All Questions
8 questions
1
vote
1
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189
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How to evaluate the following integral?
How to (analytically) calculate the following integral,
$$I = \int_{S_{2n-1}} \left( \langle z, \zeta \rangle \right)^a e^{ b \langle z, \zeta \rangle} \, d\sigma(\zeta),$$
where $\langle z, \zeta \...
- 501
2
votes
1
answer
315
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Where to find or how to establish a general formula for the improper integral $\int_{0}^{\infty}\frac{\sin t}{t}(\ln t)^k\operatorname{d\!} t$?
When I tried to establish the Maclaurin power series expansion of the reciprocal $\frac{1}{\Gamma(z)}$ of the Euler gamma function $\Gamma(z)$, I came across the improper integral $$
I_k=\int_{0}^{\...
- 1,091
1
vote
0
answers
161
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Justify $\int_0^\infty e^{-ax^2}\ \mathrm{d}x$ for complex $a$ and zero real part [closed]
(Reposted from math stack exchange)
I have searched and failed to find a rigorous proof showing that
$$\int_{0}^\infty e^{-ax^2}\ \mathrm{d}x = \frac{\sqrt{\pi}}{2\sqrt{a}}$$
is true for $\Re(a)=0$ ...
user161698
3
votes
1
answer
223
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Ratio of Selberg integral
I'm considering a ratio of incomplete Selberg integral:
$$f_n(a,b)=\frac{\int_{\Delta_a}\prod_{i=1}^nx_i^{\alpha-\frac{n+1}{2}}\prod_{i=1}^n(1-x_i)^{-1/2}\prod_{i<j}|x_i-x_j|}{\int_{\Delta_b}\prod_{...
- 1,495
0
votes
1
answer
262
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Solving or bounding the real part of the integral $\int_0^{2 \pi i m} \frac{e^{-t}}{t-a} dt$
I would be interested in finding a closed form or, at least, bounding (in terms of $m$ as it becomes larger) the real part of the following itnegral:
$$f(m,a):=\int_0^{2 \pi i m} \frac{e^{-t}}{t-a} ...
- 193
13
votes
0
answers
497
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Is it possible that the following integral is $0$?
Given any integer $n\geqslant1$, let $E,F$ be two subsets of $\{\{i,j\}:1\leqslant i<j\leqslant n\}$ such that every two sets in $F$ are disjoint.
It is not difficult to see that
$$\int_{1<|z|&...
- 1,997
2
votes
2
answers
270
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An inner product on the vector space $\mathbb{R}[x_1,\cdots,x_n]_m$
For any given integers $m,n\geq1$, let $\mathbb{R}[x_1,\cdots,x_n]_m$ be the vector space of homogeneous polynomials of degree $m$ in $x_1,\cdots,x_n$ over the field of real numbers $\mathbb{R}$. ...
- 1,997
5
votes
0
answers
437
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From Selberg integral to Dyson integral
My question is about the derivation from Selberg integral to Dyson integral in this paper:
Selberg integral :
$$ S_n(\alpha,\beta,\gamma) :=
\int_0 ^1 \cdots \int_0 ^1
\prod_{j=1}^n t_j^{\alpha-1}(...
- 395