All Questions
Tagged with cv.complex-variables integration
98 questions
3
votes
1
answer
187
views
Variation of steepest descent/Laplace methods for non-exponential integrands
I was wondering if versions of the Laplace/steepest descent methods exists for integrals of the type
$$\int_C f(z) M(\lambda g(z)) dz$$
for $\lambda >>0$ functions $f(z), g(z): \mathbb C \...
- 135
2
votes
0
answers
219
views
Integral with product of two infinite sums
I am looking for references and results on integrals with product of two infinite sums:
$$S_1=\int_0^{\infty} \sum_{n=0}^{\infty} f(nx) \sum_{k=0}^{\infty} \overline{ g(kx)} dx$$
Above integral is ...
- 1,199
13
votes
0
answers
497
views
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
$$\int_{1<|z|&...
- 1,997
3
votes
1
answer
266
views
Complex integral
We would like to compute (or bound) the following complex integral:
$$\int\limits_0^{\infty}\left|\int\limits_{-\infty}^{\infty}\frac{e^{its}}{e^s-\lambda}\,ds\right|\,dt$$
where $\lambda \notin S_{\...
2
votes
2
answers
270
views
An inner product on the vector space $\mathbb{R}[x_1,\cdots,x_n]_m$
For any given integers $m,n\geq1$, let $\mathbb{R}[x_1,\cdots,x_n]_m$ be the vector space of homogeneous polynomials of degree $m$ in $x_1,\cdots,x_n$ over the field of real numbers $\mathbb{R}$. ...
- 1,997
0
votes
0
answers
158
views
On reasonable asymptotic estimates for some integral involving the logarithm of the Riemann zeta function
Let
$$I(T) = \int_{-T}^{T} \frac{\log|\zeta(\frac{1}{2} + it|)|}{\frac{1}{4}+t^2}\mathrm{d}t$$
where $\zeta$ denotes the Riemann zeta function.
What are the reasonable asymptotic estimates for $I(T)...
user123416
1
vote
0
answers
108
views
Asymptotics of an integral by two methods
This was asked in MSE, here, but the answer was not satisfactory.
I want to compute the asymptotic behavior of the integral
$$ f(K,a)=\int_0^1 (1-x)^Ke^{iKa\frac{x}{1-x}}x^2dx$$
when $K$ is large ...
- 1,549
1
vote
1
answer
256
views
Seeking the derivation of the Fourier Sine Transform of $x^{2\nu}(x^2+a^2)^{-\mu-1}$
In this answer on math.stackexchange.com the Fourier Sine Transform of $x^{2\nu}(x^2+a^2)^{-\mu-1}$ is given in terms of the generalized hypergeometric function:
$$\frac{1}{2}a^{2\nu-2\mu}\frac{\Gamma(...
- 2,239
10
votes
1
answer
803
views
A natural residue formula
A residue formula
I have strong evindence to believe that the following identity holds:
$$
\frac{n!}{2\pi i}\oint_{|z-1|=\epsilon} \frac{z^{a-1} \mathrm{d}z}{(z^d-1)^{n+1}} = d^{-n-1}\prod_{j=1}^{n}\...
- 833
0
votes
1
answer
121
views
Integration over convex curves
Let $S$ be a noncompact closed convex proper subset of $\mathbb{C}.$
It is well-known that the boundary $\partial S$ is a rectifiable curve et hence, that we can make integration over it. Is there ...
- 193
1
vote
0
answers
101
views
Density of Geometric Stable distribution
If we define
$$
\psi(t|\alpha, \beta, \gamma, \mu) = -it\mu+|\gamma t|^\alpha(1-i\beta \mathrm{sgn}(t) \Phi)
$$
with
$$\Phi = \tan \frac{\pi \alpha}{2} \mathbf{1}_{\{ \alpha \neq 1 \} } - \frac{2}{\pi}...
1
vote
1
answer
535
views
Incoherence of Fubini therorem with integral on Fourier series
I ask this question because of the apparent incoherence of the value of following integral:
$$I=\int_{0}^{1} \int_{0}^{\infty} \left|\sum_{n=1}^{\infty} f(nx) e^{2 i \pi n y} \right|^2 dx dy$$
...
- 1,199
2
votes
0
answers
379
views
Is this double integral of Fourier series always real?
Consider $f(x)$ a function from $\mathbb{R^+}$ to $\mathbb{C}$ such that $f(x) \sim_0 x$ and $\int_{0}^{\infty} f(x) dx=\int_{0}^{\infty} x^2 f(x) dx=0$
Can we demonstrate that following integral is ...
- 1,199
5
votes
1
answer
690
views
Simple integral representation for a beta function with more than two variables
The beta function $B(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)}$ can be written for $\Re x, \Re y > 0$ symmetrically as $$ B(x,y) = \int_{-\frac12}^{\frac12}(\frac12-t)^{x-1}(\frac12+t)^{y-1}\,...
- 13.4k
6
votes
0
answers
1k
views
Evaluating $\iint_{\mathbb{R}\times \mathbb{R}^{+}}e^{-|w-c_{1}|^2-|u-c_{2}|^2}\frac{1}{w_{1}+iw_{2}-u_{1}-iu_{2}}dw_{1}dw_{2}du_{1}du_{2}$
For $c_{1},c_{2}\in \mathbb{H}:=\{Im(z)>0\}$ I want to compute the following integral or prove it doesn't exist:
$$\int_{\mathbb{R}\times \mathbb{R}^{+}}\int_{\mathbb{R}\times \mathbb{R}^{+}}e^{-|...
- 5,474
2
votes
0
answers
60
views
Finding a function in contour integration involving Riemann mapping
Let $T$ be a rectifiable Jordan curve in $\mathbb{C},$ $G$ be the interior of $T,$ and $\Phi$ be a conformal map of the unit disk $\mathbb{D}$ onto $G.$ Let $\mathcal{P}_{n}$ be the space of algebraic ...
4
votes
0
answers
261
views
Is the following integral positive or not?
Let $n$ be a given even positive integer. We have the following integral
\begin{eqnarray}
&&\int_0^1\cdots\int_0^1\prod\limits_{i=1}^n\prod\limits_{j=1}^n(x_i-y_j)dx_1\cdots dx_ndy_1\cdots ...
- 1,997
7
votes
1
answer
1k
views
The sinc function strikes again [duplicate]
Recall $\text{sinc}(x)=\frac{\sin x}x$. It's a familiar exercise that $\int_0^{\infty}\text{sinc}(x)\,dx=\frac{\pi}2$.
But, at present, I wish to ask about the following claim on a "sinc-ing" ...
- 43.1k
1
vote
0
answers
252
views
Contour integration and application of residue theorem [closed]
I found the following contour integration done in an article, but I do not fully comprehend what has actually been calculated here? Contour integration
The argument of the function $s$ is supposed to ...
- 11
1
vote
1
answer
686
views
Analytic continuation of definite integral?
This question comes from physics...
We have the following functional (or function dose not matter):
$$S\left[x\right]:=\intop_{-\frac{t_{0}}{2}}^{+\frac{t_{0}}{2}}dt\,\mathcal{L}\left[x\right]\::\:{\...
- 121
2
votes
0
answers
128
views
An estimation for holomorphic functions in the unit disc
I have proved using Green's theorem the following equality:$$\frac{1}{2\pi i}\left(\frac{1}{\zeta}-\bar{\zeta}\right)\int_{\partial \mathbb{D}}f(z+\zeta)\log\left(\frac{z}{z-\zeta}\right)dz=f(\zeta)+\...
- 121
5
votes
1
answer
752
views
Gaussian integral over a ball
How to compute the following integral?
$$\int_{\|x\|^2\leq R} \exp(-x^\ast G x+2\mathcal{Re}(x^\ast a)) \,dx,$$
where $x$ is an $M \times 1$ vector ($M\gg 1$), $G$ is a positive definite matrix, and $...
- 129
1
vote
3
answers
280
views
Evaluation of Gaussian density integral
Is there any closed form, asymptotics, and/or approximations for the following integral:
$$f(c) := \frac{1}{\sqrt{2\sigma^2\pi}}\int e^{-x^2/2\sigma^2}\frac{x^2}{1-cx^2}dx,$$ where $\sigma^2$ is real ...
- 135
-1
votes
1
answer
406
views
Topological properties of complex valued Riemann sum limit curve and a particular integral inequality
I am studying under what conditions the following integral inequality would hold ($a$ real, $a>0$):
$$ \int_{-\infty} ^{\infty} \frac{f(ix)}{a\pm ix}dx\ = 0 \ \ \ \ \Rightarrow \ \ \ \int_{-\...
- 362
3
votes
1
answer
244
views
How to compute $\int_{\mathbb S^2} e^{-i\left<t,\omega\right>} \, e^{-i\left< A(\omega)x,y\right>} \, d\sigma(\omega)$
I would like compute the following
$$I_{t,x,y} = \int_{\mathbb S^2} e^{-i\left<t,\omega\right>} \, e^{-i\left< A(\omega)x,y\right>} \, d\sigma(\omega); $$
where $\mathbb S^2$ is the two-...
- 650
4
votes
1
answer
132
views
Integral Expression in Complex Dynamics
Let $\phi\in \mathbb{C}(z)$ be a degree $d\geq 2$ rational map, which we can write as $\phi = \frac{f}{g}$ for $f,g\in \mathbb{C}[z]$. Let $\omega_{FS}$ denote the Fubini-Study form on $\mathbb{P}^1(\...
- 41
4
votes
1
answer
347
views
Two similar integrals
Let $n$ be a given even positive integer. We have the following integral
\begin{align}
\int_0^{\infty}\cdots\int_0^{\infty}e^{-(x_1+\cdots+x_n+y_1+\cdots+y_n)}\prod\limits_{i=1}^n\prod\limits_{j=1}^n(...
- 1,997
1
vote
0
answers
83
views
Complex integration over 1-singular chain
Let $f$ be a continuous function on $\mathbb{C}$ and assume that $\lim_{z\to \infty} zf(z) = \lambda.$ Let us note for all natural $n$ $$C_n = \{z \in \mathbb{C} : |z|=n\}.$$ Then, a usual fact of ...
- 1,017
5
votes
0
answers
226
views
Contour integral $\int_{\gamma}e^{-t\mu}(-\mu)^{-m}\log\sqrt{-\mu}d\mu$
Motivation: In my research I try to find the asymptotics of the heat trace $\text{tr}e^{-t\Delta}$ as $t\to0$, where $\Delta$ is Laplace operator on a manifold with singularities. First I find the ...
- 51
3
votes
1
answer
1k
views
A Gaussian integral over complex variables by a defined Green's function for a Gaussian ensemble of random matrix
We construct an $N\times N$ matrix $J$ whose elements are drawn from Gaussian distribution with zero mean and variance $\frac{1}{N}$. Since we want to have different variances for different columns, ...
- 325
13
votes
1
answer
840
views
$\pi e$ and an unfamiliar polynomial
Ever since my exposure to this integral involving $\pi e$, I've conjectured and set about evaluating the possible nature of the following integral
$$\int_0^1 x^m \sin(\pi x) x^x (1-x)^{1-x} \ dx, \...
- 1,549
4
votes
2
answers
426
views
On the search for an explicit form of a particular integral
Let $f$ be integrable over the interval $(0, 1)$, and
$$I_n = \int_0^{1} x^n f(x) \, \mathrm{d}x.$$
Suppose $f(x) = f(1-x)$; we can then show that
$$I_n = \sum_{k=0}^{n} \binom{n}{k} (-1)^k \, I_{k}...
- 1,549
-1
votes
1
answer
2k
views
Real and imaginary part of an holomorphic function
I guess this could be a very elementary question. Anyway I can not find an answer in literature.
Let $f:U\rightarrow\mathbb{C}$ be an holomorphic function on an upen subset $U\subseteq\mathbb{C}$. ...
user61586
3
votes
1
answer
225
views
A conjecture regarding the integral of the square of an entire function
Can some help me prove or disprove the following assertion which I encountered in research? Thanks!
Let $f:\mathbb R\to\mathbb R$ be an analytic function. If for $\forall c > 0$, we can find some $...
- 113
4
votes
0
answers
684
views
A difficult integral which the Risch algorithm shows is not elementary
For reasons which aren't conceptually related to the problem a few of my colleagues and I are in need of finding an expression for the following integral in terms of $a$ and $\delta$:
$$\int_{\delta}^...
- 1,431
0
votes
1
answer
342
views
Integrate Faddeeva function
I came across this integration in my studies.
$\int_{-\infty}^{\infty}|F((w_\textbf{_} - \hat{w_\textbf{_}})\tau) |^2 . d\tau$
It uses the Faddeeva function which is $F(z) = e^{-z^2}erfc(-iz)$. I ...
1
vote
1
answer
1k
views
Contour integral around semi-circle
Can one use contour integration to evaluate $\int^{\pi}_{0} \frac{1}{1-\rho*sin(\theta)}d\theta$ for $0<\rho<1$? This would be trivial if the upper limit were $2\pi$ as we could let $z=e^{i\...
- 211
4
votes
0
answers
287
views
Is there a coordinate free proof of the Morrey--Kohn--Hormander identity?
The Morrey--Kohn--Hormander identity is the key to proving vanishing/existence results on bounded pseudoconvex domains in $\mathbb{C}^n$, or more generally, Stein domains. See, for instance, the ...
- 18.7k
5
votes
1
answer
434
views
Laurent expansion of a principal value integral
Let $f(t)$ be a nice Hölder continuous function. Also, suppose that $f$ is even. I'm interested in evaluating integrals of the form:
$$\oint (1-z)^{k+1}\int_0^1 \frac{f(t)}{(1-zt)^{n+1}}dtdz,$$
...
- 4,952
4
votes
2
answers
600
views
I don't understand behavior of this integral, help!
In an answer to a question I needed the following integral:
$$
f(z):=\int\limits_0^\infty t\coth(zt)e^{-t^2}dt;
$$
it represented deviation from modularity of some other function. However I noticed ...
- 18.4k
1
vote
1
answer
278
views
Request for help with two integrals
It would be great if someone can help me do these integrals - using numerical integration on Mathematica it seems that these converge - in what follows $a \in \mathbb{R}$ and $q \in \mathbb{N}$ and $n ...
- 1,893
5
votes
0
answers
437
views
From Selberg integral to Dyson integral
My question is about the derivation from Selberg integral to Dyson integral in this paper:
Selberg integral :
$$ S_n(\alpha,\beta,\gamma) :=
\int_0 ^1 \cdots \int_0 ^1
\prod_{j=1}^n t_j^{\alpha-1}(...
- 395
2
votes
0
answers
603
views
complex contour integral calculation after Möbius transformation
Good day to everyone.
In my scientific research I've got stuck with a contour integration problem.
I would like to evaluate the following integral:
$$I=\int_0^{\infty } \frac{e^{\frac{\alpha -\mathrm ...
- 121
0
votes
0
answers
381
views
Help with an irregular integral
I am looking for help with doing the following integral :
$$\frac{1}{2\pi i}\int_{1}^{\infty}\ln\left(\frac{1-e^{-2\pi i x}}{1-e^{2\pi i x}} \right )\frac{dx}{x\left(\ln x+z\right)}\;\;\;\;z\in \...
- 181
10
votes
1
answer
432
views
Modern version of an inequality of R. M. Gabriel for contour integrals
I am currently reading the 1998 article Dynamics of the Binary Euclidean Algorithm:
Functional Analysis and Operators by Brigitte Vallée, which cites a 1928 article by R. M. Gabriel for the following ...
- 6,206
1
vote
0
answers
336
views
Residue Cancellation
I am trying to understand how to apply the residue theorem to solve
$\frac{1}{2\pi j}\int^{\gamma+j\infty}_{\gamma-j\infty}\Gamma(n-s)\Gamma(s)\Gamma(1-s) {}_1F_1(s;b;c) \left(\frac{b}{c}\right)^s\,\...
- 447
4
votes
1
answer
6k
views
Inverse of a function defined by an integral
Hi, I have a function defined by an integral as follows.
$$
z=f(w) = \int_0^w \frac{(\zeta-a_1)^{\alpha_1}(\zeta-a_2)^{\alpha_2}...}{(\zeta-b_1)^{\beta_1}(\zeta-b_2)^{\beta_2}...}\ d\zeta
$$
where $w$ ...
- 167
7
votes
2
answers
1k
views
Contour integration problem from probability
Can integrals of the form
$$
\int_{-\infty}^{\infty}{\exp\left(-\left[x - c\right]^{2}\right) \over 1 + x^{2}}\, {\rm d}x
$$
be computed in closed form using contour integration (or any other ...
- 5,217