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
35 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
30 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
39 views

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

The integral: $$J=\int_0^{\pi/2}x\tan(x)^pdx$$ has the solution: $$J=\dfrac{\pi}{4\sin\left(\frac{\pi}{2}p\right)}\left[\Psi\left(\dfrac{1}{2}\right)-\Psi\left(\dfrac{1-p}{2}\right)\right]$$ where the ...
0
votes
1answer
22 views

Can't see how this equality that involves the gamma function holds

$$\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
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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
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4answers
59 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 ...
3
votes
1answer
63 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
144 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
33 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
43 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
18 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
57 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
42 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
21 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
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0answers
21 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
48 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 ...
1
vote
1answer
53 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
20 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
58 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
1answer
100 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
74 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
62 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
40 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
24 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
83 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
46 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
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votes
0answers
45 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
113 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
20 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
96 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
37 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
27 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 ...
0
votes
1answer
26 views

Integral / Gamma Expectation

I would like to solve the following integral, $\int_{0}^{\infty}\frac{\phi}{a+b\phi} \phi^{c-1}e^{-d\phi}d\phi$. Note $\phi \sim Ga(c,d)$ is a gamma distributed random variable and the integral can ...
0
votes
1answer
36 views

Question about gamma function recurrence formula.

I know that $\Gamma(k+1)=k\Gamma(k)$. But I am not sure about $2k\,\Gamma(2k)=$? Can anyone help me out with this? Is it $\Gamma(2k+1)$ or $\Gamma(2k+2)$?
0
votes
1answer
45 views

Definite integral

So I was playing around with Euler's Reflection Formula ($\Gamma(s)\Gamma(1-s)=\frac{\pi}{\sin(\pi s)}$), trying to prove it with calculus, and was able to reduce $$ ...
1
vote
2answers
128 views

Product of Gamma functions II

What is the value of the product of Gamma functions \begin{align} \prod_{k=1}^{10} \Gamma\left(\frac{k}{10}\right) \end{align} and can it be shown that \begin{align} \prod_{k=1}^{20} ...
1
vote
2answers
48 views

Product of Gamma functions I

What is the value of the product of Gamma functions \begin{align} \prod_{k=1}^{8} \Gamma\left( \frac{k}{8} \right) \end{align} and can it be shown that the product \begin{align} \prod_{k=1}^{16} ...
3
votes
1answer
48 views

Prove $\lim_{n \to \infty} \frac{\Gamma(n+1/2)}{\Gamma(n)~n^{1/2}}=1$

Prove $$\lim_{x \to \infty} \frac{\Gamma(x+1/2)}{\Gamma(x)~x^{1/2}}=1.$$ I got this problem from Probability and Statistics by Degroot & Schervish. There is a hint to use Stirling's formula ...
2
votes
0answers
32 views

Upperbound for the definite integral $f(t)=\int_{2}^{t}\frac{\sin^{2}(\pi \frac{\Gamma ^{2}(x)}{x})}{\sin^{2}(\frac{\pi }{x})}dx$

What is a good upper bound for this function? $$f(t)=\int_{2}^{t}\frac{\sin^{2}(\pi \frac{\Gamma ^{2}(x)}{x})}{\sin^{2}(\frac{\pi }{x})}dx$$ I managed to find this. $$f(t)=\int_{2}^{t}\frac{\beta ...
3
votes
0answers
34 views

Fractional derivatives of Gamma function

For integer $n \geq 0$, we have $\dfrac{d^n}{ds^n} x^s = (\ln x)^n \,x^s$. From this it follows, for example, that $$\int_0^{\infty}e^{-x}\ln^n x \,dx= \Gamma^{(n)}(0)$$ Question: is there a way of ...
2
votes
2answers
87 views

Derivative of $\Gamma$ at $1$

I've been given two definitions of the Gamma function, the integral defintion: $\Gamma(z) = \int_0^\infty t^{z-1}e^{-t}dt$ (for $Re(z)>0$) and the product definition (for $1/\Gamma$): ...
0
votes
0answers
32 views

Laguerre polynomial defined as contour integral

Let $P_n(x)=\frac{1}{2\pi i}\oint_{\Sigma}{\frac{\Gamma(t-n)}{\Gamma(t+1)^2}x^tdt}$ be a polynomial of degree $n$ where $n\in\mathbb{N}, x>0$ and $\Sigma$ is a closed contour in the $t$-plane that ...
0
votes
1answer
130 views

Approximation of $x!$ - Proof needed

By drawing a graph of the geometric derivative of $x!$, $e^{\left(\frac{\text{d}ln(x!)}{\text{d}x}\right)}$, i guessed that $e^{\left(\frac{\text{d}ln(x!)}{\text{d}x}\right)}\sim_{+\infty}(x+1/2)$. ...
1
vote
1answer
33 views

E[Log(x+a)] when x has gamma distribution

Is there a formula for this using built-in functions in matlab or mathematica like the Gamma functions or Ei's? $$\int_0^\infty \log(x+a)e^{-\alpha x}x^\beta dx. $$ Thanks.
3
votes
1answer
52 views

Is there a simpler form for $\Re \frac{\Gamma(1/2-i)}{\Gamma(1-i)}$?

Is there a simpler (i.e. manifestly real) form for $\Re \frac{\Gamma(1/2-i)}{\Gamma(1-i)}$ or $\Im \frac{\Gamma(1/2-i)}{\Gamma(1-i)}$, or more generally for $\frac{\Gamma(1/2-ia)}{\Gamma(1-ia)}$ with ...
1
vote
4answers
111 views

Some gamma function questions…

I have shown that $\Gamma(a+1)=a\Gamma(a)$ for all $a>0$. But I'd also like to show the following 2 things: 1) Using the previous fact, I'd like to show that $\lim_{a \to 0^{+}}a\Gamma(a) = ...
1
vote
2answers
54 views

Gamma Function Result

I want to know the way to prove that $$ \Gamma(n + 1/2)= \frac{(2n)! \sqrt{\pi}}{4^n n!}. $$ I tried writing term by term and it gives the result, but I want to know how to prove it without ...