Questions on the evaluation of integrals along a locus in the complex plane.

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4
votes
0answers
90 views

Ugly-nice double series

I'm trying to evaluate the following ugly double sum (presented in raw notation as used in my calculations): $\sum _{m=1}^{\infty } \sum _{n=1}^{\infty } \frac{4 m \cos \left(\frac{2 \pi m ...
4
votes
0answers
169 views

Integral $=\int_0^\infty x^{\alpha -1}Li_n (-\sigma x) Li_m(-\omega x^r)dx$.

I am trying to calculate an integral that can be expressed in terms of infinite hypergeometric series by using transforms and Residue method, the integral is $$ ...
4
votes
0answers
90 views

Showing that $\lim_{N \to \infty} \int_{|z|=N+\frac{1}{2}} \frac{ \sinh az}{\sinh \pi z} \mathrm{e}^{ibz} \ dz =0$

To evaluate $ \displaystyle \int_{0}^{\infty} \frac{\sinh ax}{\sinh \pi x} \cos (bx) \ dx \ (a< \pi)$, you could let $ \displaystyle f(z) = \frac{\mathrm{e}^{(a+ib)z}}{\sinh \pi z} $ and integrate ...
4
votes
0answers
97 views

Determining “good” contours for evaluating integrals

This is more of a general question, but I'll lead with an example. Suppose we wish to evaluate $\displaystyle \int_0^{\infty} \dfrac{1}{1+x^7} dx$ The goal, it seems, is to find nice contours which ...
4
votes
0answers
57 views

where is the mistake with my fake proof? [duplicate]

I tried to show that $$\int_{-\infty}^{+\infty} \frac{dx}{(x^2+1)^{n+1}}=\frac {(2n)!\pi}{2^{2n}(n!)^2}$$ using contour integration so $$\int_{C} \frac{dz}{(z^2+1)^{n+1}}=\int_{-\infty}^{+\infty} ...
4
votes
0answers
565 views

contour integration of a function with two branch points .

Many of us have seen the evaluation of the integral $$\int^{\infty}_0 \frac{dx}{x^p(1+x)}\, dx \,\,\, 0<\Re(p)<1$$ It can be solved using contour integration or beta function . I thought of ...
4
votes
0answers
491 views

contour integral with singularity on the contour

I want to compute the following integral $$\oint_{|z|=1}\frac{\exp \left (\frac{1}{z} \right)}{z^2-1}\,dz$$ The integrand has essential singularity at the origin, and $2$-poles at $\pm 1$,which lie ...
3
votes
4answers
199 views

Evaluating $I(n) = \int^{\infty}_{0} \frac{\ln(x)}{x^n(1+x)}\, dx$ for real $n$

I am not sure how to handle the additional parameter $n$. I first need to find out for which real values of $n$ will the integral converge. Based on intuition and checking with mathematica, I believe ...
3
votes
2answers
918 views

using contour integration

I am trying to understand using contour integration to evaluate definite integrals. I still don't understand how it works for rational functions in $x$. So can anyone please elaborate this method ...
3
votes
2answers
217 views

Improper integral involving $e^x$

How do you show that $$\int_{-\infty }^{+\infty }{\frac{{{x}^{2}}\, \text{d}x}{\left( \beta +{{\text{e}}^{x}} \right)\left( 1-{{\text{e}}^{-x}} \right)}}=\frac{\left( {{\pi }^{2}}+{{\ln }^{2}}\beta ...
3
votes
4answers
159 views

Evaluating a complex contour

I need to show the following result: $$ \int_{-\infty}^\infty \frac{1}{(1+x^2)^{n+1}}dx\, = \frac{1\cdot 3\cdot\ldots\cdot(2n-1)}{2\cdot 4\cdot\ldots\cdot(2n)}\pi $$ With n=1,2,3,... This function ...
3
votes
2answers
162 views

Integral $\int_0^a \ln \left( \frac{b-\sqrt{a^2-x^2}}{b+\sqrt{a^2-x^2}} \right)dx$

Hi I am trying to calculate, $$ \int_0^a \ln \left( \frac{b-\sqrt{a^2-x^2}}{b+\sqrt{a^2-x^2}} \right)dx $$ where $a,b$ are positive real constants. I Know $\ln(xy)=\ln x +\ln y$, but I do not ...
3
votes
2answers
136 views

Does $\lim_{R \to \infty} \int_{C_{R}} e^{iz} \ dz \to 0$?

Let $C_{R}$ be the upper half of the circle $|z|= R$. Does $ \displaystyle \lim_{R \to \infty} \int_{C_{R}} e^{iz} \ dz \to 0 $? Jordan's lemma is not applicable here. And I'm not sure how to get a ...
3
votes
2answers
62 views

what is the proper contour for $\int_{-\infty}^{\infty}\frac{e^z}{1+e^{nz}}dz$

what is the proper contour for $$\int_{-\infty}^{\infty}\frac{e^z}{1+e^{nz}}dz:2\leq n$$ I tried with rectangle contour but the problem which I faced how to make the contour contain all branches point ...
3
votes
3answers
96 views

Find $I:=\lim\limits_{R\to \infty}\int\limits_{-R}^R \frac{x \sin(3x)}{x ^2+4}dx$ using residues

Find $I:=\lim\limits_{R\to \infty}\int\limits_{-R}^R \frac{x \sin(3x)}{x ^2+4}dx$ using residues. Let $f(z)= \frac{z \sin(3z)}{z ^2+4}$. First define two contours: $$\Gamma_1: z=t \text{ where } ...
3
votes
1answer
58 views

Complex integral of $\frac{\cos x}{x^2+4} $

I want to evaluate: $$ \int_{-\infty}^{\infty}\frac{\cos x}{x^2 +4} dx $$ Using wolfram alpha, it gave an answer of $\frac{\pi}{2e^2}$. Wolfram Alpha is never wrong. Attempt $$ ...
3
votes
1answer
94 views

$\frac{5\pi^3}{154}=\int_{\pi/6}^{\pi/2}\bigg[\Re\big(\text{Li}_2(4\sin^2\theta)\big) +\text{Li}_2\bigg(\frac{1}{4\sin^2\theta}\bigg) \bigg]d\theta$

I am trying to prove $$ \int_{\pi/6}^{\pi/2}\bigg[\Re\big(\text{Li}_2(4\sin^2\theta)\big) +\text{Li}_2\bigg(\frac{1}{4\sin^2\theta}\bigg) \bigg]d\theta=\frac{5\pi^3}{54}. $$ Clearly, this closed form ...
3
votes
2answers
131 views

erf(a+ib) error function separate into real and imaginary part

Is there an easy way to separate erf(a+ib) into real and imaginary part?
3
votes
2answers
160 views

How to compute the integral $\int_0^\infty\frac{x}{e^x+1}dx$ using the Residue theorem.

How to compute the integral $\int_0^\infty\frac{x}{e^x+1}dx$ using the Residue theorem, just as the title says. I have used rectangles, circles to do, but without any progress. By changing variable ...
3
votes
1answer
201 views

Evaluating $\int_0^\infty \frac{\cos(ax)-e^{-ax}}{x \left(x^4+b^4 \right)}dx$

How can we evaluate $$\int_0^\infty \frac{\cos(ax)-e^{-ax}}{x \left(x^4+b^4\right)}dx \quad a,b>0$$ using Complex Analysis? This problem was given in a Complex Analysis book which I was reading. ...
3
votes
4answers
142 views

Contour integration of $\int \frac{dx} {(1+x^2)^{n+1}}$

I want to compute $$ \int_{-\infty}^\infty \frac 1{ (1+x^2)^{n+1}} dx $$ for $n \in \mathbb N_{\geq 1}$. If I let $$ f(z) := \frac 1 {(z+i)^{n+1}(z-i)^{n+1}} $$ then I see that $f$ has poles of ...
3
votes
2answers
118 views

Complex integration help

The integral given is $$\int_{-\infty}^{\infty} \frac{\cos(x)-1}{x^2}\,dx $$ Ok, so, I've used the upper semi circular contour with the function $$f(z) = \frac{e^{iz}-1}{z^2}$$ Now the residue I ...
3
votes
3answers
543 views

contour integral computations

Let $C$ be the boundary of the square whose vertices are $1+i$, $1-i$, $-1 + i$ and $-1 -i$. Suppose that $C$ is oriented counterclockwise. How to compute a) $$\int_C \frac{e^z}{z-1/2} \, dz$$ b) ...
3
votes
1answer
94 views

Integrating $e^{a/x^2-x^2}/(1-e^{b/x^2})$

I want to solve the following two integrals analytically \begin{aligned} I_1 = & \int\limits_0^{\infty}\frac{e^{a/x^2}}{1-e^{b/x^2}}e^{-x^2}dx \\ I_2 = & ...
3
votes
2answers
57 views

Infinitely real-differentiable function with $f(0)=0$ but $\int_{\partial B_1(0)}\frac{f(z)}{z}dz\ne0$

I'm searching for a infinitely real-differentiable function $f:\mathbb{C}\to\mathbb{C}$ with $f(0)=0$ but $$(*)\;\;\;\;\;\int_{\partial B_1(0)}\frac{f(z)}{z}dz\ne0$$ where ...
3
votes
1answer
116 views

Contour method to solve $\int^\infty_0\frac{\ln(1+x)}{1+x^2}\,dx$

Prove the following using complex analysis $$\tag{1}\int^\infty_0\frac{\ln(1+x)}{1+x^2}\,dx=\frac{\pi}{2}\ln(2)$$ I found this problem in Schaum's outlines of complex variables. I thought that we ...
3
votes
2answers
143 views

Integral $\int_0^\infty e^{imx^2}dx$

In evaluating an integral in path integrals in QFT, I am stuck with this integral (that came up from evaluating a functional integral), $$I = \bigg( \frac{m}{2\pi i\tau}\bigg) \int ...
3
votes
3answers
93 views

Contour integration problem

I am to evaluate $\displaystyle\int_0^{\infty} \dfrac{\sin x}{x(x^2+1)}dx$ via contour integration. Now I used an indented semicircular contour, and the parts lying on the real line and the big arc ...
3
votes
1answer
166 views

Understanding Dogbone contour example

I am trying to understand example VI on the wikipedia page http://en.wikipedia.org/wiki/Methods_of_contour_integration, but one particular point has mystified me for hours. After it is shown that ...
3
votes
3answers
222 views

Are the integration contours of this improper integral properly selected?

I have been recently trying to review some topics on improper integrals. The Integral I am trying to solve is: $$ \int_0^\infty {log(x) \over x^2 -1} dx $$ The branch cut of the $log(x)$ is ...
3
votes
3answers
127 views

Evaluating $\int_{0}^{\infty} \frac{1}{1+x^{3/2}}\,\textrm{d}x$

I'm trying to evaluate this integral using contour integration (over a Riemann surface), but I'm stuck at the step where I need to calculate the residues. The roots of $1+z^{3/2}$ are $1$ and ...
3
votes
1answer
484 views

Change of variables in a complex integral

I want to evaluate this integral using Residue Theorem $$\int_C^\ \frac{4z} {z^4 +6z^2 +1} dz = $$ $$ C : |z| = 1 $$ so I substitute letting $$\ W = z ^ {2 } $$ $$ dw = 2z dz $$ and the ...
3
votes
3answers
412 views

Complex analysis: contour integration

Evaluate by contour integral: $$\int_0^1{ dx\over (x^2-x^3)^\frac 13}$$ Should I go for some kind of substitution so that the limit changes to $0$ to $\pi/2$?
3
votes
1answer
103 views

contour integration of $f(z) = z^{a-1}(z+z^{-1})^{b-1}$

I want to evaluate $\displaystyle f(z)= z^{a-1} \left(z + z^{-1}\right)^{b} \ (a >b >-1)$ counterclockwise around the right half of the circle $|z|=1$. So I close the contour with the vertical ...
3
votes
2answers
278 views

Contour integration of $\int_{-\infty}^{\infty} \frac{1-b+x^2}{(1-b+x^2)^2 + 4bx^2}dx = \pi$

Given $0< b <1$, derive the equality: $\displaystyle \int_{-\infty}^{\infty} \frac{1-b+x^2}{(1-b+x^2)^2 + 4bx^2}dx = \pi$ by integrating the function $(1+z^2)^{-1}$ around the rectangle with ...
3
votes
1answer
64 views

Contour integration of $\frac{(\ln z)^2}{z^2+1} $

I'm supposed to take the principal branch of $\ln z$ and evaluate this integral: $$ \oint \frac{(\ln z)^2}{z^2+1} $$ Attempt I suppose the integral they are talking about is something like ...
3
votes
1answer
182 views

LogSine Integral $I=-\int_0^{\pi/3} \ln^2\big(2\cos \frac{\theta}{2}\big) d\theta$

These are known as LogSine integrals at $2\pi/3$, so I will call the integral Ls as this is common in the literature. I am trying to prove $$ Ls=-\int_0^{\pi/3} \ln^2\big(2\cos \frac{\theta}{2}\big) ...
3
votes
2answers
200 views

Improper Integral of $x^2/\cosh(x)$

I need to compute the improper integral $$ \int_{-\infty}^{\infty}{\frac{x^{2}}{\cosh\left(x\right)}\,{\rm d}x} $$ using contour integration and possibly principal values. Trying to approach this as ...
3
votes
1answer
141 views

Would like help with a contour integral.

Disclaimer: the knowledge I have about contour integration is solely from the book "Mathematical Methods in the Physical Sciences" by Mary L. Boas. I am trying to understand how the following ...
3
votes
2answers
134 views

How to find this integral using Cauchy integral formula

How to obtain that $$\int\limits_{|z|=r} (\bar{z})^{-m} z^{-n-1}\, dz = \begin{cases} 2\pi ir^{-2m} &\text{if}\,\,n=m, \\ 0 &\text{if}\,\,n \neq m, \end{cases}$$ for $r>0$. I suppose I ...
3
votes
1answer
226 views

Analytic continuation of the Riemann zeta function using contour integration

To find the analytic continuation of the Riemann zeta function using contour integration one can integrate $\displaystyle f(z) = \frac{z^{s-1}}{e^{-z}-1}$ around a contour that consists of rays just ...
3
votes
1answer
43 views

Integration of exponential with square

It is known that $\int_\mathbb{R}e^{-tx^2}dx=\sqrt{\pi/t}$. What about $\int_\mathbb{R}e^{-t(x+ai)^2}dx$ for $a\in\mathbb{R}$? Is it still also $\sqrt{\pi/t}$? I can't simply change the variable ...
3
votes
3answers
216 views

Evaluating an improper integral that involves $\exp(-|x|)$

I am trying to prove that the function $f:\mathbb C\setminus\mathbb R\rightarrow\mathbb C$ defined by $$ f(z) := \frac{1}{2\pi i}\int_{-\infty}^\infty\frac{\exp(-|x|)}{x-z}dx $$ is holomorphic. I ...
3
votes
1answer
767 views

Inverse Laplace transform using contour integration

I want to show by contour integration that $\displaystyle\mathcal{L}^{-1} \{\text{arccot}(s) \}(t)= \frac{\sin t\ }{t}$. In other words, I want to evaluate $\displaystyle \frac{1}{2 \pi i} \int_{a - ...
3
votes
1answer
244 views

Cauchy principal value of $\int_{-\infty}^{\infty} \frac{1}{x^{3}} \ dx $

By definition $\displaystyle\text{PV} \int_{-\infty}^{\infty} \frac{1}{x^{3}} \ dx = \lim_{\epsilon \to 0 ,\ a \to \infty} \Big(\int_{-a}^{0 - \epsilon} \frac{1}{x^{3}} \ dx + \int_{o+ \epsilon}^{a} ...
3
votes
2answers
655 views

contour integration and branch points

An exercise in a textbook says to evaluate $\displaystyle \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \cos (ax) \cos^{b} (x) \ d x \ (a > b > -1)$ by letting $\displaystyle f(z) = z^{a-1} ...
3
votes
2answers
74 views

And another real integral to be solved by contour integration

I want to solve $$\int_0^\infty\frac{1}{x^3+x^2+x+1}dx$$ and i have really learned a lot already by failing to solve it. I want to solve it using a clever contour. It is possible to do it using ...
3
votes
2answers
50 views

find $\int _\gamma \frac{1}{z+\frac {1}{2}}dz$

I'm asked to find $$\int _\gamma \frac{1}{z+\frac {1}{2}}$$ where $\gamma (t)=e^{it}, 0\leq t\leq 2\pi$. To do this I deal with two different logarithms, one without the negative imaginary axis ...
3
votes
1answer
264 views

contour integration of logarithm function

I'm new to contour integral involving branch point and stuck on this particular integration. Here is the problem: $$\int_{\mathcal{C}}\log z\,\mathrm{d}z,$$ where $\mathcal{C}$ is a closed square ...
3
votes
1answer
337 views

Contour Integration: $\int_0^\infty\frac{1}{x^a(1-x)}\,dx$ for $0<a<1$.

I've been trying to calculate $$\int_0^\infty\frac{1}{x^a(1-x)}\,dx\quad\text{with }0<a<1.$$I haven't had much luck. I tried taking the branch cut with of the positive reals and estimating that ...