distribution)

A question on this probability paper is asking to find the moment generating function of a gamma distribution with parameters $(\alpha,\theta)$. The pdf is given as $$f_{\alpha,\theta}(x)=\frac{x^{\...
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2answers
131 views

What is $\int_0^1{\ln\Gamma(x)\sin(\pi x)}\mathop{\mathrm{d}x}$?

Hello I am stuck with this integral: $$\int_0^1{\ln\Gamma(x)\sin(\pi x)}\mathop{\mathrm{d}x}$$ My questions are: What is the integrand, $\ln(\Gamma(x)\sin(x\pi))$ or $\ln(\Gamma(x))\:\sin(x\pi)$? ...
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1answer
33 views

customer service time problem

In a disc shop two employees are working. When we get inside the shop, we see that the two employees are already serving two customers (one customer for each employee), with the service time being a ...
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1answer
44 views

Evaluating an integral in terms of the gamma function

I need to evaluate the following integral $$\int_0^\infty x^{\mu}\exp(-{\lambda}x^\kappa)\,dx$$ (where $\mu$, $\lambda$, and $\kappa$ are are all real) in terms of the Gamma function $\Gamma(t)=\int_0^...
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1answer
33 views

Integrating the lower incomplete gamma $\int_0^\infty x^{a-1}e^{-s x} \gamma(b,x) \mathrm{d}x$

I need to prove that $$\int_0^\infty x^{a-1}e^{-s x} \gamma(b,x) \mathrm{d}x = \frac{\Gamma(a+b)}{a(1+s)^{a+b}}F(1,a+b,1+b, 1/(1+s))$$ where $F(a,b,c;x)$ is the hypergeometric function. To show this,...
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3answers
278 views

Integral $\int_{-\infty}^\infty\frac{\Gamma(x)\,\sin(\pi x)}{\Gamma\left(x+a\right)}\,dx$

I would like to evaluate this integral: $$\mathcal F(a)=\int_{-\infty}^\infty\frac{\Gamma(x)\,\sin(\pi x)}{\Gamma\left(x+a\right)}\,dx,\quad a>0.\tag1$$ For all $a>0$ the integrand is a smooth ...
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2answers
66 views

Limit of gamma and digamma function

In my answer of the previous OP, I'm able to prove that \begin{align} I(a)&=\int_0^\infty e^{-(a-2)x}\cdot\frac{1-e^{-x}(1+x)}{x(1-e^{x})(e^{x}+e^{-x})}dx\tag1\\[10pt] &=\int_0^1\frac{y^{a-1}}...
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1answer
47 views

What is the Taylor series expansion of incomplete gamma function $\gamma(s,-x)$ when $x\to +\infty$?

What is the Taylor series expansion of the lower incomplete gamma function $\gamma(s,-x)$? The standard answer from wikipedia is interms of $x^k$: $$\gamma(s,-x)=\sum_{k=0}^{\infty}\frac{(-x)^se^{x}(-...
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13 views

Möbius inversion for Taylor series and simple computations related with $\sum_{k=4}^\infty\frac{\mu(k)}{k^s}$ for $\Re s>\frac{1}{2}$

I've known an example of Möbius inversion for Taylor series (see the reference in my previous recent post, Benito, Navas and Varona, Möbius inversion from the point of view of arithmetical semigroup ...
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18 views

Möbius inversion for Taylor series and the logarithmic integral

See the first paragraph in page 77 of this article from Universidad de la Rioja (in english )Benito, Navas and Varona, Möbius inversion from the point of view of arithmetical semigroup flows (...
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1answer
57 views

Integral of incomplete gamma function and limit of hypergeometric function

Let $a > 0$ and consider the integral $$\int_x^\infty \frac{\Gamma(a,t)}{t}\,dt$$ where $\Gamma(a,t)$ is the upper incomplete gamma function $$\Gamma(a,t) = \int_t^\infty x^{a-1} e^{-x} \, dx.$$ ...
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1answer
11 views

Gamma function's mean and standard deviation through shape and rate

I'm currently learning how to program neural networks. In the most basic ones, you initialize your matrix with random numbers that are distributed using Gamma function. I'm using a math library, that ...
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1answer
31 views

The Bessel function and finding expression

The Bessel Function $J_v$ of the first kind of order $v$ can be defined by the series expresion $$J_v(x)=\sum_{n=0} ^{\infty} \frac{(-1)^n}{n!\Gamma{(1+v+n)}}\left(\frac{x}{2}\right)^{2n+v}$$ (i) if ...
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1answer
31 views

How sum of exponential variables is a gamma variable [duplicate]

I have the task to calculate $P(S_{100}\geq 200)$ where $S_{100}=\sum^{100}_{i=1} X_i$ and $X_i$, $i=1,2, \cdots, 100$ are independent $exp(\lambda)$ random variables. One method is to use the fact ...
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1answer
26 views

Integral with exponential function

I'm trying to determine the order of an integral involving the derivative of the heat kernel on the real line. Based on some numerics, it appears as though the following identity holds: $$ \int_{\...
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1answer
140 views

What are some common ways to express in mathematical notation the indefinite integral of the $\Gamma$ function? [closed]

I checked on WolframAlpha (I am not good at calculating integrals) and it said there are no elementary functions to express it and gave me a horribly complex (I think... I didn't examine it very ...
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1answer
56 views

How to prove that $e^{-\gamma}=\prod_{n=1}^\infty\left(1+\frac1n\right)e^{-1/n}$

Suppose we defined the Gamma function $$\frac1{\Gamma(z)}=ze^{\gamma z}\prod_{n=1}^\infty\left(1+\frac zn\right)e^{-z/n}$$ where $\gamma$ is just a constant. I want to prove that $\Gamma(1)=1$, so I ...
3
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1answer
42 views

Detailed explanation of the Γ reflection formula understandable by an AP Calculus student

In my recent question about the Fransén-Robinson constant, an answer was given using the Gamma reflection formula. However, as an AP Calculus student, I didn't quite understand how the reflection ...
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61 views

Proving this formula for the Zeta function?

Could some one link me to a proof of this integral? $\zeta{(s)} = \frac{1}{\Gamma{(s)}}\int_{0}^{\infty} \frac{x^{s-1}}{e^x - 1} dx$ All the sites I've seen so far just introduce with the definition ...
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3answers
95 views

Evaluating an integral using the gamma function

My question regards an integral $$\int_0^\infty \frac{\sin(x^p)}{x^p}\mathrm{d}x$$ The answer should be $$\frac{1}{p-1}\cos(\frac{\pi}{2p})\Gamma(\frac{1}{p})$$ and I roughly know that I should apply ...
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1answer
98 views

How was the difference of the Fransén–Robinson constant and Euler's number found?

I recently ran across the following integral: $$ \int_{0}^{\infty}\frac{1}{\Gamma(x)}dx $$ Which I learned is equal to the Fransén-Robinson constant. On the linked wikipedia page for the Fransén-...
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0answers
32 views

Power series representation with Gamma function

This is taken from Stein and Shakarchi's Complex Analysis (Chapter 6, Exercise 4): Prove that if we take $$f(z) = \frac{1}{(1-z)^\alpha}$$ for $|z|<1$ (defined in terms of the principal branch ...
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50 views

Integrating lower incomplete gamma function raised to the power $k$

I'm trying to solve the following integral: $$\int_0^\infty \gamma(t,x)^k x^t e^{-x} \mathrm{d} x$$ I'm fighting with it for quiet a while and didn't get any result. Though, I do have the ...
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1answer
67 views

About prefactor in book's Gamma function identity

In "Mathematical Methods for Physicists" (Arfken & Weber, 7th ed.), exercise 13.1.16 says the following, Prove that $$|\Gamma (\alpha+i\beta)|=|\Gamma(\alpha )|\prod_{n=0}^{\infty}\left[1+\...
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0answers
31 views

multifactorial of non-integer

I want to calculate 12.1!!!!!! , Just for curiosity. (One of my friend texted the term to me for some complex reason.) I searched for multifactorial in terms of gamma function or equation, and found ...
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25 views

Bounding of series

Please help - I tried writing the LHS term as a sum and was hoping to find a 'known' convergent sequence that would bound it, but haven't had any luck. Let R > 1. Show that there is some M > 0 such ...
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0answers
19 views

Bound for series in Gamma proof

Please help. I am rather stuck with this problem (and our lecturer says she doesn't know how to do this either). It is part of a proof that the Gamma function is holomorphic, but was omitted from the ...
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1answer
62 views

Anti-treta function in terms of standard special functions

Define treta$^*$ function as $$ \tau(\alpha_1,\alpha_2,\alpha_3) = \iint_{0< x_1< x_2<1} x_1^{\alpha_1-1}(x_2-x_1)^{\alpha_2-1}(1-x_2)^{\alpha_3-1}\, d(x_1,x_2).\tag{1} $$ Similarly to the ...
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0answers
76 views

Irrationality of $\min$ and $\arg\,\min$ of $\Gamma|_{[1, 2]}$

The Gamma function achieves a local minimum at $x^* \approx 1.46163$, where $\Gamma(x^*) \approx 0.88560$. Can $x^*$ and $\Gamma(x^*)$ easily be proven irrational? Are they transcendental?
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1answer
71 views

Definite integral problem of $\frac{x^n}{n!}$

I want to evaluate the following definite integral. $$\int_0^\infty\frac{x^n}{n!}dn$$ Where we have $n!=\Gamma(n+1)=\int_0^\infty t^ne^{-t}dt$ so that we can have $n\in\mathbb{R}$. I don't think ...
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22 views

representation of $\zeta(s)$ with line integral

Let $0<\varepsilon<2\pi$ and $\gamma_{\varepsilon}:\mathbb{R}\rightarrow\mathbb{C}, \gamma_{\varepsilon}(t)=\begin{cases}-1-t+\varepsilon i&t\le-1\\\varepsilon\exp(\pi i(1+\frac{t}{2}))&-...
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1answer
33 views

Bound for series

Please help. I am rather stuck with this problem (and our lecturer says she doesn't know how to do this either). It is part of a proof that the Gamma function is holomorphic, but was omitted from the ...
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1answer
23 views

Differential entropy of $\Gamma$

Let $X \sim Gamma(\alpha,\beta)$ be gamma distributed random variable with probability distribution function $$ f_{X}(x)=\frac{\beta^{\alpha}x^{\alpha-1}e^{-\beta x}}{\Gamma(\alpha)},\;x>0 $$ ...
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1answer
52 views

Deriving $ \frac{1}{n^s} = \frac{1}{\Gamma(s)} \int_0^{\infty}t^{s-1} e^{-nt} \, dt $ Backwards?

Is it possible to start with $\dfrac{1}{n^s}$ and then, without knowing the Gamma function in advance, naturally (with reasons!) derive that $$ \frac{1}{n^s} = \frac{1}{\Gamma(s)} \int_0^{\infty}t^{s-...
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1answer
114 views

Is a hypersphere of non-integer dimension a fractal?

Thanks to the gamma function the formula for the surface of a unit http://mathworld.wolfram.com/Hypersphere.html $$ S(n) = \frac{2 \pi^{n/2}}{\Gamma(n/2)} $$ allows to calculate the surface of a ...
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3answers
679 views

Special Gamma function integral

I'm trying to evaluate this integral $$\int_{0}^{1} \sin (\pi x)\ln (\Gamma (x)) dx$$ and I got to the point, when I need to find $\displaystyle \int_{0}^{\pi } \sin (x)\ln (\sin (x)) dx$ but ...
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113 views

Finding the singularities and residues of a Gamma/Riemann Zeta function.

The function I have is $f(z)=\zeta(z)\Gamma(z − 1)\sin(\pi z)$ and I need to find all singularities and their residues so I can evaluate a clockwise contour integral for the contour $\left\lvert z+\...
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1answer
78 views

How to compute $\int_0^1x^a(1-x)^be^{cx}dx$?

How to compute the integral $I(a,b,c) = \int_0^1x^a(1-x)^be^{cx}dx$ ? I know that, $\int_0^1{x^a(1-x)^b}dx = \frac{\Gamma(a+1)\Gamma(b+1)}{\Gamma(a+b+2)}$. Using this result, I tried integration by ...
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93 views

Two complementary continued fractions that are algebraic numbers

Define the two similar continued fractions, $$x=\cfrac{1}{km\color{blue}+\cfrac{(m-1)(m+1)} {3km\color{blue}+\cfrac{(2m-1)(2m+1)}{5km\color{blue}+\cfrac{(3m-1)(3m+1)}{7km\color{blue}+\ddots}}}}\tag1$$...
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1answer
56 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 $\int_{0}^...
2
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1answer
29 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: $$...
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0answers
15 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 ...