For questions about integration methods that use results from complex analysis and their applications

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6
votes
0answers
201 views

Complex contour integral: How does the stationary point method used in this case?

I was reading a paper which has the following integral in order to do the inverse Laplace transformation: $$ I=\frac{1}{2\pi i}\int_{-i\infty+\gamma}^{i\infty+\gamma} ...
4
votes
0answers
51 views

Integrate complex function over $\mathbb{C}^2$

I have a question in mind and I would appreciate your help. Usually in complex analysis we consider integrals of the form $\int_\gamma f(z) dz$ where $\gamma $ is a contour and ...
3
votes
0answers
23 views

Complex line integration with assumptions

Let $f: \mathbb{C} \to \mathbb{C} $ be a holomorphic function with $$ \lim_{\lvert z \rvert\to\infty} \frac{f(z)}{z^{n-1}} = 0$$ for some $n\in\mathbb{N}$. How can I prove that $$ \lim_{r\to\infty} ...
3
votes
0answers
44 views

Inverse Mellin of the exponential of the digamma function

(Cross-posted from mathoverflow: No answers yet; bounty there expires in less than 24 hours) I'm looking for a function $f_p(x)$ with real parameter $p>0$ satisfying $$ \int_0^\infty ...
3
votes
0answers
3k views

Integrate complex conjugate

I'm doing some exercises on complex integration, and stumbled over this: $$\int_\gamma \bar{z} dz,$$ Where $\gamma$ is any closed $C^1$ curve which is the boundary of a bounded, connected, open $U ...
2
votes
0answers
16 views

If $f(z)=\sum_{n=0}^{\infty}c_{n}(z-z_{0})^{n}$, then $c_{n}r^{n}=\frac{1}{2\pi}\int_{0}^{2\pi}f\left(z_{0}+re^{-2nt}\right)dt$

Let $U\subseteq \mathbb{C}$ be an open set, $z_{0}\in U$ and $R>0$ such that $\mathcal{B}_{R}(z_{0}) \subseteq U$. Let $f:U\rightarrow \mathbb{C}$ be a holomorphic function with Taylor's serie ...
2
votes
0answers
71 views

Why is this integral finite?

I am looking at this integral $\displaystyle\int_{-\infty}^\infty dx \, \frac{e^{iax}}{\sinh^2{bx}}$. Now dividing the integral into real and complex parts, respectively $\int_{-\infty}^{\infty} dx ...
2
votes
0answers
25 views

Evaluating functions similar to the Bessel functions

In the problem there are two integrals, and one is asked to evaluate them by taking an integral over a unit circle of some chosen function. The integrals are $$\int_0^{2\pi}e{^{\sin n\theta}}\cos ...
2
votes
0answers
75 views

Integral of $e^{ix^2}$

How does one evaluate $$\int_{-\infty}^{\infty} e^{ix^2} dx$$ I know the trick how to evaluate $\int_{-\infty}^{\infty} e^{-x^2}dx$ but trying to apply it here I get a limit which does not converge: ...
2
votes
0answers
90 views

Why does this example of global residue theorem not work?

This question is related to and inspired by a previous question What is the residue obtained from this integral? , but note that the appearing functions are slightly different. Consider the following ...
2
votes
0answers
30 views

Complex integral difficulty

We need some sort of analytic expression for the integral: $$\int^\infty _{-\infty} \frac{\sqrt{\alpha^2 + 1}}{(\mathrm{i}\alpha)^{\frac{3}{4}}}\mathrm{e}^{\mathrm{i}\alpha \chi} \mathrm{d}\alpha$$ ...
2
votes
0answers
45 views

Moving limit inside a contour integral

I'm trying to compute this integral as part of a larger problem I'm working on. I'm trying to solve the integral $\int_0^\infty \frac{\sin(x)}{x}dx$ and to do it I'm using the method where you ...
2
votes
0answers
171 views

Complex line integral for closed homotopic curve.

I try to show that the following analogous result for $\textbf{closed curve}$ : Let $\gamma_0, \gamma_1$ be continuous paths in a domain $D \subseteq \mathbb{C}$ and $\omega$ be a closed form on $D$. ...
2
votes
0answers
36 views

How can I solve $\int \left| \sum_{i=0}^{N} f_i(x) \right|^2 d x $ for complex functions $f_i$?

The $f_i(x)$ are complex functions and I know that $\int_{-\infty}^{\infty} f_i (x ) \, d x=c_i$. How can solve this integral? $$\int_{-\infty}^{\infty} \left| \sum_{i=0}^{N} f_i(x) \right|^2 d x $$ ...
2
votes
0answers
64 views

Imaginary part of Log laplacian

I'm confused about how to calculate $\nabla^2 \log z$, where $z=re^{i\theta}$ is a complex number. My calculations return $$ \nabla^2 \log z = 2\pi\frac{\delta(r)}{r} [\delta(\theta) + i ...
2
votes
0answers
35 views

$ g(u;s_1,s_2)=\int_{(\alpha+\beta)^{-1}}^{\alpha+\beta}b^{s_2}(u^2-b)^{-\dfrac{s_1+s_2}{2}}(u^2b-1)^{-\dfrac{s_1+s_2}{2}} \mathrm {d}b$

Do you have any idea how to compute this function? $$ \displaystyle g(u;s_1,s_2)=\int_{(\alpha+\beta)^{-1}}^{\alpha+\beta}b^{s_2}(u^2-b)^{-\dfrac{s_1+s_2}{2}}(u^2b-1)^{-\dfrac{s_1+s_2}{2}} \mathrm ...
2
votes
0answers
291 views

Laplace transform of a product of functions

While trying to compute the Laplace transform of a certain product, part of the calculation leaves me with a Bromwich integral which has the form: ...
2
votes
0answers
137 views

Show Smoothness by Morera

I'm trying to show smoothness on $(0,\infty)(\Re)$ of the following function: $$ f(t,x)=\sum_{n=-\infty}^\infty e^{-\large \frac{(x-2\pi n)^2}{2t}}\frac{1}{\sqrt{2\pi t}} $$ The function is ...
1
vote
0answers
18 views

Complex integration by substitution

Integrate $ f(z) $ counterclockwise around the unit circle. $$ f(z) = 1/(4z-3) $$ My solution C(contour) : $ z(t) = \cos{t} + i\sin{t} = e^{it}, 0<t\leq 2\pi $ $$ \oint_C \frac{1}{4z-3} dz = ...
1
vote
0answers
50 views

Complex integration with infinitely many poles on imaginary axis

I'm trying to integrate with a closed contour on the upper-half of the complex plane. $I = \displaystyle\int_{-\infty}^\infty \dfrac{z\,\mathrm{sech(z)}}{[(z-a)^2+b^2][(z+a)^2+b^2]} dz$ There are ...
1
vote
0answers
40 views

Integral of complex logarithm on a disk in the plane

Let $a$ be a complex number and $D$ the disk centered around $0$ and of radius $R$. I would like to compute the integral I=$\int_D \log(|z-a|)d^2z$. I am interested in particular in the case $R\gg ...
1
vote
0answers
14 views

Is there a general algorithm to determine new contours for multivariable change of integration variables

Is there a general algorithm to determine the new region of integration upon a multivariable change of variables (where the old variables are a function of all the new variables). I have to do a ...
1
vote
0answers
53 views

integral of $e^{-\imath x}$ from $a$ to $\infty$ and characteristic functions

In the following result: $$\frac{1}{2 \pi} \int_{-\infty}^{\infty} \Phi(u) \left[\int_a^{\infty} e^{-i u x}dx\right] du = \frac{1}{2} + \frac{1}{ \pi} \int_0^{\infty} \Re\left[\frac{e^{-i u a} ...
1
vote
0answers
102 views

Differentiating a period of an elliptic curve under the integral sign

Let $$g = \frac{27J}{1 - J},$$ where $J$ is the absolute invariant, and define $$\Omega = \int_{\gamma(J)} \frac{dz}{\sqrt{4z^3 + g(z + 1)}}.$$ Here, $\gamma(J)$ is a contour in the complex plane that ...
1
vote
0answers
37 views

Complex integration upper semi-circle $\int f(z)dz$ when $r>1$

Question: Let $r$ be a real number, $r > 0$ and let $Lr$ be the line from the point $−r$ to $r$ in $C$. Let $γr$ be the upper half circle with radius $r$ and center in $0$. $$ \ f(z) = ...
1
vote
0answers
27 views

Evaluate $ \int_{\gamma} (z + \frac {1}{z - 1}) \, dz $ where $ \gamma $ is the perimeter of the parallelogram with vertices $ i, -i, 2 + i, 2 - i $

I'm learning about the theory of integration in the complex plane and need to verify my work to this problem since my textbook is using a different method of resolution : $ $ Evaluate the line ...
1
vote
0answers
31 views

Average Value of an Analytic Function on a Circle

I have encountered a pretty classic statement, but in the book I am working through it is presented prior to the Cauchy Integral Theorem. Here is the exercise that I'm sure most of you are familiar ...
1
vote
0answers
44 views

Area integral over complex plane of non-holomorphic gaussian $e^{-z\bar{z}}$

Let $A$ be the two-dimensional area integral, over the complex plane, of a gaussian function: $$A = \int_\text{plane} \frac{i}{2} d\bar{z} \wedge dz \ \exp[-z\bar{z}]$$ Of course one could evaluate ...
1
vote
0answers
52 views

How to solve an integral with a fractional order.

How should I find a value of these integrals: $$ A:=\int_{0}^{\infty}\frac{\sin(x)}{(x+1)^{2-\nu}x^{\nu}}dx \quad\text{and}\quad B:=\int_{0}^{\infty}\frac{\sin(x)}{(x+1)^{1-\nu}x^{\nu}}dx, $$ where ...
1
vote
0answers
41 views

Some questions about integration in a complex plane

On a two-dimension infinite plane we can always denote a complex number z which satisfies: $$ z=x+iy\\ \bar{z}=x-iy $$ and write down the surface element $dxdy$ as $\frac{1}{2}dzd\bar{z}$..Then my ...
1
vote
0answers
31 views

Estimate on the exponential integral of a complex argument (a reference needed)

Consider the exponential integral of the complex argument defined by $$ Ei( z ) = \gamma + \ln(-z) +\sum\limits_{ n = 1 }^{ \infty } \frac{ z^n }{n n!}, $$ where $ z \in \mathbb{C} \backslash ( ...
1
vote
0answers
15 views

How to perform this complex integration

How to perform this integration:- $\int\limits_{-\infty}^{\infty}\frac{(Az^2+\mathrm{i}Bz+C)\mathrm{e}^{-\mathrm{i}za}}{z^2(A_0z^2+\mathrm{i}B‌​z+C)}\mathrm{d}z\quad$ where, $\quad a,A,A_0,B,C>0$
1
vote
0answers
31 views

Multi-dimensional Gauss integral with complex non-symmetric coefficients

Is there a closed formula that evaluates the integral, $$ I = \int_{-\infty}^{\infty} dx_1 .. \int_{-\infty}^{\infty} dx_D \exp\left(-\sum_{\mu=1}^D \sum_{\nu=1}^D a_{\mu\nu} x_{\mu} x_{\nu} ...
1
vote
0answers
65 views

Multivariate Residue Theorem

Let $G(s,t)$ be a complex valued function in two variables that converges absolutely for $Re(s), Re(t)>1$. Suppose we can analytically continue $G$ in such a way that $$G(s,t) = f(s)g(t)H(s,t)$$ ...
1
vote
0answers
55 views

Integral of ratio of complex polynomials

Let $p(z),q(z) \in \mathbb{C}[z]$ two polynomials with coefficients in $\mathbb{C}$ s.t. $deg(p) = m$, $deg(q) = n$ and $n \ge m +2$. I need to show that $$ \lim_{R \to \infty} \int_{|z| = R} ...
1
vote
0answers
73 views

Integrals of the type $f'(z)/f(z)$

I am having trouble understanding integrals of the form: $$\int_\gamma\frac{f'(z)}{f(z)}\,{\rm d}z$$I am aware that there are problems with the complex logarithm, and we have the formula: ...
1
vote
0answers
31 views

Contour integration of logarithm: $g(\omega) \log[1 - \chi(q,\omega)]$

I'm trying to calculate the integral $$ \frac{1}{2\pi i} \int_\mathcal{C} g(\omega) \log[1 - \chi(q,\omega)], $$ where $g(\omega) = (e^{\beta \omega}-1)^{-1}$ has an infinite number of evenly spaced ...
1
vote
0answers
49 views

How to calculate this residue which has a pole of order n-r?

So I have this complex integral: $$ \oint \frac{dz}{2\pi}\frac{e^{iz(br-(n-r)a)}}{\left(1-(1-q)e^{-ik_{1}-iza}\right)^{n-r}}$$ b,r,q,a,n are all constants in this context. However I'm not entirely ...
1
vote
0answers
73 views

Using residue theorem find $\int_{\theta-\pi}^{\theta-2\pi}1/(r^2+1-2r\cos u)$

Find the value of $I$, using residue theorem. $$ I= \int_{\theta-\pi}^{\theta-2\pi} \frac{du}{r^2+1-2r\cos(u)} - \int_{\theta}^{\theta-\pi} \frac{du}{r^2+1-2r\cos(u)} $$ with $r<1$ and ...
1
vote
0answers
39 views

calculate complex integral $\int_{0}^ {\pi} \frac{(x \sin x)dx }{1-2a \cos x+a^2}, a>0$

I don't know how to calculate this complex integral: $$\int_{0}^ {\pi} \frac{(x \sin x)dx }{1-2a \cos x+a^2}, a>0$$
1
vote
0answers
47 views

What is the meaning of this integral?

Does anyone know the meaning of this type of integral? $\displaystyle{\int f(z) \,\overline {dz}}$. I think this means $\displaystyle{\int u\,dx + v\,dy+i\int v\,dx -u\,dy}$ where $f=u + iv$
1
vote
0answers
67 views

Using Multiple Branch Cuts in a Contour Integral

I have the integral $$I=\int_0^{\infty}\!\frac{x^{1/2}}{\left(x^3+a^3\right)^{1/2}\left(2x^3+a^3\right)^{1/6}}\,\mathrm dx$$ which I am trying to integrate using complex integration. I know that ...
1
vote
0answers
53 views

Complex integration, limits, arctan

$\left.\frac12i\;\text{Log}\frac{1-(-i+e^{it})}{1+(-1+e^{it})}\right|_0^{2\pi}=\frac12i\left(\log\left|\frac{i}{i}\right|+i\arg 1-\log|1|-i\arg1+2\pi ik\right)$ could someone explain how this is ...
1
vote
0answers
49 views

If f is analytic in the unit disk and $|f(z)|\leq \frac{1}{1-|z|}$ then $|f'(0)|\leq 4$

I need to prove that if $f$ is analytic in the unit disk and $|f(z)|\leq \frac{1}{1-|z|}$ for z all $z\in D_1(0)$ then $|f'(0)|\leq 4$. This is my proof and I need to verify this. Let $n\in ...
1
vote
0answers
53 views

Sum of Lomax random variables

Suppose $X_1,X_2,\cdots X_n$ are $n$ i.i.d Lomax random variables with pdf $f(x)=\frac{m}{(1+x)^{m+1}},x\geq 0,m\in \mathbb N$. I need to determine the pdf (or cdf) of the sum $S_n=\sum_{i=1}^{n}X_i$. ...
1
vote
0answers
29 views

Find similar estimate for complex numbers

According to Borwein, page 356 Prop. 2, $\left|\ln\left(\frac{4}{k}\right)-\operatorname{I}(1,k)\right|\leq 4k^2\left(8+\left|\ln k\right|\right)$ holds for $k\in\left(0,1\right]$. ...
1
vote
0answers
28 views

Explicit computation of integrals along loops

I am learning a bit about integration on one-dimensional complex tori. It's exciting stuff, but I have some trouble making things a bit explicit. Let's consider the elliptic curve $E = \mathbb ...
1
vote
0answers
34 views

Find natural number to satisfy inequality given with integral

Find the number $N\in \mathbb{N} $ for which the following inequality is true, if $\sigma(t)=|z|=1 $, with $0\leq t\leq \pi/2$ and $\left | \int_{1}^{i} e^{z^{-1}} \right | \leq N \frac{\pi}{2}$. ...
1
vote
0answers
83 views

Calculating this integral: $\int_{\partial_{10}(1-2i)}\frac{39!\require{cancel}\cancel{z^z}z^z}{(z+4)^{42}}dz$

Please take a look at $$\int_{\partial B_{10}(1-2i)}\frac{39!\require{cancel}\cancel{z^z}ze^z}{(z+4)^{42}}dz$$ At a first glance, this looks like a case for Cauchy's differentiation formula, which ...
1
vote
0answers
52 views

Second Derivative of Holomorphic function According to Cauchy's

I am asked to prove that, supposing f is holomorphic on the region G, w $\in$ G and $\gamma$ is a positively oriented, simple, closed, smooth, G-contractiblecurve such that w is inside $\gamma$ , we ...