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.
3,225 questions
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Question on Proposition 5.4 of Iwaniec Kowalski
I have a relatively involved question about Proposition 5.4 of Iwaniec-Kowalski. The Proposition states the following:
Proposition 5.4. Suppose $\textrm{Re}(s+\kappa_j)\geq 3\alpha>0$ for $1\leq j\...
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Mellin transform of the power of the reciprocal gamma function
Let $$f(z)=\frac{1}{\Gamma^n(z)}\text{, $n>0$: integer, and $z>0$}.$$ How to obtain the Mellin transform of $f(z)$? I couldn't find it in any table of Mellin transforms. thanks in advance.
My ...
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Gamma times confluent hypergeometric function for large arguments
I need to evaluate the following expression for large $|a|$
$\Gamma(a)U(a, 1, x)$,
where $a$ and $x$ are real and $x\geq0$. For moderate values, I can just use mpmath. However, in the limit of large $|...
- 69
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Evaluating $\int\arctan\left(x!\right)\mathrm dx$
What I've done so far:
By parts:
$$\int\arctan\left(x!\right)\mathrm dx=x\arctan(x!)-\int\frac{\Gamma(x+1)\psi(x+1)}{1+(x!)^2}\mathrm dx$$
I cannot get any further than this.
Does anyone know any ...
- 29
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Formula in "On the Number of Prime Numbers less than a Given Quantity"
In Riemann's Famous paper "On the Number of Prime Numbers less than a Given Quantity" in page 6 he writes:
$$-\log\Pi\left(\frac{s}{2}\right)=\lim\left(\sum_{n=1}^{n=m}\log\left(1+\frac{s}{...
- 133
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Principal value integral of the negative side of the gamma function
Since the Gamma function has a pole on 0 and all negative integers, with the signs alternating, out of curiosity I want to investigate the area under gamma in the negative half-plane with all the ...
- 399
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Show $\int_{0}^{1} \theta ^k (1- \theta)^l d \theta= \frac{\Gamma(k+1) \Gamma(l+1)}{\Gamma(k+l+2)}$ (Mixture of Bernoulli dists)
To prove this $\int_{0}^{1} \theta ^k (1- \theta)^l d\theta = \frac{\Gamma(k+1) \Gamma(l+1)}{\Gamma(k+l+2)}$ (for $k,l > -1$) I know you can use the definition of the Beta distribution - however I ...
- 515
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Is Gamma function just an analytical continuation of factorial or is there more?
Factorials are the number of arrangements of things. Is there a similar interpretation for the Gamma function or is it just an extension of factorials? Does $\left(\frac{1}{2}\right)!$ mean anything?? ...
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Prove $\prod_{n=1}^\infty \frac{n(n+a+b)}{(n+a)(n+b)} = \frac{\Gamma(a+1)\Gamma(b+1)}{\Gamma(a+b+1)}$
Note that you may think it's duplicate but please read the details to see that there is reason for posting this:
My question is $95$% related to this solution (but I noticed this after writing the ...
- 1,610
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Prove equivalence of sums with Gamma functions
Consider the following sum:
$$
\sum_{n=0}^{a-1} \frac{(-1)^n}{\Gamma(1+n)^2}\frac{\Gamma(b+1)\Gamma(c+1)}{\Gamma(-b-c+n)\Gamma(a-n)\Gamma(b-n+1)\Gamma(c-n+1)},
$$
where $a,n\in\mathbb{N}$ and $b,c\in\...
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Use $\Gamma(z)$ (the Gamma function) and $\frac{1}{(i\xi + 0)^{1+\alpha}}$ to represent the Fourier transform of distribution $u = x_+^\alpha$
Assume $\alpha > -1$. Prove that
$$
\frac{1}{(i\xi + 0)^{1+\alpha}} \overset{\mathcal{D}'}{=} \lim_{\varepsilon \to 0^+} (i\xi + \varepsilon)^{-1-\alpha}
$$
defines a distribution on $\mathbb{R}$. ...
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Numerical integration algorithm for $\int_0^{\infty} dx x^p \sin^2 x f(x)$
I would like to introduce the following kind of integrals:
$$
\int_0^{\infty} dx x^p \sin^2 x f(x),
$$
where $-3 < p < -1$, and $f(x)$ is an arbitrary `good' function which does not introduce ...
- 529
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Magnitude of a complex number's factorial.
In a problem I was doing the magnitude of this complex number appeared:
$$q=\left(-x^{\frac{1}{3}}\frac{\left(1+\sqrt{3}i\right)}{2}\right)!$$
I know that there's an analytic solution for the ...
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complex Gamma function represented as integral on complex semi-line.
Let $s \in \mathbb{C}$ satisfying $|s| \geq 1$ and $|\text{arg}(s)| \leq \pi - \delta$.
The notes I am reading claim that Cauchy's residue theorem can be used to write $$\Gamma(s) = \frac{1}{s}\int_{L}...
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Gamma function at negative integers and neutrix calculus
From what I learnt earlier, $\Gamma(-j)$ is not defined for $j=1,2,\ldots$. However,from "Some Results on the Gamma Function for Negative Integers" I get that $\Gamma(-j)$ is defined using ...
- 365
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How to calculate the ratio of the Gamma-function and its derivative $\frac{ \Gamma'(x) }{ (\Gamma (x))^2 }$ around their poles
I stumbled across the the ratio $ \lim_{x \rightarrow 0} \frac{ \Gamma'(x) }{ (\Gamma (x))^2 }$.
$\Gamma$ is the Gamma
-function $\Gamma(x) = \int_0^\infty dt \ e^{-t} \ t^{x-1}$ and it's derivative $\...
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Fractional Pascal's Triangle [closed]
Is there a fractional version of Pascal's triangle for binomial expansion similar to the standard triangle used for binomial expansion? If so, is it related to the Gamma function?
\begin{array}{cccccc}...
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Integral (convolution) equation with singular kernel
I have stumbled upon an integral equation whose kernel involves a quotient of Gamma functions:
$$ x(2-x)f(x)+g(x)=-\int_{0}^{x}dy\;f(y)\;g(x-y), $$
where
$$g(x)=\frac{\Gamma(1-x)}{(1+x)\Gamma(x)}.$$
...
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Problem related to distribution function of $X\sim \operatorname{Gam}(n,\lambda)$ and the incomplete gamma function
Let $X\sim \operatorname{Gam}(n,\lambda)$. Show that the the distribution function of $X$ is
$$
F(x)=\begin{cases}
1-\sum_{k=0}^{n-1}e^{-\lambda x}\frac{(\lambda x)^k}{k!} & \text{if $x>0$},\\
...
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Closed-form of the hypergeometric function?
Is there a closed-form of the following generalized hypergeometric?
$${}_3F_2\left(\left.\begin{array}{c}1,1,\frac13\\\frac54,\frac74\end{array}\right|1\right) ;\quad{}_3F_2\left(\left.\begin{array}{c}...
- 403
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Numerically Integrating $\Gamma(i+1)$
I have been trying to solve $i!$ for some time now, but I need a nudge in the right direction. Of course, you can Google "i!" and be given the value $0.498015668 - 0.154949828 i$ but I want ...
- 2,206
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Sum of the reciprocal of $(3n+1)!$
I was playing around and came across the following sum
$$S=\sum_{n=0}^{\infty}\frac{1}{(3n+1)!}$$
$$S\approx 1.04187$$
My only intuition was the product form of the reciprocal gamma function, but that ...
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Properties of the function $ f(x):= \displaystyle\prod_{n=0}^{\left\lfloor x\right\rfloor-1} (x-n). $
The following function on $[1,\infty)$ is clearly continuous and increasing (although admittedly I have not provided a proof of these things), as small increases in $x$ give small increases in $f$:
$$ ...
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Why the gamma function? [duplicate]
If you ask someone to evaluate $\left(-\frac{1}{2}\right)!$ they would probably provide you with $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$. Similarly, $\left(-\frac{3}{4}\right)!$ would get called $\...
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Extending the Euler-Mascheroni Constant to higher dimensions $\gamma_n$.
The classical Euler-Mascheroni Constant is
$$
\gamma = \lim_{n \to \infty} \left( \sum_{k=1}^n \frac{1}{k} - \ln(n) \right)
$$
I'm trying to extend $\gamma$ to higher dimensions denoted by $\gamma_n$ ...
- 2,113
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Closing the contour in Mellin transform
Good morning, I am computing some Mellin transforms and I encountered the following integral
$$ \int_{-i \infty}^{i \infty} \frac{ds}{2\pi i} \frac{z^{-s}}{s-d} \frac{\Gamma(s)\Gamma(a-s)\Gamma(b-s)}{\...
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Properties of $L(s, \chi)$ for primitive/non-primitive $\chi$
Let $q$ be an integer $\geq 2$ and $\chi$ a primitive character mod $q$. Using Theorem 4.3.6 and Lemma 4.3.1, prove the following:
a) $L(s, \chi)$ has an analytic continuation to $\mathbb{C}$.
b) If $\...
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How to solve a definite integration of a high order exponential function combined with rational function $x^{-1}$
$$\displaystyle \int_{0}^\infty \left(e^{\large-x}- e^{\large -Ax-Bx^{c}}\right) \cdot x^{-1} \mathrm dx,$$
with $A>0,\,\, B>0,\,\, 1>C>0$
I have attempted to apply the Taylor expansion ...
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How to evaluate the gamma function $ \int_{0}^{\infty} e^{-h^2x^2}\,dx $?
$$ \int_{0}^{\infty} e^{-h^2x^2}\,dx $$
How can I evaluate this integral? I was thinking about making the replacement $ u = h^2x^2$ but then I get $du = 2h^2x\, dx$, I am blind after this...
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Integrating $\int_0^1\frac{x-x^2}{\ln x}dx$ using gamma function
$$\int_0^1\frac{x-x^2}{\ln x}dx$$
How to solve this integral without using exponential integral Ei? like using the definition of the gamma function.
I tried to solve it using gamma function, beta ...
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Incomplete proof in Higher Trascendental functions
I am currently studying the book "Higher Trascendental functions" by Erdélyi et al. On page 27 he defines the function:
$$\Psi(z,s,v)=\sum_{n=0}^{\infty}(v+n)^{-s}z^n$$
and then claims that ...
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Solving $\int_0^{\infty}\frac{e^{-x}}{\Gamma(\frac32+x)\Gamma(\frac12-x)}dx$
I found the following integral on Instagram and found it rather interesting so I gave it a shot. However, I got stuck in the middle and needed some guidance to continue.$$\int_0^{\infty}\frac{e^{-x}}{\...
- 2,206
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Sum of ratio of gamma functions
I needed to solve the following expression:
$$
\sum_{x=a+1}^{b-1} \frac{\Gamma(x)}{\Gamma(x-a)}, \hspace{5mm} x,a,b \in \mathbb{N} .
$$
Now, I tried to find an alternative expression for it by myself, ...
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Whittaker function and gamma function
I am not an expert in special function. I would like to know if one of the following expressions can be simplify in some way, maybe using some other special functions (I tried with Coulomb wave ...
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Recursive Incomplete Gamma Function for $a=1$
It can be shown that the incomplete Gamma function:
$$
\Gamma_X(a) = \int_0^x t^{a-1} e^{-t} \, dt
$$
Can be expressed as a recursive relationship:
$$
\Gamma_X(a) = (a-1) \ \Gamma_X(a-1) - x^{a-1} \ e^...
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An asymptotic for a sum involving gamma functions
I have a rather scary sum
$$S(n) = \sum _{k=0} ^{\infty} \frac{\Gamma^A\left( k+an+1 \right)\Gamma^B\left( k+bn+\frac12 \right)}{\Gamma^{A-1}\left( k+1 \right)\Gamma^{B-1}\left( k+\frac12 \right)}\...
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Large $x$ behavior for the sum $\sum _{m=1} ^{\infty} \frac{x^{m(A+B)}}{\Gamma(m+1)^A\cdot \Gamma \left( m+\frac12 \right)^B}$
Consider the sum $S$ given by
$$S_{A,B}(x) =\sum _{m=1} ^{\infty} \frac{x^{m(A+B)}}{\Gamma(m+1)^A\cdot \Gamma \left( m+\frac12 \right)^B}, \quad \text{where}~~ A,B \in \mathbb{N}$$
Wolfram Mathematica ...
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Sitrling's formula for gamma function with negative argument
We have the classical Stirling's approximation formula for the $\Gamma(\cdot)$ function in the form:
$$
\Gamma(x)\approx\sqrt{2\pi}\exp{(-x)}x^{x-1/2}
$$
for $x >0 $ as $x\to\infty$.
I am ...
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Evaluation of an integral with result as special functions and matching Mathematica's results
This question is about calculating the integral
\begin{equation}
I = \int_\varepsilon^\infty v^{\frac m2} (v-\varepsilon)^{\frac k2+j-1} \exp{\left(-\frac{v}{2\varsigma}\right)}dv
\end{equation}
where ...
- 365
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Relation between the Gauß Integral and the gamma function
If one takes a more "generalised" form of the famos Gauß integral, you can show the following:
$$
G(q)=\int \limits_{0}^{\infty}e^{-x^{q}}dx
\\
u=x^q \implies \frac{du}{dx}=qx^{q-1}\implies ...
- 129
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Equivalence product form of the Gamma function
From the LinesThatConnect video https://youtu.be/v_HeaeUUOnc?si=dUJfeHD9D0O_f5hL we could derive the factorial function
$$
\lim _{N \rightarrow \infty} N^x \prod_{k=1}^N \frac{k}{k+x}
$$
and it’s ...
- 21
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55
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Extending an integral in the complex plane
I'm having trouble with this exercise, I have to compute the integral:
$$
\int^{\infty}_{0}t^{z - 1}{\rm e}^{{\rm i}t}
\,{\rm d}t\quad
\mbox{where}\ \Re\left(z\...
- 133
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votes
0
answers
69
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Formula relating the Riemann zeta function and Euler's constant
In the digital library of mathematical formulas I have found the equation:
$$\zeta(1+s)=\frac{\sin(\pi s)}{\pi}\int_{0}^{\infty}x^{-s-1}\left(\gamma+\psi(1+x)\right)dx$$
for $0<Re(s)<1$, where $\...
- 133
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The golden ratio $\phi$ for $_2F_1\big(\frac16,\frac16,\frac23,-2^7\phi^9\big)$ and $_2F_1\big(\frac16,\frac56,1,\frac{\phi^{-5}}{5\sqrt5}\big)$?
I. Context
Given the golden ratio $\phi$, then we have the nice closed-forms,
\begin{align}
_2F_1\left(\frac16,\frac16,\frac23,-2^7\phi^9\right) &= \frac{3}{5^{5/6}}\phi^{-1}\\[6pt]
_2F_1\left(\...
- 56.2k
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1
answer
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Computing a seemingly "$i$th" order integral
I recently learned about the cauchy repeated integration formula, which is stated as follows
$$\int_{a}^{t}f(x)\hspace{1mm}dx^n = \frac{1}{\Gamma(n)}\int_{a}^{t}(t-x)^{n-1}f(x)dx$$
I decided to have ...
- 33
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$\int_0^1 \sqrt{\frac{2-x^4}{1-x^4}}\, dx$ in terms of the gamma function or elliptic integrals
Question
Can
$$\int_0^1 \sqrt{\frac{2-x^4}{1-x^4}}\, dx$$
be expressed in closed form in terms of the gamma function at rational arguments or in closed form in terms of elliptic integrals?
Thoughts
...
- 649
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Equation relating the Riemann zeta function and the digamma function
While going through a few functional equations for the Riemann zeta function I stumbled upon one connecting it to the Digamma function $\psi(x)$, the formula is:
$$\zeta(s)=\frac{1}{s-1}+\frac{\sin(\...
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On the tribonacci constant, $K(k_{163})$, and gamma functions
I. Some results
Let $T$ be the tribonacci constant, the real root of $x^3-x^2-x-1=0$. Then the exact value of the complete elliptic integral of the first kind $K(k_d)$ for $d=11$ is,
$$K(k_{11}) = \...
- 56.2k
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On the supergolden ratio and the general closed-form of $\prod_{m=1}^{d}\Gamma\left(\frac{m}{d}\right)^{\big(\frac{-d}{\; m}\big)}$?
I. Some results
In a previous post, we considered the complete elliptic integral of the first kind $K(k_{58})$ which is not in Mathworld's limited list of $K(k_d)$. Their list is short, and there is ...
- 56.2k
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1
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Equation connecting The Riemann Zeta function and the Gamma function [closed]
I am studying indipendently analitic number theory (I started with Apostol's book) and have now developed an interest in the Riemann Zeta function.
It is often possible while studying the Riemann zeta ...
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