All Questions
Tagged with cv.complex-variables integration
42 questions with no upvoted or accepted answers
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
$$\int_{1<|z|&...
- 1,997
9
votes
0
answers
1k
views
How complicated can an elementary antiderivative get?
I asked this question on MSE here.
I recently learned that there are many very large numbers that have been defined, such as $\operatorname{TREE}(3)$ and many others that are too big to be written ...
- 541
6
votes
0
answers
130
views
Complex beta function $\int_{\mathbb{R}^2} (x^2+y^2)^{\alpha-1}((1-x)^2+y^2)^{\beta-1} \,dx\,dy$
I am interested in showing that the integral
\begin{align}
& \int_{\mathbb{C}} |z|^{2\alpha-2}|1-z|^{2\beta - 2} \,dA(z) \\[8pt]
= {} & \int_{\mathbb{R}^2} (x^2+y^2)^{\alpha-1}((1-x)^2+y^2)^{\...
- 177
6
votes
0
answers
1k
views
Evaluating $\iint_{\mathbb{R}\times \mathbb{R}^{+}}e^{-|w-c_{1}|^2-|u-c_{2}|^2}\frac{1}{w_{1}+iw_{2}-u_{1}-iu_{2}}dw_{1}dw_{2}du_{1}du_{2}$
For $c_{1},c_{2}\in \mathbb{H}:=\{Im(z)>0\}$ I want to compute the following integral or prove it doesn't exist:
$$\int_{\mathbb{R}\times \mathbb{R}^{+}}\int_{\mathbb{R}\times \mathbb{R}^{+}}e^{-|...
- 5,474
5
votes
0
answers
650
views
Nature of function as $x\rightarrow\infty$
I'm studying the limits and applicability of Abel Plana summation for different test functions (class of functions). In doing so this just pops out and couldn't handle the said integral so asked here (...
- 774
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
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 :
$$ S_n(\alpha,\beta,\gamma) :=
\int_0 ^1 \cdots \int_0 ^1
\prod_{j=1}^n t_j^{\alpha-1}(...
- 395
5
votes
1
answer
752
views
Gaussian integral over a ball
How to compute the following integral?
$$\int_{\|x\|^2\leq R} \exp(-x^\ast G x+2\mathcal{Re}(x^\ast a)) \,dx,$$
where $x$ is an $M \times 1$ vector ($M\gg 1$), $G$ is a positive definite matrix, and $...
- 129
4
votes
0
answers
73
views
Saddle point approximation for multiple contour integrals
General Question: Is there a reference where the saddle point approximation is applied to multiple contour integrals?
In particular, say we have the integral
$$ I_N = \frac{1}{(2\pi i)^N} \oint \left[\...
- 241
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
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 $\delta$:
$$\int_{\delta}^...
- 1,431
4
votes
0
answers
287
views
Is there a coordinate free proof of the Morrey--Kohn--Hormander identity?
The Morrey--Kohn--Hormander identity is the key to proving vanishing/existence results on bounded pseudoconvex domains in $\mathbb{C}^n$, or more generally, Stein domains. See, for instance, the ...
- 18.7k
3
votes
0
answers
65
views
How to find or approximate (e.g. using method of steepest descent ) integral?
Can you give any advice on how to find or approximate the following integral
$$
F(t,y) = \int_{0}^{y}\frac{i e^{-\frac{3 t^2 \left(x^2+1\right)}{2 \left(9 x^2+1\right)}-i \frac{4 t^2 x}{9 x^2+1}}}{\...
- 31
3
votes
0
answers
646
views
On properties on a certain functional
Consider the following function:
$$F(z) = \omega(z)\sin^2\left(\frac{c\Gamma(z)}{z}\right)$$
Here, $\omega(z)$ is a weight we have to construct and $c$ is a constant.
The following three conditions ...
- 375
2
votes
0
answers
90
views
Computing a complex integral with many poles
For an integer $k\geq 1$, let $f:\mathbb{C}^k\to\mathbb{C}$ be such that $f$ is analytic in the region $\text{Re}(u_i) > -1$ (say) for each $1\leq i \leq k$, and decays rapidly on vertical lines (i....
- 1,990
2
votes
0
answers
94
views
Computing $\int_0^{2\pi}\frac{e^{ikt}}{|e^{it}-e^{it_0}|^m}~\text{d}t$, where $k\in\mathbb{Z}$, $t_0\in\mathbb{C}$, and $m=1,3,5,\dots$
I am working on a project on accurate numerical quadrature where I need to compute the following integral in order to find my quadrature weights,
$$
\int_{0}^{2\pi}\frac{{\rm e}^{{\rm i}kt}}{\,\left\...
- 21
2
votes
0
answers
219
views
Integral with product of two infinite sums
I am looking for references and results on integrals with product of two infinite sums:
$$S_1=\int_0^{\infty} \sum_{n=0}^{\infty} f(nx) \sum_{k=0}^{\infty} \overline{ g(kx)} dx$$
Above integral is ...
- 1,199
2
votes
0
answers
379
views
Is this double integral of Fourier series always real?
Consider $f(x)$ a function from $\mathbb{R^+}$ to $\mathbb{C}$ such that $f(x) \sim_0 x$ and $\int_{0}^{\infty} f(x) dx=\int_{0}^{\infty} x^2 f(x) dx=0$
Can we demonstrate that following integral is ...
- 1,199
2
votes
0
answers
60
views
Finding a function in contour integration involving Riemann mapping
Let $T$ be a rectifiable Jordan curve in $\mathbb{C},$ $G$ be the interior of $T,$ and $\Phi$ be a conformal map of the unit disk $\mathbb{D}$ onto $G.$ Let $\mathcal{P}_{n}$ be the space of algebraic ...
2
votes
0
answers
128
views
An estimation for holomorphic functions in the unit disc
I have proved using Green's theorem the following equality:$$\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
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:
$$I=\int_0^{\infty } \frac{e^{\frac{\alpha -\mathrm ...
- 121
1
vote
0
answers
144
views
integral over the unit sphere of $\Bbb C^n$
Please, is there a way to calculate this integral
$$\int_{S_{2n-1}} \frac{e^{a \langle z, \zeta \rangle}}{|z - \zeta|^{\beta}} \, d\sigma(\zeta)$$
where $ z $ is a fixed point in the complex unit ball ...
- 501
1
vote
0
answers
80
views
An integral containing modified Bessel functions
During my studies I am facing the following problem. Let $I_\nu(x)$ be the modified Bessel function for $\nu\in(0,1/2]$.
I want to compute the following integral (it is are resolvent)
$$
R(z) = \frac{...
- 11
1
vote
0
answers
338
views
Recognizing when a $2\pi$-periodic function is a shifted sine
Consider a real analytic function $f(x)$ with a period of $2\pi.$ Let $t\in \operatorname{Range}(f(x))$, $$I(t)=\int_{0}^{2\pi}\log |t-f(x)|dx.$$ Prove that $I(t)$ is equal to a constant if and only ...
- 21
1
vote
0
answers
59
views
Extension to all dimensions of complex line integral
Let $\Gamma$ be a smooth curve in $\mathbb{C}^d$. Since $\mathbb{C}^d$ can be seen as $\mathbb{R}^{2d}$, one can define the line integral of functions $f:\Gamma\to \mathbb{C}$ using for instance ...
- 311
1
vote
0
answers
196
views
Asymptotic of a functional as $x\rightarrow \infty$
Consider the following functional :
$$
I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy, s) − F(x −\mathrm iy, s)}{\mathrm e^{2πy}-1},
$$
where $ F(z, s) = \dfrac{\sinh(\sin^2[π\Gamma(z)/(2z)])...
- 375
1
vote
0
answers
69
views
How to fix multi-valued function on contour?
I am sorry to ask such an embarrassingly simple question here. My question is about contour integral of the multivalude function.
I want to calculate the Fourier transformation of a muti-valued ...
- 395
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
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
$$\Phi = \tan \frac{\pi \alpha}{2} \mathbf{1}_{\{ \alpha \neq 1 \} } - \frac{2}{\pi}...
1
vote
0
answers
83
views
Complex integration over 1-singular chain
Let $f$ be a continuous function on $\mathbb{C}$ and assume that $\lim_{z\to \infty} zf(z) = \lambda.$ Let us note for all natural $n$ $$C_n = \{z \in \mathbb{C} : |z|=n\}.$$ Then, a usual fact of ...
- 1,017
1
vote
0
answers
336
views
Residue Cancellation
I am trying to understand how to apply the residue theorem to solve
$\frac{1}{2\pi j}\int^{\gamma+j\infty}_{\gamma-j\infty}\Gamma(n-s)\Gamma(s)\Gamma(1-s) {}_1F_1(s;b;c) \left(\frac{b}{c}\right)^s\,\...
- 447
0
votes
0
answers
77
views
Completeness of a normed space
We consider the set $\mathcal{PC}([-r,0],X)$
$$\mathcal{PC}([-r,0],X):=\{\varphi:[-r, 0] \rightarrow X: \varphi \text{ is continuous everywhere except
for a finite number
of points } t_* \text{ ...
- 39
0
votes
0
answers
120
views
How to prove an equality involving Laguerre polynomials
Assume $\mu<0$. Let $L^\alpha_n(x)$ be Laguerre polynomials of type $n$ and $f\in L^2(\Bbb C)$.
How to prove that $$\sum^\infty_{k=0}\int_{\Bbb C} f(z)\frac{1}{2(2k+1)-\mu}L^0_k(\lvert z\rvert^2)...
- 501
0
votes
0
answers
103
views
Computing $\int^{4b}_0 {e^{-tx}\biggl(\frac{\sqrt{4bx-x^2}}{(2b-2c)^2+4cx}\biggr) dx}$
I am a PhD student working on complex analysis. After integrating over a keyhole contour to obtain the inverse of a particular Laplace transform, I ended up with the following integral:
$$\frac{1}{\pi}...
- 9
0
votes
2
answers
682
views
On integral relating logarithm of absolute value of Zeta function
Sorry for such a direct question:
Consider the following integral:
$$I(t)=\int_{1/2}^{1} {\log|\zeta(a+it)|}da.$$
How to find the nature of $I(t)$ as $t\rightarrow\infty$?
- 774
0
votes
0
answers
83
views
The loss of double periodicity (ellipticity)
Consider a meromorphic function $f(\mathfrak{a}_1, \mathfrak{a}_2)$, such that
$$
\begin{align}
f(\mathfrak{a}_1, \mathfrak{a}_2) = f (\mathfrak{a}_1 + 1, \mathfrak{a}_2) = f(\mathfrak{a}_1 + \tau, \...
- 857
0
votes
0
answers
266
views
Bounding the absolute value of a complex integral
I'm working on some problems involving Fourier transforms and convolution problems and there is one problem I cannot solve. In my situation we have a complex valued function $f(ix)$, with $x\in\mathbb{...
- 129
0
votes
0
answers
47
views
Oscillatory integral independent of a parameter
Let $h: \mathbb{R} \mapsto \mathbb{C}$ be a positive definite function, continuous at the origin. (In fact, $h$ is the Fourier transform of a finite measure). Define the oscillatory integral
$$Q(t) :=...
- 222
0
votes
0
answers
43
views
integrating multivariable rational function over a product of disks
Suppose I have a rational function of $k$ complex variables:
$$ R(x_1,...,x_k) = P(x_1,...,x_k)/Q(x_1,...,x_k) $$
where $P$ and $Q$ are polynomials. Now I'd like to compute the integral of this ...
- 169
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
0
votes
0
answers
381
views
Help with an irregular integral
I am looking for help with doing the following integral :
$$\frac{1}{2\pi i}\int_{1}^{\infty}\ln\left(\frac{1-e^{-2\pi i x}}{1-e^{2\pi i x}} \right )\frac{dx}{x\left(\ln x+z\right)}\;\;\;\;z\in \...
- 181
-1
votes
1
answer
406
views
Topological properties of complex valued Riemann sum limit curve and a particular integral inequality
I am studying under what conditions the following integral inequality would hold ($a$ real, $a>0$):
$$ \int_{-\infty} ^{\infty} \frac{f(ix)}{a\pm ix}dx\ = 0 \ \ \ \ \Rightarrow \ \ \ \int_{-\...
- 362