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

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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
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0answers
26 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
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1answer
27 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)} ...
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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
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1answer
87 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) - ...
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1answer
60 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 ...
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2answers
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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) = ...
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1answer
31 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: ...
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3answers
183 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 ...
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2answers
26 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 ...
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0answers
41 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 ...
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1answer
25 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 $\phi_{n} = \Gamma(n+2) \, \theta_{n}$ ...
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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 ...
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0answers
42 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{$*$} ...
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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: ...
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0answers
11 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!
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0answers
28 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 ...
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1answer
28 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, ...
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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 ...
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1answer
52 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.
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1answer
38 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.
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3answers
138 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 ...
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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 $\Gamma(2p,qt)$ ...
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0answers
63 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} ...
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0answers
35 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} ...
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1answer
19 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
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1answer
59 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 ...
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1answer
38 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 ...
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2answers
119 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
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2answers
76 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: ...
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1answer
27 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)! $. ...
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0answers
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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 ...
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0answers
59 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, $$f(x,y)=\left({\Gamma(4x+y)\Gamma(y)\over {\Gamma(2x+y)}^2},{\Gamma(4x+y)\Gamma(2x+y)\over {\Gamma(3x+y)}^2}\right)$$ ...
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2answers
40 views

expectation of lgamma of gamma distribution

Is there a closed-form expression for $E[\log(\Gamma(X))]$, where $X \sim Gamma(k, \theta)$? Edit: Note the gamma function inside the log. Edit 2: If there's no closed-form expression, is there a ...
6
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2answers
84 views

Limit involving Zeta and Gamma function

Can someone help me evaluate this limit? $$\lim_{x\to +\infty}\frac {\zeta(1+\frac 1x)}{\Gamma(x)}$$ I never came across this kind of limit so I don't even know where to start.
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0answers
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Integral over $B_1^n(0)$

Evaluate $I = \int_{B_1^n(0)} (a_1x_1 + \cdots + a_nx_n)^{2/3}$ where $a_j \in \mathbb{R}$. Here's where I'm at, following a hint: Consider an orthonormal transformation $T$ with the first row equal ...
12
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2answers
139 views

What is the exact value of $\eta(6i)$?

Let $\eta(\tau)$ be the Dedekind eta function. In his Lost Notebook, Ramanujan played around with a related function and came up with some of the nice evaluations, $$\begin{aligned} \eta(i) &= ...
2
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1answer
45 views

Ratio of Gamma Functions

Is it possible to show that: \begin{align} \frac{\Gamma\left(\frac{1}{78}\right) \Gamma\left(\frac{29}{78}\right) \Gamma\left(\frac{35}{78}\right) \Gamma\left(\frac{53}{78}\right) ...
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1answer
51 views

$\lim_{x \to \infty} \Gamma(x+1)/\Gamma(x+1+1/x^2)$

Can someone please help me with the calculation of this limit? $\lim_{x \to \infty} \Gamma(x+1)/\Gamma(x+1+1/x^2)$ I tried wolframalpha and seems to be 1, yet there are no "detailed steps" as to how ...
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1answer
47 views

Integral using gamma and beta functions

I can't solve this, no matter how I try $$\int_{-\infty}^{+\infty} \frac{e^{2x}}{4e^{3x}+9}\,dx$$ Thanks in advance
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53 views

Can this relationship be expressed algebraically?

$\frac{\left(x-1\right)!+1}{x}=\frac{\left(y-1\right)!+1}{y}$ When I graphed it, I noticed that it bears a resemblance with the equation (which could of course be completely coincidental): ...
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1answer
81 views

Evaluating Integral with $\ln^3$ in the Integrand

Over the past week I've come across several 'Natural Logarithm Integrals': $$\int_0^{\pi/2}\ln(\sin(x))\tan(x)dx, \int_0^{\pi/2}\ln(\sin(x))\ln(\tan(x))dx$$ and so on so forth. This lead to me ...
4
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1answer
79 views

Mistake with Integration with Beta, Gamma, Digamma Fuctions

Problem: Evaluate: $$I=\int_0^{\pi/2} \ln(\sin(x))\tan(x)dx$$ I tried to attempt it by using the Beta, Gamma and Digamma Functions. My approach was as follows: $$$$ Consider ...
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0answers
20 views

Books about Gamma, Polygamma and other related functions

I'd like to know the most possible things about Gamma function, Polygamma functions, their derivatives and logarithms and other generalizations. But searching on the internet is too annoying for such ...
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1answer
90 views

Solve this equation : $(2x)! = (x)! (x+2)!$ [closed]

Solve this equation : $(2x)! = (x)! (x+2)!$
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2answers
110 views

What do you think about this conjecture?

Beforehand, please know that I'm a little bit of an amateur mathemitician, so this could be very wrong. But, I have tested it over and over and over again and it seems to be plausible, but there's ...
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0answers
9 views

Decay of reciprocal gamma function and similar functions

It is known that the reciprocal gamma function $1/\Gamma$ is entire of order 1 meaning, in particular, that for any $\varepsilon > 1$ there exists $C_\varepsilon > 0$ such that $$ \left| ...
3
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2answers
41 views

Given the p.d.f. of X, find the mean and variance of $X$.

Suppose that $X$ has probability density function, $$f(x)=\begin{cases} \dfrac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1} & \text{for }0\leq x\leq1 \\[8pt] 0 ...
8
votes
2answers
172 views

How to Integrate $ \int^{\pi/2}_{0} x \ln(\cos x) \sqrt{\tan x}\,dx$

Evaluate $$\displaystyle \int^{\frac{\pi}{2}}_{0} x \ln(\cos x) \sqrt{\tan x}dx$$ Unfortunately, I have no idea on how to integrate this and thus cannot provide any inputs on my own. The only ...
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0answers
36 views

What is the inverse of factorial? [duplicate]

Please write a function in any form it can be expressed in, that is the inverse function of the factorial function. By factorial function I mean $\Gamma(z)$.