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

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1answer
64 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 $\Gamma(2p,qt)$ ...
2
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0answers
46 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
28 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
18 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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2answers
51 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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0answers
49 views

Is there a way to solve this integral? $\int_0^4 \frac {120x^ky^{4-k}}{\Gamma(5-k)\Gamma(1+k)}dk$ [closed]

Is there a way to evaluate this integral? $$\int_0^4 \frac {120x^ky^{4-k}}{\Gamma(5-k)\Gamma(1+k)}dk$$
1
vote
1answer
37 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
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2answers
112 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
69 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
45 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
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0answers
47 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, $$f(x,y)=\left({\Gamma(4x+y)\Gamma(y)\over {\Gamma(2x+y)}^2},{\Gamma(4x+y)\Gamma(2x+y)\over {\Gamma(3x+y)}^2}\right)$$ ...
0
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2answers
38 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
78 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.
2
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0answers
13 views

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
127 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
votes
1answer
44 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) ...
2
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1answer
50 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 ...
1
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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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0answers
52 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
77 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
19 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
87 views

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

Solve this equation : $(2x)! = (x)! (x+2)!$
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2answers
108 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 ...
2
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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
36 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
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2answers
164 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
34 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 $\Gamma(z)$.
2
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1answer
60 views

Is there any intuitive way to think about the gamma function?

Is there a way to realize the gamma function intuitively? My first (and probably correct) guess is no, because, for example, $\Gamma(\frac 12)=\sqrt{\pi}$ doesn't make any intuitive sense at all. ...
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1answer
77 views

Analog to Euler's Reflection Formula (Connecting the Gamma Function to Cosines)

I'm wondering if there is an analogous equation describing the relation between the Gamma function and cosines, as Euler's Reduction Formula does for the Gamma function and sines: $\Gamma (z) \Gamma ...
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0answers
16 views

Second partial moment of the Gamma pdf

I would like to rewrite the following integral in terms of the (incomplete) Gamma function: $$\int_r^\infty (x-r)^2f(x;k,\theta)\,dx$$ where $f(x;k,\theta)$ is the Gamma probability density function ...
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0answers
23 views

Joint transformation of a gamma distribution

I have a question regarding the transformation of a gamma distribution. I think I solved the problem, but I am not sure whether it is correct. Let $X$ and $Y$ be independent and Gamma distributed ...
2
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1answer
29 views

Rewrite Beta Functions in terms of Gamma Functions

Consider the following result in terms of Beta functions $B(\cdot, \cdot)$. $$ \mathbb E \left( \frac{Y_i^2}{\sigma^2} \right) = \frac{1}{r^2}\frac{n!}{(i-1)!(n-i)!} [B(n-i+1, i) -2B(n-i+r+1, i) + ...
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3answers
60 views

The derivative of $x!$ and its continuity

is the factorial of fractions and negative numbers defined? If yes, then what is its graph? Also please find its domain. Our teacher said the factorial of a fraction is the fraction itself. He also ...
2
votes
1answer
61 views

Bounding infinite series derived from polygamma functions

Let $f(x) = 2 \psi^{(1)}(x+1) + x \psi^{(2)}(x+1) $ for $ x > 0 $, where $\psi^{(i)}(x)$ is the $i^{th}$ derivative of the digamma function $\psi(x)$. The goal is to prove that $ f(x) < ...
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0answers
17 views

Finding zeros of a function involving Gamma function.

I am looking for the zeros of following function ($a$ and $b$ are real): $$ F(a,b) = 4^{a+ib} \Gamma(a+ib) \Gamma(-a) \Gamma(-ib) + \Gamma(-a-ib) \Gamma(ib)\Gamma(a) $$ and I have no idea on ...
1
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1answer
50 views

Alternate formula for $Γ(t + 1)$

I believe that: $$\frac{d^{n}}{dx^{n}}[x^{n}] \equiv Γ(n + 1) \equiv n!$$ Would this have any application, if it has not already been discovered, which I am almost certain that it has?
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2answers
68 views

how can I show this integral diverges?

I want to show $E(T_a)=\infty$ $$E(T_a)=\int_0^{\infty}{{x|a|}\over\sqrt{2\pi}}x^{-3/2}e^{-a^2/x}dx$$ to show this I need to show this integral diverges. I know gamma function that $$\Gamma ...
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0answers
30 views

convergence and holomorphic function - explanations and proofs

I haven't had analysis for a long time and I've forgotten plenty. Could you please explain me and prove that the following converges: $$ \sum_{n \geq 1}e^{-n^2t\pi}, t>0 c $$ and explain why the ...
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0answers
26 views

Decomposing a series

When I insert the following function \begin{equation} F(X,Y)=-\frac{1}{Y^{2/3}}\sum _{m=0}^{\infty } \frac{\Gamma \left(\frac{m+2}{3}\right)}{m! \Gamma (m+1)}\left(-\frac{X^2}{2^2 ...
1
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1answer
55 views

Gamma and Beta function proof.

I'm trying to proof the equality $B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ when $x,y>0,$ without using calculus in many variables. I've investigated about the topic but all references make ...
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1answer
30 views

How to obtain the Laruent expansion of gamma function around $z=0$?

I want to prove, the laurent expansion of gamma function. \begin{align} \Gamma(z) = \frac1z-\gamma+\frac12\left(\gamma^2+\frac {\pi^2}6\right)z-\frac16\left(\gamma^3+\frac {\gamma\pi^2}2+2 ...
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1answer
31 views

Gamma and Beta Functions [closed]

\begin{equation*} \int \limits _0 ^\infty x^m \mathbb e ^{-x^n} \mathbb d x = \frac 1 n \Gamma (\frac {m+1} n), \space m>-1, \space n>0. \end{equation*} \begin{equation*} \int \limits _0 ^1 ...
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0answers
31 views

Derivation of Gamma distribution characteristic function reference?

I was wondering if there was a derivation of the Gamma distribution characteristic function without expanding the $e^{itx}$ into an infinite summation?
3
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1answer
77 views

Concerning Hurwitz Zeta function, how to prove the following identity?

It is claimed that $$\zeta'(0,s)=\ln\left(\frac{\Gamma(s)}{\sqrt{2\pi}}\right)$$ where the derivative is meant by the first argument (as usual with Hurwitz Zeta). How to prove this? Wolfram Alpha ...
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1answer
38 views

Express each of the following integrals in terms of the gamma or beta functions and simplify when possible

$$\begin{align} \\ &= \int_{0}^{1} \ ( {(1/x) - 1} ) ^ {1/4} dx \\ &= \end{align}$$ I try to solve but not sure if correct or not and need help in these also ,I try in first one ...
0
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1answer
48 views

Proving $\int_0^\infty e^{-ax}x^n\,dx = \frac{1}{a^{n+1}} \Gamma(n+1)$

Prove that $$ \int_{0}^{\infty} \ e^ {-ax} x^{n} dx = \frac{1}{a^{n+1}} \Gamma(n+1) \qquad (n>-1, \, a>0). $$ My try: Let $dv = e^{-ax}$ and $u = x^n$. Then $v = -\frac{1}{a}e^{-ax}$ ...
0
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0answers
24 views

Can I calculate the upper limit of the definite integral in this equation?

while working on an engineering thesis I came to this equation: $$\sum_{k=1}^N\int_0^Te_k(t)dt=s$$ The $e_k(t)$ is erlangian distribution with shape k, N is an integer (its value can be assumed to any ...