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

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3
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2answers
56 views

Proof of Functional Equation Zeta

$$ \pi^{-s/2}\zeta(s)\Gamma(s/2)=\pi^{-(1-s)/2}\zeta(1-s)\Gamma((1-s)/2) $$ (That's the equation that want to prove) Hello guys, so I'm trying to prove the functional equation of Riemann Zeta, ...
3
votes
2answers
65 views

How to calculate $\displaystyle \lim_{\alpha \to -1}\left(2\Gamma(-\alpha-1)+\Gamma\left(\frac{\alpha + 1}{2}\right) \right)$

How can I compute the following limit? $$\displaystyle \lim_{\alpha \to -1}\left(2\Gamma(-\alpha-1)+\Gamma\left(\frac{\alpha + 1}{2}\right) \right)$$ The answer appears to be about -1.73 I would be ...
4
votes
1answer
63 views

Yet another Gamma function approximation

I know I have asked a similar question a couple of days, ago, but I still have a problem. I need a upper bound for: $$ ...
11
votes
3answers
239 views

Intuition behind $(-\frac{1}{2})! = \sqrt{\pi}$

It can be shown that using the definition of the Gamma function as: $$\Gamma(t) = \int_0^\infty x^{t-1} e^{-x} dx $$ that $$\Gamma(\tfrac{1}{2}) = \sqrt{\pi}$$ or slightly abusing notation, that ...
1
vote
1answer
30 views

Lower and upper bound of the Stirling's approximation

Perhaps everybody has heard of the Stirling's approximation, namely: $$ \Gamma(z)\approx\sqrt{\frac{2\pi}{z}}\left(\frac{z}{e}\right)^z $$ Thus (the very basic example): $$ ...
0
votes
2answers
62 views

Approximation of the Gamma function

I am having trouble obtaining a lower bound for the following formula: $$ \ln\frac{\Gamma\left(\frac{x}{3}\right)}{\Gamma\left(\frac{x}{4}+1\right)\Gamma\left(\frac{x}{12}+1\right)}. $$ I tried using ...
4
votes
2answers
35 views

How to get to this equality $\prod_{m=1}^{\infty} \frac{m+1}{m}\times\frac{m+x}{m+x+1}=x+1$?

How to get to this equality $$\prod_{m=1}^{\infty} \frac{m+1}{m}\times\frac{m+x}{m+x+1}=x+1?$$ I was studying the Euler Gamma function as it gave at the beginning of its history, and need to ...
3
votes
0answers
31 views

Evaluating Elliptic Integrals in terms of Gamma Function

Some complete elliptic integral of first and second kind $E(k)$ and $K(k)$ can be evaluated for some particular values of $k$ in terms of Euler Gamma function. For example, for $k = \sqrt{2}/2$, ...
4
votes
2answers
85 views

Evaluate the integral $\int_0^\infty x^{t-1}e^{-\beta x}dx$

I want to evaluate the following integral $$\int_0^\infty x^{t-1}e^{-\beta x}dx$$ where $\beta$ is a complex number. Now, if $\beta$ was real, we could just set $y = \beta x$ and we will reduce to ...
3
votes
4answers
98 views

Why does $\sqrt{3x} \left(\dfrac {x}{2} \right)^x$ approximate $x!$ pretty well?

I was just messing around and trying out things in the desmos calculator and found that $\sqrt{3x} \left(\dfrac {x}{2} \right)^x$ is pretty close to $x!$ most of the time, here is a graph. Why does ...
0
votes
0answers
13 views

constrained optimization including sum of two upper incomplete gamma function in both fitness function and constraint

i'm trying to solve this constrained optimization problem the constraint is $$\zeta=\frac {\Gamma \left(M,\frac {\lambda}{\left |\sum_{i=1}^Nw_i*h_{ei} \right|^2 + \sum_{i=1}^N\left |w_i \right|^2} ...
9
votes
2answers
216 views

Fourier transform of $\Gamma \left (\frac{1}{2}-i \frac{p}{2 \pi} \right) /\sqrt{ \cosh(p/2)}$

Is it possible to compute the following Fourier transform analytically? $$\psi(x) = \frac{1}{\sqrt{4 \pi}}\int \Gamma \left (\frac{1}{2}-i \frac{p}{2 \pi} \right) \frac{e^{i p x}}{\sqrt{ \cosh(p/2)}} ...
11
votes
1answer
147 views

Proof that the factorial is nonelementary

Is there a proof that the factorial function $!:\mathbb N\to\mathbb N$ is nonelementary? If it were equal to an elementary function (call it $P(n)$), then it would extend the factorial function to ...
1
vote
1answer
33 views

Is $\Gamma(\alpha k+1)t^{\beta k}-\Gamma(\beta k+1)t^{\alpha k}>0$ true?

$$\Gamma(\alpha k+1)t^{\beta k}-\Gamma(\beta k+1)t^{\alpha k}>0$$ where $0<\alpha<\beta,$ $0<k,t$, $\Gamma(z)$ is a gamma function, $\int_0^\infty t^{z-1}e^{-t}dt$, $Re$ $z>0$. I ...
1
vote
2answers
26 views

Simplifying Series of Gamma Function

I am trying to simplify summation of gamma function $$\sum_{k=1}^\infty\frac{\Gamma(k+b)}{(k-1)!\Gamma(b)} = \sum_{k=1}^\infty\frac{\Gamma(k+b)}{\Gamma(k)\Gamma(b)} , b > 1$$ I need ...
4
votes
2answers
152 views

Help with C is Euler's constant and $\Gamma(0)=\infty$ in paper

I am referring to a paper by S. Nadarajah & S. Kotz. The notation is simple enough to understand, however i having trouble with $C$ is Euler’s constant and $\Gamma(0)=\infty$ by equation (2.3) I ...
1
vote
1answer
54 views

Use stirlings approximation to prove inequality.

I have come across this statement in a text on finite elements. I can give you the reference if that will be useful. The text mentions that the inequality follows from Stirling's formula. I can't ...
0
votes
0answers
11 views

Scaling of the incomplete gamma function

I'm interested in the real function $y(x)$ given by the solution to the equation $\Gamma(y) = A \Gamma(y,x)$, with$\Gamma(y,x)$ the incomplete Gamma function and $A>1$ real. Any idea how to go ...
2
votes
0answers
40 views

Integral and derivatives of the gamma function

Here is my question: Starting from the relation $$\int_{0}^{+\infty}t^{a-1}e^{-nt}\,dt=n^{-a}\Gamma(a)\qquad a>0$$ and differentiating $m-$times under the integral sign we can get to ...
2
votes
1answer
28 views

Relation of gamma function to a factorial mimic function

The gamma function is an analytic extension of the factorial function. For real and positive $x$, with $x=int(x)+frac(x)$, where consider the function: $$f(x)=\prod_\limits{i=1}^{int(x)} ...
0
votes
0answers
6 views

Implementing the incomplete gamma function under libmatheval

I am considering libmatheval for a task in C. The problem is I need the incomplete gamma function. 1) How hard is it to implement new functions in libmatheval? 2) Any instructions on how to? 3) ...
2
votes
1answer
90 views

Improper integral of $\sqrt{x^4 + 1} - x^2$ [duplicate]

I'm having a little trouble with this integral: $\int^\infty_0 (\sqrt{x^4+1} - x^2)\,dx$. Using the likes of Maple, I can easily find that it takes the form $-\frac{2}{3}\sqrt{2}(1+i)K(i) - ...
2
votes
1answer
65 views

How could I solve $x^{t-1}e^{-x} = a$ for $x$?

Consider this equation: $$x^{t-1}e^{-x} = a$$ I am aware that this is what you integrate from $0$ to $\infty$ in respect to $x$ to get the Gamma Function, but I do not want to worry about it here. I ...
0
votes
2answers
28 views

How to compute the incomplete Gamma function with an negetive parameter in MATLAB?

I need to compute the upper incomplete gamma function $$\Gamma(-0.5,0.5r)$$ But in MATLAB, the upper incomplete Gamma function "gammainc" is defined as $$\Gamma\left(\alpha ; x\right) = ...
0
votes
1answer
33 views

I have question about incomplete gamma functions?

the question is about definition of upper and lower incomplete gamma functions in [1] we can see : lower case : $ \gamma(a,x)=\int_{0}^{x}e^{-t} \ t^{a-1}dt \ \ \ \ , \ \ \ Re(a)>0 $ Upper case: ...
6
votes
3answers
189 views

Why is the Gamma function off by 1 from the factorial? [duplicate]

Why didn't they define it as $$ \tilde \Gamma(x) = \int_0^\infty t^x e^{-t} \, dt ?$$ Then the definition would have two less characters than the standard definition of $\Gamma(x)$, and we would have ...
0
votes
2answers
31 views

Logarithmic derivative of Polygamma functions

While studying Gamma function and related functions I noticed that its logarithmic derivative (the so-called Digamma function) is studied more than its "normal" derivative but on the other hand I ...
1
vote
0answers
44 views

Expected distance of biased 1d random walk with 0 drift

There were many question about biased 1d random walks earlier, but as far as I can tell none of these are directly related. Let $p,q>0$ such that $p+q=1$. Let $X_0=0$ and $X_{i+1} = X_i+1$ with ...
1
vote
1answer
26 views

Variable coefficient difference equation I

Consider the difference equation \begin{align} n \, \phi_{n+1} &= (2 \, n^{2} + 2 \, n -1) \, \phi_{n} + (n+1) \, \phi_{n-1}. \end{align} It is seen that if $\phi_{n} = \Gamma(n+2) \, \theta_{n}$ ...
1
vote
0answers
38 views

Gamma function is too difficult [duplicate]

During my study to gamma function I find this form but I try to prove but I can't $$\Gamma(x) = \lim_{n\to \infty} \frac {n! n^{x-1}}{x(x-1)(x-2)........(x+n-1)} $$ So I want to know how I can prove ...
5
votes
0answers
50 views

Looking for functions “similar” to the $ \Gamma $ function

According to wikipedia: https://en.wikipedia.org/wiki/Functional_equation The $\Gamma$ - function is the only solution that suffices the following three equations: $$ f(x)={f(x+1) \over x}\tag{$*$} ...
0
votes
2answers
37 views

Other forms for the derivative of the Gamma function

When I searched for the derivative of the Gamma function I got something of the form: $$\Gamma'(x)=\Gamma(x) \psi(x)$$ But from the definition of the Digamma function to me it's like writing: ...
0
votes
0answers
14 views

incomplete gamma function with negative arguments

Does the Gamma function with negative arguments exist? $\Gamma(\Delta,-\Lambda x)$ where $\Delta$ is a negative integer and $\Lambda$ is a positive real number. thanks!
0
votes
0answers
29 views

Re-Expressing the Digamma

I was reading some articles on the digamma function, and I was wondering if anyone knows how to express the digamma function $\psi^{(0)}(n)$ in terms of a trigonometric function or a logarithmic ...
1
vote
1answer
32 views

Questions about a dominated convergence theorem problem

The problem is to find the derivative of Gamma function $\Gamma (y) = \int_0^{ + \infty } {{e^{ - x}}{x^{y - 1}}dx} $ using dominated convergence theorem. Although the following content is lengthy, ...
0
votes
0answers
22 views

How to compute $\Gamma(z) $ with Re(z) > 0, by hand.

I was wondering, if there were a way to compute some special values of the Gamma function by hand. For example: $\Gamma(1+i) = i*\Gamma(i) $ - in fact there are really a lot of nice identities one ...
0
votes
1answer
57 views

Power series representation of gamma function?

I am looking for a power-series expression of the form $\Gamma(z)=b+\sum_{k=0}^\infty a_kz^k$ where the $a_k$ can be calculated as some function of k.
3
votes
1answer
39 views

Proving that the Gamma function $\Gamma(y)$ converges for $y>0$.

How can I justify that $$\Gamma(y)=\int_0^\infty t^{y-1}e^{-t} \, \mathrm{d}t$$ exists for all $y>0$? I'm struggling to compare it to a known convergent integral.
2
votes
3answers
143 views

How do i evaluate this integral $ \int_{\pi /4}^{\pi /3}\frac{\sqrt{\tan x}}{\sin x}dx $?

Is there some one show me how do i evaluate this integral :$$ \int_{\pi /4}^{\pi /3}\frac{\sqrt{\tan x}}{\sin x}dx $$ Note :By mathematica,the result is : $\frac{Gamma\left(\frac1 ...
0
votes
0answers
20 views

Condition on the limit of the upper incomplete gamma function

I am trying to find a lower bound on $q$ such that $$\Gamma(2p,qt)=\int_{qt}^{\infty}{x^{2p-1}e^{-x}\,dx}>0$$ for all $t>0,p>0$, and $q<0$. At first, I tried expanding $\Gamma(2p,qt)$ ...
2
votes
0answers
64 views

Gamma function still hard for me

During my study I find a form for gamma function it was $\Gamma (x) = \lim_{n\to\infty} \frac{n! n^{x-1}}{x(x-1)(x-2)........(x+n-1)}$ And then by simplify this limit I get $$\lim_{n\to\infty} ...
1
vote
0answers
40 views

An integral involving the Gamma function

By utilizing the results of two previously asked questions, Series involving Laguerre polynomials and Integral of binomial coefficients, what is a resulting value of the integral \begin{align} ...
0
votes
1answer
24 views

gamma distribution probability

Am I doing this gamma distribution correctly? Calculate $P(Y>4)$ while $Y\sim \Gamma(a,b) \text{ with } a = 2, b =3$ $P(Y > 4) = 1 - P(Y \leq 4)$ with pdf $f(y)$ given $$f(y) = ...
6
votes
1answer
66 views

closed form for $\int_{0}^{\infty}\frac{ \beta(a+ix,a-ix)}{\beta(b+ix,b-ix)}\frac{dx}{(b^2+x^2)}$

closed form for : $$\int_{0}^{\infty}\frac{ \beta(a+ix,a-ix)}{\beta(b+ix,b-ix)}\frac{\mathrm{dx}}{(b^2+x^2)}$$ where $\beta$ is beta function I tried with the definition of beta and i got ...
1
vote
1answer
39 views

Transcendence of Values of Beta Function

Wikipedia mentions that the number $$a = \dfrac{\Gamma\left(\dfrac{1}{4}\right)}{\pi^{1/4}}$$ is transcendental. Since $\Gamma(1/2) = \sqrt{\pi}$ the above number $a$ seems to connected to a ...
9
votes
2answers
124 views

Integral of binomial coefficients

Let the integral in question be given by \begin{align} f_{n}(x) = \int_{1}^{x} \binom{t-1}{n} \, dt. \end{align} The integral can also be seen in the form \begin{align} f_{n}(x) = \frac{1}{n!} \, ...
3
votes
2answers
90 views

How to prove Raabe's Formula [duplicate]

For quite some time, I've been trying to prove Raabe's Formula, or in other words: $$\int_a^{a+1} \ln\bigg(\Gamma(t)\bigg)dt=\dfrac{1}{2}\ln(2\pi)+a\ln(a)-a$$ This is how I tried: ...
0
votes
1answer
28 views

How do I calculate values for Gamma function with complex arguments?

I can calculate the values of Gamma function for positive integer arguments using the formula $\Gamma (t) = \displaystyle\int_0^{\infty} e^{-x} x ^ {t-1} \mathrm{d}x $. Which is equal to $ (t-1)! $. ...
0
votes
0answers
47 views

Factorial Series I

Given the factorial series \begin{align} \beta_{1}(x) = \sum_{s=1}^{\infty} \frac{(-1)^{s} \, s!}{x(x+1)(x+2)\cdots(x+s)} \end{align} then what are the coefficients, $a_{s}$, of the square of ...
3
votes
0answers
69 views

Proving injectivity of a multivariable function

Let I denote the interval $(0,\infty)$, we define the function $f:I^2\to I^2$ by, $$f(x,y)=\left({\Gamma(4x+y)\Gamma(y)\over {\Gamma(2x+y)}^2},{\Gamma(4x+y)\Gamma(2x+y)\over {\Gamma(3x+y)}^2}\right)$$ ...