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

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5
votes
1answer
87 views

Expressing $\int_{-\infty}^\infty dx/(x^2+1)^n$ in terms of Gamma function

How to prove this identity for $n>1/2$? $$\int_{-\infty}^{\infty}\frac{dx}{(x^2+1)^n}=\frac{\sqrt{\pi}\cdot \Gamma(n-\frac{1}{2}) }{\Gamma (n)}$$
1
vote
1answer
22 views

Probability: Gamma Function vs Gamma Distribution

Could someone help me with setting up the function of this question. I've been setting it up with the gamma distribution function but kept getting the wrong answer. What I did was I used the Gamma ...
1
vote
2answers
70 views

Solving integral that contain exponential function and lower incomplete gamma function

I have the following integral; $$y=\int_0^\infty\frac{e^{-xf}}{m+x}\gamma(a,hx)~dx$$ where $f,m,h\in\mathbb{R}^+$ , $a\in\mathbb{N}$ , $\gamma\left(a,h x\right)$ is the lower incomplete gamma ...
13
votes
1answer
161 views

A Gamma limit $\lim_{n\rightarrow+\infty}\sum_{k=1}^n \displaystyle \left( \Gamma\bigl(\frac{k}{n}\bigr)\right)^{-k}=\frac{e^\gamma}{e^\gamma-1}$

Show that $$\lim_{n\rightarrow+\infty}\sum_{k=1}^n \displaystyle \left( \Gamma\bigl(\frac{k}{n}\bigr)\right)^{-k}=\frac{e^\gamma}{e^\gamma-1}$$ where $\gamma$ is the Euler-Mascheroni Constant. ...
8
votes
2answers
82 views

Trigamma identity $4\,\psi_1\!\left(\frac15\right)+\psi_1\!\left(\frac25\right)-\psi_1\!\left(\frac1{10}\right)=\frac{4\pi^2}{\phi\,\sqrt5}.$

I heuristically discovered the following identity for the trigamma function, that I could not find in any tables or papers or infer from existing formulae (e.g. [1], [2], [3], [4], [5], [6]): ...
3
votes
0answers
51 views

Analytic solution for system of Digamma function equations

Peace be upon you, There is a while, I am seeking for the analytic solution of $\alpha$ and $\beta$ in this system \begin{align*} \begin{cases} \psi(\alpha)-\psi(\alpha+\beta)=c_1\\ ...
2
votes
1answer
31 views

Computing of the Gamma Function

I have stumbled upon Gamma functions when dealing with Gamma distributions on my studies with basic statistics. However, I have not understood how its computation expands factorials to real and ...
6
votes
1answer
83 views

Fixed points of the Gamma Function?

I am interested in complex values of $z$ such that $$ \Gamma (z) =z$$ Clearly, the one trivial value of $z$ is 1. Also, looking at a graph of the gamma function on the real axis, I can tell that there ...
2
votes
1answer
56 views

Asymptotics of coefficients $[x^n] \frac{1}{\Gamma(1+x)}$ as $n$ is great

I am interested in the behaviour, as $n$ is great, of the coefficients $g_n$ in the Maclauren expansion of $\displaystyle \frac{1}{\Gamma(1+x)} $. We have $$ \frac{1}{\Gamma(1+x)}=\sum_{n=0}^\infty ...
9
votes
1answer
293 views

Complex Factorial Equaling One

For what complex values of $z$ is $$z! =1? $$ Are they even all known? Are there finitely many or infinitely many? (Yes, the trivial $z$ are 0 and 1. )
10
votes
3answers
160 views

Scary contour integral, but is also an integral representation for $\Gamma$-function

This problem is supposed to be from an old Acta Mathematica volume I circa 1880's, and is attributed to Bourguet. By using a parabola with its focus on the origin as a contour, show that: ...
2
votes
2answers
26 views

Convergence of $\Gamma(z)=\lim_{n\to\infty}\frac{n^zn!}{\prod_{m=0}^n(z+m)}$

This question is a place to store proofs of the convergence of Euler's product formula for the gamma function: $$\Gamma(z)=\lim_{n\to\infty}\frac{n^zn!}{\prod_{m=0}^n(z+m)}$$ which is convergent for ...
1
vote
1answer
39 views

Approximation for lower incomplete gamma function

Does any one know approximations for the lower incomplete gamma function $\gamma(a,bx)$. The problem is this: I want to find the quantile function for the CDF of the gamma distribution. The CDF of the ...
1
vote
2answers
31 views

Integral related to the gamma function: $\int_0^{\infty} y^{\alpha-1}e^{-y}dy=(\alpha-1)\int_0^{\infty} y^{\alpha-2}e^{-y}dy$

I have difficulty with the gamma function: $\int_0^{\infty} y^{\alpha-1}e^{-y}dy=(\alpha-1)\int_0^{\infty} y^{\alpha-2}e^{-y}dy$ How do we go from left to right? Thanks!
6
votes
1answer
63 views

A generalization of Bell numbers to arbitrary complex arguments

For $n\in\mathbb N$, the Bell number $B_n$ is a number of ways to partition the integer range $[1,\,n]$ into pairwise disjoint non-empty subsets. E.g. $B_3=5$ because ...
2
votes
1answer
38 views

Show that $\frac{\Gamma(\frac 1 3)^2}{\Gamma(\frac 1 6)}=\frac{\sqrt {\pi}\sqrt[3] 2}{\sqrt 3}$.

Show that $$\frac{\Gamma(\frac 1 3)^2}{\Gamma(\frac 1 6)}=\frac{\sqrt {\pi}\sqrt[3] 2}{\sqrt 3}$$ Since there's $\sqrt {\pi}$, I suspect I have to related it to $\Gamma(1/2)$. Please give me some ...
1
vote
0answers
48 views

A sum with binomial coefficients in the numerator and denominator.

I am struggling with a combinatorial sum as apart of a long statistical-mechanics derivation. I would appreciate any help. I seek the result of the following summation, for integer $\ell,n$, and ...
2
votes
1answer
59 views

Evaluation of the definite integral: $\int_0^{\pi/2}x\tan(x)^pdx$

The integral: $$J=\int_{0}^{\pi/2}x\,\tan^{p}\left(x\right)\,{\rm d}x$$ has the solution: ...
0
votes
1answer
38 views

Understanding an equality that involves the gamma function

$$\frac{1}{\sqrt{2\pi}}\sigma^m \Gamma\left(\frac{m+1}{2}\right) 2^{\frac{m+1}{2}} = 2^{\frac{m}{2}}\sigma^m \left(\frac{m-1}{2}\right) \left(\frac{m-3}{2}\right)\cdots \left(\frac{3}{2}\right) ...
1
vote
1answer
42 views

Gamma function and Gauss sums

In this Wikipedia article appears this : "Gauss sums are the analogues for finite fields of the Gamma function." What was the relation between gamma functions and non-finite fields?
3
votes
4answers
62 views

Elementary Proof of sum of alternating factorials

so I came across the rather interesting identity $$ \sum_{i = 0}^{\infty}\left[\sum_{j=0}^{i}\left[\frac{(-1)^j}{j!} \right] \prod_{j=0}^{i}\left[ (x-j)\right] \right] = x! $$ For positive integers ...
4
votes
1answer
70 views

Is this another version of the gamma function?

I know that $\Gamma \left( x \right) $ is the unique function on $x \in (0, \infty)$ such that $f \left( 1 \right) =1$ $f(x+1)=xf(x)$ ${\frac {d^{2}}{d{x}^{2}}}ln(f \left( x \right))>0$ ...
7
votes
4answers
154 views

A closed form of $\sum_{k=0}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)$

I am looking for a closed form of the following series \begin{equation} \mathcal{I}=\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right) \end{equation} I have no idea how to ...
1
vote
0answers
36 views

Gamma Function Removable Singularity

So apparently: xΓ(x) = Γ(x+1) But when I plug them both into Mac's Grapher, Γ(x+1) is defined at 0 (it equals 1), whereas xΓ(x) is not. Can I define 0 * Γ(0) to ...
2
votes
0answers
51 views

Factorial of Complex Values

Since the gamma function is an analytic continuation of the factorial function, we can find the factorial of complex values. How does one go about doing so? I've looked far and wide on the internet ...
2
votes
1answer
21 views

A formula occurring in Dirichlet's proof of the infinity of primes in an AP.

While studying Dirichlet's proof of an infinity of primes in any AP with first term and common difference coprime, the formula below involving the gamma function was quoted as being well known. ...
2
votes
1answer
63 views

Integral involving Gamma Function

I am solving the following integral: $$ \int_{-1}^{K} u^B e^{-u} du $$ The solution of the integral is a lower incomplete Gamma Function if -1 is replaced with 0. Can anybody help me in solving the ...
3
votes
2answers
46 views

Do the centroid of a unit n-hemisphere and that of the whole n-sphere coincide when $n \to \infty$?

It is known that the distance between the centroid and the center of a unit semicircle is $\displaystyle\frac{4}{3\pi}$, whereas that of a unit hemisphere is $\displaystyle\frac{3}{8}$. I am ...
-1
votes
1answer
26 views

Syntax of Upper Incomplete Gamma Function in MATLAB [closed]

How is the Upper Incomplete Gamma Function implemented in MATLAB? I am confused about the syntax and it is not properly explained in MATLAB documents. If the limit is from x to $\infty$ , and i have ...
0
votes
1answer
44 views

Gamma distribution power series identity

As part of giving a computationally efficient means of calculating the gamma distribution, Williams in 'Weighing the Odds', pg 149 asserts the following identity, where $\Gamma$ denotes the Gamma ...
0
votes
0answers
22 views

Gamma function whose argument is a reciprocal power with integer base and exponent

Consider the analytic continuation of the factorial function $n!$ given by $\Gamma(z)$ (note $n!=\Gamma(n+1)$), and suppose $z=a^{-n}$, where $a,n\in\mathbb{N}$ are positive integers. Are there any ...
0
votes
1answer
51 views

Integrate the integral. Where $({x_1},{y_1}),({x_2},{y_2})\in[0,a] \times[0,b],x_1<x_2,y_1<y_2$

What is the integration of the below integral? $\|N(u)(x_1,y_1)-N(u)(x_2,y_2)\|\le\|\mu(x_1,y_1)-\mu(x_2,y_2)\|+L_1\|u(x_1,y_1)-u(x_2,y_2)\|+\|\dfrac{L_2}{\Gamma (r_1)\Gamma ...
0
votes
1answer
58 views

Evaluating $\dfrac{1}{\Gamma (r)}\int_{0}^{x}(x-t)^{\alpha -1}t^{\lambda}dt$ [closed]

How can I evaluate the following integral $$\frac1{\Gamma(r)}\int_0^x(x-t)^{\alpha-1}t^\lambda\ dt$$
0
votes
0answers
21 views

zeros of Incomplet Gamma function

for which values of complex variable $z$ let us getting the zeros of incomplet gamma function ($\Gamma(0.5,z)$) ? I would be interest for any replies or any comments
1
vote
1answer
60 views

Is there a nice solution to the equation $\Psi(x)=\ln(\pi)$ with a positive real $x$?

I tried to find a nice solution to the following equation: $$ \Psi(x)=\ln(\pi) $$ with $x\in\Bbb R_{\ge0}$ and where $\Psi(x)=\frac{\Gamma'(x)}{\Gamma(x)}$. Is there a nice expression for x satisfying ...
1
vote
2answers
145 views

Evaluating $\lim_{s\to0}\sin(s)\,\Gamma(s)$

How do I evaluate: $\displaystyle\lim_{s\to0}\sin(s)\Gamma(s)$ kk, i understand now.
1
vote
2answers
75 views

Compute $\int_{0}^{\infty}e^{-tz}(z+d)^{n-1}dz$ as a function of $\Gamma(n)$

Is it possible to compute this integral $$\int_{0}^{\infty}e^{-tz}(z+d)^{n-1}dz$$ as a function of complete gamma $\Gamma(n)$. If possible, I'm looking for a closed form solution. Thanks!
1
vote
1answer
64 views

Gamma functions

This question is from this following derivation from pages 204-205 of PDE Evans, 2nd edition. This is from the same proof of Example 9 on those pages of the textbook, and I asked a question about that ...
2
votes
0answers
42 views

Why does the Riemann Xi function $(\xi(s))$ have order of growth 1

Why does $s(s-1)\xi(s)$, have order of growth 1? In other words, why is it that $\forall \epsilon > 0 $ $\exists A_{\epsilon},B_{\epsilon} \in \mathbb R_+$ so that $\forall s \in \mathbb C$, ...
1
vote
1answer
41 views

Evaluating improper integral expression

Can anybody please guide me in evaluating this expression, for my research work. I have tried a lot but in vain. Both the integrals involve gamma functions and even wolfram said that the time limit ...
0
votes
1answer
26 views

Two definitions of the Incomplete Gamma Function - are they equivalent?

From Loss Models, 4th ed., by Klugman et al.: Definition 5.5 The incomplete Gamma function with parameter $\alpha > 0$ is denoted and defined by $$\Gamma\left(\alpha ; x\right) = ...
1
vote
1answer
86 views

Duplication formula for gamma function

Using the Weierstrass definition for $\Gamma(x)$ and $\Gamma\Big(x + \frac12\Big)$, how can I prove the duplication formula? This is problem $10.7.3$ in the book Irresistible Integrals, by Boros and ...
0
votes
2answers
47 views

Values of Incomplete gamma function

Look this claim : Does $\Gamma(0.5,-x^2)= i\alpha$, for $x$ large real number? i=unity imaginary part $\alpha$ is real number I would like someone to prove me this if it's a true claim
0
votes
0answers
53 views

Solving integral that contain upper incomplete gamma function, exponential, and powers

I have this integration formula; $ f=\int\limits_{0}^{\infty}\frac{e^{-b~z}}{\sqrt{z}} \Big(\frac{\beta}{\beta+z}\Big)\Big(\frac{\beta+z}{z}\Big)^L ...
4
votes
2answers
121 views

Euler's limit formula for the factorial function

In Euler's first (known) letter to Goldbach on October 13, 1729 he mentions an infinite product that interpolates the factorial function: ...
0
votes
0answers
22 views

Simplifcation involving Upper and Lower Incomplete Gamma Function

What is the simplifcation of the following mathematical equation? $$ \gamma(B+1,N) - \Gamma(B+1,N) $$ where B and N are variables quantity, and $\Gamma(x), \gamma(x)$ are the Upper and lower ...
1
vote
2answers
119 views

Evaluating integral using Beta and Gamma functions.

$$\int_0^{\infty} x^{-3/2} (1 - e^{-x})\, dx$$ Evaluate the above integral with the help of Beta and Gamma functions. I'm badly stuck. I'm getting Gamma of a negative number and have no clue how ...
1
vote
1answer
48 views

Prove the following identity (gamma function)

Prove the following identity $$\sum _{i=0}^{k} (-1)^{i} \binom{\alpha}{i} = \frac{\Gamma (k+1-\alpha)}{\Gamma(1-\alpha) \Gamma(k+1)}$$ I tried to expand the left side $$\binom{\alpha}{0} - ...
3
votes
0answers
44 views

Partial Fractions Decomposition of the Gamma Function

I'm currently dealing with a problem my professor raised (since I just studied the Mittag-Leffler's Partial Fractions Theorem). The problem is to derive a partial fractions decomposition of the Gamma ...
0
votes
2answers
28 views

Gamma and Exponential distribution question?

The working time of one bank has an exponential distribution with a parameter λ=0.1 (in minutes). You came in the bank, but there were already 35 people before you. What's the probability that all of ...