Questions tagged [gamma-function]

Questions on the gamma function $\Gamma(z)$ of Euler extending the usual factorial $n!$ for arbitrary argument, and related functions. The Gamma function is a specific way to extend the factorial function to other values using integrals.

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-4
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1answer
58 views

Why is gamma function defined such that $\Gamma (n)=(n-1)!$ rather then $\Gamma(n)=n!$ [duplicate]

Why is gamma function defined such that $\Gamma (n)=(n-1)!$ rather then $\Gamma(n)=n!$, The latter ssems far more logical.
0
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0answers
21 views

Prove that $\gamma<\int_{0}^{1}\frac{-\operatorname{li}(x)}{\Gamma(x)}dx<\frac{1}{3\gamma}$

$$\gamma<\int_{0}^{1}\frac{-\operatorname{li}(x)}{\Gamma(x)}dx<\frac{1}{3\gamma}$$ Where $\operatorname{li}(x)$ is the Logarithmic integral function $\Gamma(x)$ is the Gamma function $\gamma$ ...
0
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0answers
23 views

Special function expression for the (truncated) moment of the generalized Gamma distribution

The, kind of, truncated moment of the generalized Gamma distribution for positive $a,d$ and $p$ is $$\int_0^\infty (x-k)_+^a x^{d-1}e^{-x^p}dx=\frac1p\int_0^\infty (y^{\frac1p}-k)_+^a\,y^{\frac dp-1}e^...
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0answers
21 views

Comparision between the incomplete gamma function and the gamma function for complex argument

Is the lower incomplete gamma function (https://en.wikipedia.org/wiki/Incomplete_gamma_function#Definition) bounded by the gamma function in the right half plane or in a strip parallel to the ...
2
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1answer
16 views

Limit of ratio of incomplete gamma function

In order to derive Sterling's approximation, I need to show that the following integral decays quicker than at least $\mathcal{O}(n^2)$: $\lim_{n\to\infty}\dfrac{\int_{2n}^\infty x^ne^{-x}dx}{\int_{0}^...
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1answer
35 views

“generalised” gamma-like integral $\int_0^\infty x^ne^{-f(n)x}dx$

I have noticed that if we have an integral of the form: $$I[f]=\int_0^\infty x^ne^{-f(n)x}dx=\frac{1}{f^{n+1}(n)}\int_0^\infty x^ne^{-x}dx=\frac{n!}{f^{n+1}(n)}$$ I was wondering what kind of ...
2
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0answers
12 views

Integral of upper incomplete gamma function

Can anyone please help me with the integral below. I would like to know whether the following relation is correct? $\int\big(\frac{t}{A}\big)^{n-1}\exp\big(-\frac{t}{A}\big) dt = -A\Gamma(n, \frac{t}{...
1
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1answer
20 views

How to calculate the following Laplace transform: $ \mathcal{L}[\frac{1-J_0(t)}{t}] $?

I'm trying to calculate the Laplace transform of this function. $$ \mathcal{L}[\frac{1-J_0(t)}{t}] $$ where $J_0(t)$ is the zeroth Bessel function. Solution Attempt The p-Bessel function is ...
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0answers
27 views

Meaning of Generalized Binomial Coefficients

The title speaks for itself. I have seen and understood that a common way to write binomial coefficients in general is $${n \choose k} = \frac{\Gamma{(n + 1)}}{\Gamma{(k + 1)}\Gamma{(n - k + 1)}},$$ ...
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0answers
31 views

Is this $a_n=(1-\frac{1}{2!})^{(\frac{1}{2!}-\frac{1}{3!})^{…^{(\frac{1}{n!}-\frac{1}{(n+1)!})}}}$ have a finit limit?

My question here is related to telescopic sum using factorial and it is related to my question here, I have computed some values of $a_n=(1-\frac{1}{2!})^{(\frac{1}{2!}-\frac{1}{3!})^{...^{(\frac{1}{n!...
6
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1answer
112 views

What number's factorial is $i$?

I am trying to find the solution to the equation- $$\Gamma(z)=i$$ I have tried doing it the following way- LHS is- $$\displaystyle \int_{0}^{\infty}t^ze^{-t}\ dt$$ Taking $z=a+ib$, we get- $$\...
2
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1answer
55 views

On a log-gamma definite integral

A very famous log-gamma integral due to Raabe is $$\int_0^1 \log \Gamma (x) \, dx = \frac{1}{2} \log (2\pi).$$ Several proofs of this result can be found here. I would like to known about the ...
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0answers
31 views

can you help me solve this problem [closed]

Use $J_n(x)=\sum\limits_{k=0}^{\infty}\frac{(-1)^r}{r!\Gamma (n+r+1) }(x/2)^{n+2r}$ to prove that: $\begin{align} \int_{0}^{\infty} J_{0}(bx) e^{-ax} \, dx &= \sum_{0}^{\infty}(-1)^{r} \frac{b^...
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0answers
34 views

Prove using Bessel function [closed]

use \begin{align} J_n(x)=\sum\limits_{r=0}^{\infty}\frac{(-1)^r}{r!\Gamma (n+r+1) }(x/2)^{n+2r} \end{align} to prove that: \begin{align} \frac{d}{dx} [x^{2}J_{n-1}(x)J_{n+1}(x)] &= 2x^{2}J_n(x)...
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2answers
30 views

Calculating $\Gamma (n+5/2)$

Hey I'm pretty new to the gamma function and was trying to calculate $\Gamma (n+\frac52)$. I got to the integer $\int_0^\infty t^{n+\frac32}e^{-t}\mathrm dt\\$, and I really don't know how to go on ...
2
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1answer
44 views

Trying for an alternate solution of integration of {$\tan x$}

I was trying to see if I could find an alternate solution to the below problem (just playing around, not getting ambitious). I came at some point which I thought was interesting - so thought of ...
0
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1answer
40 views

Show that $\Gamma(x) \sim \sqrt{2 \pi} e^{-x}x^{x-\frac12}$.

The gamma function is defined by $$\Gamma(x) = \int_0^\infty t^{x-1} e^{-t} dt $$ where $x > 0$. Show that $\Gamma(x) \sim \sqrt{2 \pi} e^{-x}x^{x-\frac12}$. $\sim$ denotes that the ratio ...
-1
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1answer
33 views

How do we prove that $\int_0^1 (x\ln x)^{-1+n}\,dx = -\left( -\frac{1}{n} \right)^n \Gamma(n)$

I need this solution to prove that $$\int_0^1 (x\ln x)^{-1+n}\,dx = -\left( -\frac{1}{n} \right)^n \Gamma(n)$$ Thank you!
2
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1answer
21 views

The modulus of Gamma function $\left| \Gamma(x+iy) \right|$ is strictly decreasing when $x \in (0,\frac{1}{2})$ for a fixed $y \in \mathbb{R_+}$

Let $\Gamma(s)$ be the Gamma function, extension of the factorial function to complex numbers. My question is: Fixed $y \in \mathbb{R_+}$ is it true that $\left| \Gamma(x+iy) \right|$, the modulus ...
2
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0answers
26 views

Binomial coefficients hierarchy extended to real numbers

I am working with the binomial coefficient but I'm a little stuck when it involves real numbers instead of only naturals. I know that if $0\leq k<k'<\frac{a}{2}$ and $k,k'\in \mathbb{N}_0$, ...
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2answers
27 views

Efficient methods to calculate incomplete beta $B[a,b;x]$ for $b=0$

I am looking for an efficient numerical method (or a module) to calculate the incomplete $\beta-$function for $b=0$. e.g. https://www.wolframalpha.com/input/?i=incomplete+beta%5B4%2F5%2C1.5%2C0.0%5D+...
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0answers
15 views

Find the misterious function $F(r,a,q)$.

I've the pattern bellow where we can spot the binomial coefficients. Observing the pattern, for each row, it seems that it arises from $$ \left(\left(r a^{q} \right)^{\frac{1}{q}} +F(r,a,q)\right)^{n} ...
2
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1answer
53 views

Is this a valid approach to proving the Sylvester-Schur Theorem

I have recently been asking a lot of questions regarding the Gamma Function and a well-known upper bound of the prime counting function. I really appreciate everyone's assistance in helping me to ...
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1answer
50 views

Finding a lower bound for $\frac{\Gamma\left(2x+2 - \frac{1.25506(x+1)}{\ln(x+1)}\right)}{\Gamma\left(2x - \frac{1.25506(x)}{\ln x}\right)}$

Does it follow that for $x > e^3$, $\dfrac{\Gamma\left(2x+2 - \frac{1.25506(x+1)}{\ln(x+1)}\right)}{\Gamma\left(2x - \frac{1.25506(x)}{\ln x}\right)} > \dfrac{\Gamma\left(x+8 - \frac{1.25506}{\...
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0answers
48 views

Special integral, Help me solve it please!

When solving a problem i met an integral that: $$\int_0^\infty ln(x)x^ae^{-x}dx$$ Where a is a complex number with a negative real part. Any ideas for solving it? And other one is: $$\int_0^\infty x^...
1
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1answer
17 views

Comment on the plots of two fitted densities on a histogram

What possible comments can I draw on this following plot? It contains plot of two fitted densities. One estimating the parameters using MLE and other using MME, that I calculated from a set of data ...
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0answers
20 views

Computation of incomplete confluent hypergeometric function of the first kind

In summary I need to compute "upper incomplete confluent hypergeometric function of the first kind": $U(a, b, x, z) = \sum_{n=0}^{\infty} \frac{ x^n }{n!} \frac{\Gamma(a + n, z)}{\Gamma(b + n)}$ ====...
3
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2answers
67 views

Does it follow for $x \ge 785$, that Gautschi's Inequality implies that $\frac{\Gamma(2x + 3 - \frac{1.25006}{\ln n})}{\Gamma(2x+1)} > x^2$

Does it follow for $x \ge 785$, that Gautschi's Inequality implies that $\frac{\Gamma(2x + 3 - \frac{1.25006}{\ln n})}{\Gamma(2x+1)} > x^2$ Here's my reasoning. Please let me know if I made any ...
-3
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1answer
26 views

Prove that $\Gamma(1+1/x)=\int_{0}^{\infty} e^{{-t}^{x}} dt$ [closed]

Please help me how to do it. I have no clue about how to begin.
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0answers
28 views

Does it follow from Gautschi's Inequality that $x^s > \frac{\Gamma(x+s)}{\Gamma(x)} > x(x+1)^{s-1}$

Here's my thinking. Please let me know if any of my assumptions are incorrect. (1) From Gautschi's Inequality, if $x > 0$ is real and $0 < s < 1$: $$x^{1-s} < \frac{\Gamma(x+1)}{\Gamma(...
0
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3answers
51 views

Sum with Riemann zeta function

I found on Instagram an interesting series: $$\sum_{n\geq1} \zeta(n+1)\frac{n}{2^n}$$ In a first moment I noticed that it can be written as $$\sum_{n\geq1}\zeta(n+1)\Gamma(n+1)\frac1{2^n}$$ and so to ...
1
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1answer
43 views

What is the meaning of $x \in [0,1]$ and $x \in (0,1)$

Apologies if this is an obvious question. I am asking mostly to make sure that my assumptions are correct. I find this notation in many Wikipedia articles without any definition, for example, see ...
3
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3answers
60 views

Solving for an unknown $c$ in relation to a ratio of gamma functions

I have been working with ratios of gamma functions and I am surprised how difficult it is to make even elementary conclusions. I am hoping it is just the learning curve. Consider the following ...
1
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1answer
45 views

Inverse Fourier transform of $\Gamma(z)\zeta(2z)$

I wish to calculate the following series: $$f(t)=\sum_{n=-\infty}^\infty e^{2\pi itn}\Gamma(a+ibn)\zeta(2(a+ibn))$$ I hope to somehow use the Poisson summation formula, since I already know that - $$\...
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2answers
29 views

Proof of the expression of upper incomplete gamma function as an finite summation.

I am trying the proof the one of the finite summation expression of the incomplete upper Gamma function $$ \Gamma(m,x) = \int_{x}^\infty v^{m-1}e^{-v}d$$ So, using the equality (also given in Tab. of. ...
1
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1answer
26 views

Proving an integral identity similar to the beta function, but without using the beta function

I'm trying to calculate the volume of an $n$-dimensional hypersphere. The text I'm working out of breaks down the calculatin into a few different steps, and I'm stuck on the following one: By ...
1
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1answer
60 views

How to evaluate $\int_0^\infty e^{-x^a}dx$

please help me how to do it.I have no clue about how to begin.I attempted to replicate how the Gaussian integral is solved but that would require 'a' number of equations relating spherical and ...
2
votes
1answer
53 views

Definite integral using Euler integrals

I really need to solve this integral using Euler integrals (Gamma & beta functions). $$I=\int_0^\infty \frac{\sinh^2 bx }{\sinh^2 cx} dx $$ I have tried all that I could, but... $$I=\int_0^\...
1
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1answer
44 views

Comparing a ratio of gamma functions to a simple polynomial

I am still struggling to build my intuition as far as reasoning with ratios of gamma functions. Reasoning with factorials is significantly clearer. Consider this example. I would appreciate if ...
1
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0answers
31 views

Nested Gamma functions and $\lim_{n\to\infty} a_n=1$

Hi it's a problem that I cannot prove : Let $1\leq a_0\leq 3$ and defines the following sequence : $$\Gamma(a_n)=a_{n+1}$$ Where $\Gamma(x)$ is the Gamma function and $n\geq 0$ a ...
1
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0answers
49 views

What could be said about $\Gamma(2n)=\Gamma(n)^n$?

How do you solve $$\Gamma(2n)=\Gamma(n)^n$$ The trivial solution is of course $n=1$, but there is another one according to WolframAlpha and it gives the numerical approximation $$n ≈ 0....
1
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2answers
79 views

Is this a correct way to use the digamma to analyze whether a ratio of gamma functions is increasing?

Let: $\pi(x)$ be the prime counting function $\psi(x)$ be the digamma function Given that $\pi(x) < \dfrac{1.25506n}{\ln n}$ (see here), let: $$f(n) = \frac{\Gamma\left(2n + 1 - \frac{1.25506n}{\...
0
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0answers
31 views

Triple product identity for the Gamma function

The well-known reflection formula for the Gamma function relates its values at $z$ and $1-z$ for any $z\in\mathbb{C}$, and has the explicit form \begin{equation} \Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin{\...
1
vote
1answer
52 views

Is the following ratio of gamma functions increasing: $\frac{\Gamma(2n - \frac{1.25506n}{\ln n})}{\Gamma(n)^2}$?

For $n > 1$, is the following ratio of gamma functions increasing: $\dfrac{\Gamma(2n - \frac{1.25506n}{\ln n})}{\Gamma(n)^2}$ I suspect that it is at some point where $n > 1$. I would like ...
0
votes
2answers
64 views

Does a simpler form exist for $\frac{\Re \left(\zeta(\frac12+it)\,\Gamma(\frac12+it)\right)}{\Im \left(\zeta(\frac12+it)\,\Gamma(\frac12+it)\right)}$?

I am aware that: $$\frac{\Re \zeta(\frac12+it)}{\Im\zeta(\frac12+it)}=\cot \left( \frac12\,t\ln \left( \pi \right)+\frac{i}{2}\ln \left( {\frac {\Gamma \left( \frac14+\frac{it}{2}\right) }{\Gamma \...
2
votes
1answer
63 views

Antiderivative of the reciprocal Gamma function

I have a question about the evaluation of the following function for $x>1$: $$\Omega(x)=\int_0^x\frac{1}{\Gamma(s)}\;ds$$ In order to try to evaluate $\Omega(x)$ I used the reflection formula, ...
0
votes
2answers
54 views

Prove inequality on quotient of $\Gamma$-functions [closed]

$$\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n + 1}{2}\right)} > \frac {\sqrt{2n + 1}}{n}, \space\space\forall{}n\in\Bbb{N}$$
5
votes
1answer
146 views

How do we prove this continued fraction for the quotient of gamma functions

Given complex numbers $a=x+iy$, $b=m+in$ and a gamma function $\Gamma(z)$ with $x\gt0$ and $m\gt0$, it is conjectured that the following continued fraction holds $$\frac{\displaystyle4\Gamma\left(\...
0
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0answers
25 views

How can I show that this product is equal to a product of Gamma functions?

$\prod_{n=0}^{x-1}\left( 1+\frac{a}{an+b}\right) = \frac{\Gamma\left(\frac{a}{b}\right)\Gamma\left(x+\frac{a+b}{b}\right)}{\Gamma\left(\frac{a+b}{b}\right)\Gamma\left(x+\frac{a}{b}\right)}$ I found ...
0
votes
0answers
27 views

Is $g(x)$ a decreasing function toward a limit? If yes, what is the limit?

Let: $f(x) = \left(\Gamma(x+1)\right)^{\left(\frac{\ln x}{x\ln x - 1.25506x}\right)} + x - 1$ $g(x) = \dfrac{f(x)}{x}$ Is $g(x)$ a decreasing function toward a limit? If yes, what is the limit? I ...

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