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2 votes
1 answer
359 views

Defining integrals by residue theorem

I have always been interested in alternative definitions of mathematical objects. I wonder if one can craft an useful definition of definite integral by using the Residue Theorem from complex analysis....
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 ...
0 votes
1 answer
262 views

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} ...
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 \...
1 vote
1 answer
756 views

An integral involving the argument of the Gamma function and the Riemann Hypothesis

Evaluate $$I=\int_{0}^{\infty} \frac{t\arg \Gamma(\frac{1}{4}+\frac{it}{2})}{(\frac{1}{4}+t^2)^2}\mathrm{d}t$$ where $\Gamma(s)=\int_{0}^{\infty}e^{-x}x^{s-1}\mathrm{d}x.$ Note that $I$ converges ...
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 ...
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 ...
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|&...
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 $\mathbb{R}$. ...
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)...
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 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(...
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 ...
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}\,...
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: $$I=\int_{0}^{1} \int_{0}^{\infty} \left|\sum_{n=1}^{\infty} f(nx) e^{2 i \pi n y} \right|^2 dx dy$$ ...
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 ...
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 $...
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 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}(...
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 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 ...
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]\::\:{\...
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)+\...
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 ...
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-...
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(\...
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 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 ...
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, ...
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 ...
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 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 $U\subseteq\mathbb{C}$. ...
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 $...
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 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\...
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 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 ...
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 ...
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 ...
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 ...
2 votes
0 answers
604 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 ...
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 \...
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 ...
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 $w$ ...
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\,\...

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