All Questions
28
questions
0
votes
0
answers
74
views
Is it possible to find a closed form for $i!$? [duplicate]
I am curious is there a closed form for $i!$? I tried to search for any closed form for this but I didn't find any.
$$z! := \lim_{n \to \infty } n^z \prod_{k=1}^n \frac {k}{z+k}$$
$$i! =\lim_{n \to \...
4
votes
2
answers
188
views
A closed form for integrals of the type $\Gamma\left(\sigma+it\right)$?
Numerical evidence strongly suggests that:
$$\int_{-\infty}^{\infty}\Gamma\left(\sigma+it\right) \,\mathrm{d}t = 2\cdot\frac{\pi}{\mathrm{e}} \qquad \sigma \in \mathbb{R}, \sigma > 0$$
and
$$\int_{-...
- 3,191
3
votes
1
answer
163
views
Does a closed-form expression exist for $ \int_0^\infty \ln(x) \operatorname{sech}(x)^n dx $?
I am trying to find a closed-form expression for the following integral
$$
\int_0^\infty \ln(x) \operatorname{sech}(x)^n dx
$$
There are specific values that I would like to generate
(Table of ...
3
votes
1
answer
125
views
Neat function $I(x,y)= \sum\limits_{n=1}^{\infty}\bigg|\int_0^1 \frac{1}{t~(\log t)^y}\exp\bigg(\frac{n^x}{\log t}\bigg) ~dt~\bigg| $. Closed form?
Consider the following function:
$$ I(x,y)=\sum_{n=1}^{\infty}\bigg|\int_0^1 \frac{1}{t~(\log t)^y}\exp\bigg(\frac{n^x}{\log t}\bigg) ~dt~\bigg|$$
For $x=0$ and letting $y$ vary we get the Gamma ...
- 756
4
votes
2
answers
394
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 know about the ...
- 11.8k
1
vote
3
answers
212
views
Evaluate $\int_0^\infty x^{n+\frac12}e^{-\frac x2}\log^2x\,dx$ and $\int_0^\infty x^ne^{-\frac x2}\log^2x\,dx$
Determine the closed forms of $$\mathfrak I_1=\int_0^\infty x^{n+\frac12}e^{-\frac x2}\log^2x\,dx\quad\text{and}\quad\mathfrak I_2=\int_0^\infty x^ne^{-x/2}\log^2x\,dx$$ where $s>0$ is an integer.
...
- 27.1k
2
votes
1
answer
118
views
Compute in a closed form the following sum : $\sum_{n=1}^{+\infty}\frac{\Gamma^{4}(n+\frac{3}{4})}{(4n+3)^{2}\Gamma^{4}(n+1)}$
Today Im going to find the closed form of :
$\sum_{n=1}^{+\infty}\frac{\Gamma^{4}(n+\frac{3}{4})}{(4n+3)^{2}\Gamma^{4}(n+1)}$
My attempt :
We know that : $\Gamma(z)=\int_0^{+\infty}t^{n-1}e^{-t}...
- 205
6
votes
3
answers
322
views
Prove that $\int_0^1 \frac{x^2}{\sqrt{x^4+1}} \, dx=\frac{\sqrt{2}}{2}-\frac{\pi ^{3/2}}{\Gamma \left(\frac{1}{4}\right)^2}$
How to show
$$\int_0^1 \frac{x^2}{\sqrt{x^4+1}} \, dx=\frac{\sqrt{2}}{2}-\frac{\pi ^{3/2}}{\Gamma \left(\frac{1}{4}\right)^2}$$
I tried hypergeometric expansion, yielding $\, _2F_1\left(\frac{1}{2},\...
- 8,654
5
votes
1
answer
205
views
Prove $\sum_{n=1}^{\infty}\frac{\Gamma(n+\frac{1}{2})}{(2n+1)(2n+2)(n-1)!}=\frac{(4-π)\sqrt{\pi}}{4}$
Prove
$$S=\sum_{n=1}^{\infty}\frac{\Gamma(n+\frac{1}{2})}{(2n+1)(2n+2)(n-1)!}=\frac{(4-π)\sqrt{\pi}}{4}.$$
I don't know how to evaluate this problem .
At first I used partial fraction but I got ...
- 751
4
votes
1
answer
171
views
closed form of the following integral :$\int_{0}^{\infty}- \sqrt{x}+ \sqrt{x}\coth (x) dx$?
I have tried to evaluate this:$\int_{0}^{\infty}- \sqrt{x}+ \sqrt{x}\coth (x)$ using the the following formula
$$2 \Gamma(a) \zeta(a) \left(1-\frac{1}{2^{a}} \right) = \int_{0}^{\infty}\Big( \frac{x^{...
- 7,195
1
vote
0
answers
37
views
Closed form for an integral involving an incomplete Gamma function?
I am trying to find a closed form for this integral:
$$\int_{0}^{\infty}\int_{0}^{\infty}e^{-d_{p,s}^v\,x-d_{s,p}^v\, y+d_{p,p}^v\,\frac{\xi_1\,\sigma^2\,xy}{P_p\,\xi_2\,xy+\sigma^2}}\mathrm dy\...
- 11
3
votes
2
answers
152
views
Integration problem related to Gamma function: $ \int_{0}^{\infty} u^{\alpha + b - 1} \exp\left(-ub + u^{\alpha}c\right)du $
During my work on some statistics problem, I stumbled across the following integral:
$$ \int_{0}^{\infty} u^{\alpha + b - 1} \exp\left(-ub + u^{\alpha}c\right)du,\qquad \alpha, b, c>0 $$
I tried ...
- 131
2
votes
1
answer
130
views
On a closed form for $\int_{-\infty}^\infty\frac{dx}{\left(1+x^2\right)^p}$ [duplicate]
Consider the following function of a real variable $p$ , defined for $p>\frac{1}{2}$:
$$I(p) = \int_{-\infty}^\infty\frac{dx}{\left(1+x^2\right)^p}$$
Playing around in Wolfram Alpha, I have ...
- 3,309
3
votes
1
answer
284
views
Definite Gamma function integral
Curiosity Question
It's very well known that
$$\int_a^{a+1} \ln\Gamma(x)\ dx = \frac{1}{2}\ln(2\pi) - a - \ln(a) - (a+1)\ln\Gamma(a) + (a+1)\ln\Gamma(1+a)$$
Clearly provided that $\Gamma(a)\geq ...
- 26.3k
3
votes
2
answers
177
views
Evaluate $\int_0^{\pi/2}\frac{1-\sqrt[18]{\cos u}}{1-\cos u}du$, in terms of particular values of special functions and constants
I wondered about this question when I've considered similar integrals with different integrand as an analogous of a formula for harmonic number.
As example that I know that can be calculated using ...
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