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

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2
votes
1answer
48 views

Determining asymptotic behavior through generating functions

I need to determine the asymptotic behavior of $$a_n=\sum_{k=2}^{n-2}\frac1{\ln k\ln(n-k)}$$ as $n\to\infty$. I know some elementary methods that might help. For example, split the index $\lvert ...
4
votes
2answers
159 views

Integral $ \int_0^1 \frac{\ln \ln (1/x)}{1+x^{2p}} dx$…Definite Integral

Calculate $$ I_1:=\int_0^1 \frac{\ln \ln (1/x)}{1+x^{2p}} dx, \ p \geq 1. $$ I am trying to solve this integral $I_1$. I know how to solve a related integral $I_2$ $$ I_2:=\int_0^1 \frac{\ln \ln ...
1
vote
2answers
76 views

Contour Integral for Cosine and a rational function

I've been trying to figure out this integral via use of residues: $$\int_{-\infty}^{\infty} \displaystyle \frac{\cos{5x}}{x^4+1}dx$$ The usual semicircle contour wont work for this guy as the ...
4
votes
0answers
163 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 $$ ...
6
votes
4answers
117 views

Computing $\int_0^\infty\mathrm{d} x\frac{x}{e^x+1}$ with contour integration

Let's set: $$ \int_0^\infty\mathrm{d}x\frac{x}{e^x+1}=I. $$ I would like to compute it using, presumably, the methods of complex analysis and contour integration.
11
votes
2answers
375 views

Integral $\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d \theta$.

I am trying to calculate $$ I=\frac{1}{\pi}\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d \theta=\frac{11\pi^4}{180}=\frac{11\zeta(4)}{2}. $$ Note, we can expand the log in the integral to ...
2
votes
1answer
33 views

Is there no analytic form of $\int_b^c\frac{\sqrt{x}e^x\text{erfc}(\sqrt{x})}{\sqrt{a-x}}dx$ ?

I am trying to find an analytic answer for $\int_b^c\frac{\sqrt{x}e^x\text{erfc}(\sqrt{x})}{\sqrt{a-x}}dx$ but it doesn't seem to be in any of the integral tables that I've looked in. I don't think ...
3
votes
2answers
167 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 ...
0
votes
0answers
44 views

Integration using Cauchy Integral Formula

If $f(z)$ is analytic in $|z-a|<R$ and $0<r<R$, then $f'(a) = \frac{1}{r\pi}\int_{0}^{2pi}F(\theta)e^{-i\theta}\,d\theta$, where F($\theta$) is the real part of $f(a+re^{i\theta})$. I ...
0
votes
1answer
132 views

Clarification of Cauchy Principal Value and use of Contour Integration

I am evaluating the improper integral $\int_{-\infty}^\infty{\frac{\sin^3 x}{x^3}dx}$; I am also told to show that this is equal to its principal value, and use this fact to evaluate the integral. I ...
0
votes
1answer
32 views

Closed contour within a closed contour integral

Let $C$ and $D$ be two closed contours, $D$ lying completely within $C$, and let $a$ be a point between $C$ and $D$. Show that: $$ f(a) = \frac{1}{2\pi}\int_{C}\frac{f(z)}{z-a}dz - ...
3
votes
2answers
114 views

Trouble with $\int_0^\infty e^{-ix^2}\mathrm{d}x$

I'm trying to evaluate $$ \int_0^\infty \mathrm{d}x\ e^{-ix^2}. $$ I tried to integrate on the following contour $\Gamma_R$: the frontier of a circular sector, centered at the origin, of angle $\pi / ...
2
votes
1answer
88 views

Why does the Cauchy-Goursat theorem not apply here?

Let $C$ denote the positively oriented boundary of the half disk $0 \le r \le 1, 0 \le \theta \le \pi$, and let $f(z)$ be a continuous function defined on that half disk by writing $f(0) = 0$ and ...
6
votes
2answers
185 views

Computing the integral $ \int_0^{\infty} e^{-\phi^2+\phi}\cdot \phi^{2} \ln(1-2x\cos\phi+x^2)\, d\phi. $

Integrate $$ \int_0^{\infty} e^{-\phi^2+\phi}\cdot \phi^{2} \ln(1-2x\cos\phi+x^2) \, d\phi. $$ Something that may help $(1-2x\cos\phi+x^2)=(1-xe^{i\phi})(1-xe^{-i\phi})$. And using the series ...
5
votes
3answers
216 views

Differentiation wrt parameter $\int_0^\infty \sin^2(x)\cdot(x^2(x^2+1))^{-1}dx$

Use differentiation with respect to parameter obtaining a differential equation to solve $$ \int_0^\infty \frac{\sin^2(x)}{x^2(x^2+1)}dx $$ No complex variables, only this approach. Interesting ...
0
votes
0answers
42 views

Evaluating this complex integral, how?? [duplicate]

Looking through past papers to prepare for my exam and found this: You are asked to integrate this around the following contour: and show that it's equal to: I have found the residue of this ...
0
votes
1answer
38 views

Cauchy Riemann equations, do these satisfy it??

I have this question and am unsure of my approach. I have applied the Cauchy Riemann conditions to it: and found that this condition is true. Is that sufficient and does it make sense?
1
vote
1answer
74 views

definiteinteggral

The integral is given by $$\int_0^1 \frac{\ln (1-x)\ln x}{1+x} dx = \frac{1}{8}\big(-\pi^2\ln(4) +13\zeta(3)\big).$$ Any ideas how to prove? We cannot solve the integral so easily because we cannot ...
0
votes
1answer
110 views

Find the residue(s) of this function at each pole that lies in the contour?

Going through past papers and found this residue question I can't do. The question asks you to find the residue at each pole that lies in the contour shown. I've got as my answer for the poles ...
0
votes
1answer
59 views

Complex integrals over an ellipse instead of circle?

I was looking through past papers and found this integral: Which should be evaluated over an ellipse with I've done these plenty of times over a circle with |z| = 2 etc, but where do I start in ...
1
vote
0answers
118 views

evaluate the integral $I =\int_0^{+\infty} e^{ix^2}dx$

"Evaluate the integral $I= \int_{0}^{\infty} e^{ix^{2}}\, dx$. Let R > 0 and consider the closed contour $C_R = C(1)_R + C(2)_R + C(3)_R$ where $C(1)_R$ is the segment of the positive real axis from ...
4
votes
1answer
113 views

1 dimensional integral, definite integral.

Trying to show $$ \int_0^1 \frac{\ln x \cdot \ln(1+x)}{1-x}dx=-\frac{1}{4}\pi^2 \ln(2)+\zeta(3). $$ I am unsure how to approach this integral as I do not know how to use a power series representation ...
4
votes
5answers
254 views

Integral, definite integral

How can we prove $$ \int_0^1 \frac{\ln x \cdot \ln(1+x)}{1+x}dx=-\frac{\zeta(3)}{8}? $$ This has been one of the integrals that came out of an integral from another post on here, but no solution to ...
2
votes
1answer
92 views

Integral, 1 dimension

Re-doing last post since it was incorrect and corrected by many people on here. $$ \int_0^1 \frac{\tanh^{-1}(x)\ln x}{x(1-x^2)}dx $$ I have tried substitutions since $\tanh^{-1}(x)$ and $1-x^2$ are ...
0
votes
0answers
82 views

Integral Involving Trigonometric Functions and Exponential (Related to Marcum Q-function)

I want to solve this integral $$ \int_{0}^{\infty}\int_{0}^{2\pi}\exp(-ar^2)\exp(r\,b(\cos\theta+\sin\theta))r^{m}\cos^{m}(2\theta)d\theta \,dr,$$ where $a$ and $b$ are constants. I know how to ...
0
votes
1answer
96 views

Using residue theorem to solve double integral, involving singular roots?

I am a physics grad student (high energy), I've come across a problem while doing a certain loop integral that I don't understand. I've removed as much of the physics as I can so that this is just a ...
4
votes
1answer
173 views

Laplace transform of and impulse sampled function using “frequency” convolution

This is a long question, but assume we have this: The book uses the frequency convolution theorem to solve this problem. To solve the integral, it uses a contour + residue theorem to solve it. The ...
1
vote
1answer
66 views

Bromwich integral of $1/s^k$ with k real (non integer) and $1<k$

Is there a simple way to compute the inverse laplace transform of $1/s^k$ with k non integer using Bromwich integral (basically without using the known laplace transform of $t^n$)?
0
votes
1answer
46 views

Computing $\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t$

I'd like to calculate the following integral on the interval $[0,2\pi]$: $$ I=\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t = 2\pi. $$
3
votes
1answer
61 views

Find the analytic continuation of the $ f(z) = \int_{0}^{\infty} \frac{exp(-zt)}{1+t^2} dt$

Find the analytic continuation of the function $f(z)$ defined by $ f(z) = \int_{0}^{\infty} \frac{\exp(-zt)}{1+t^2} dt$ , $ |\arg(z)| < \pi/2$ to the domain $ -\pi/2 < \arg(z) < \pi$ I ...
1
vote
1answer
384 views

Complex integration around a branch point

I am confused about the "deformation" of a closed contour that my book is doing. For reference, it is example 2.4.3 on pg. 75-76 from this free online book. The example is the integration of 1/z ...
5
votes
3answers
228 views

How to compute $I_n=\int_{-\infty}^{+\infty}\mathrm{d}x\frac{x^{2n}}{\cosh^2 x}$?

I'd like to compute: $$ I_n = \int_{-\infty}^{+\infty}\mathrm{d}x\frac{x^{2n}}{\cosh^2 x}. $$ We have, quite easily: $$ I_0 = \int_{-\infty}^{+\infty}\mathrm{d}x\frac{1}{\cosh^2 x}=\left[\tanh ...
4
votes
1answer
115 views

Using Cauchy integral formula to calculate $\int_\gamma \frac{\cos{z}}{z^n}$

Let $\gamma(\vartheta)=\mathrm{e}^{i\vartheta},\,\vartheta\in[0,2\pi]$, and consider the integral $$I(n)=\int_\gamma \frac{\cos{z}}{z^n},$$ where $n\in \{0,2,4,6,...\}$. Is there any way to prove ...
3
votes
0answers
66 views

Branch-point order and Cauchy representation

This question is about the nature of branch points which arise in certain Cauchy-integral representations of functions of a single complex argument, $z$. Suppose we have the following representation: ...
11
votes
1answer
309 views

integral $\int_{0}^{\infty}\frac{\cos(\pi x^{2})}{1+2\cosh(\frac{2\pi}{\sqrt{3}}x)}dx=\frac{\sqrt{2}-\sqrt{6}+2}{8}$

Here is a seemingly challenging integral some may try their hand at. $$\int_{0}^{\infty}\frac{\cos(\pi x^{2})}{1+2\cosh(\frac{2\pi}{\sqrt{3}}x)}dx=\frac{\sqrt{2}-\sqrt{6}+2}{8}$$ It appears to be ...
0
votes
1answer
50 views

Finding the complex integral along an arc

How can we evaluate complex expressions like these$\int_C(Z-Z^2)dZ$ where $C$ is the upper half of the circle $|Z-2|=3$
0
votes
1answer
121 views

Very difficult contour integral

I have to compute this integral and I don't have any idea how to get further on: $$\frac{1}{2 \pi i} \int_{\mid z \mid = 1} \frac{6z^{98}}{23z^{99}-2z^{81}+z^4-7}dz$$ I tried Rouché to maybe ...
6
votes
1answer
136 views

Line contour integral of complex Gaussian

Say I have the entire function $$f(z)=e^{-\frac{1}{2}z^2}.$$ I would like to consider the integral $$I=\int_\Gamma f(x)dz,$$ where $\Gamma$ is a line with negative slope $<1$ in $\mathbb{C}$ (so if ...
6
votes
1answer
114 views

Contour integration of a meromorphic function

Given a meromorphic function $f$ which is uniformly bounded on the upper half plane. Assume that $\int_{-\infty}^{+\infty} f(x)dx$ is absolutely integrable. Then Cauchy's integral theorem suggests ...
2
votes
1answer
313 views

Integrating $\int \frac{e^{ipx}}{(\cos x)^{a}} \frac{dx}{x- \xi}$

Let me quote the passage from the book, and then I'll explain the notation. Let us integrate $$ (i) \ \int \frac{e^{ipx}}{(\cos x)^{a}} \frac{dx}{x- \xi}$$ $$ (ii) \ \int ...
11
votes
2answers
201 views

Evaluating $\int_{0}^{\infty} \frac{x^{3}- \sin^{3}(x)}{x^{5}} \ dx $ using contour integration

EDIT: Instead of expressing the integral as the imaginary part of another integral, I instead expanded $\sin^{3}(x)$ in terms of complex exponentials and I don't run into problems anymore. $$ ...
4
votes
1answer
151 views

Integral Using Harmonic Functions

Evaluate the integral: $$\int^{2 \pi}_0 \dfrac{\cos^2 \theta}{|2e^{i\theta}-z|^2} \, d \theta \qquad \mbox {when} \, |z| \neq 2.$$ Now, I thought about trying to change this to look like a Poisson ...
0
votes
0answers
68 views

Contour integral (inverse Laplace transform) with arctan

I have what I think is a relatively simple contour integral involving arctan, but it is giving me difficulty. I would really appreciate any help. The integral itself is, with τ, λ, and k all real and ...
0
votes
0answers
75 views

Evaluate the contour integral $\int_{\gamma(0,1)}\frac{e^z+e^{-z}}{z^n}dz \hspace{10mm} n=1,2,3,\cdots .$

Let $\gamma(z_0,R)$ denote the circular contour $z_0+Re^{it}$ for $0\leq t \leq 2\pi$. Evaluate $$\int_{\gamma(0,1)}\frac{e^z+e^{-z}}{z^n}dz \hspace{10mm} n=1,2,3,\cdots .$$ Using Cauchy's formula: ...
1
vote
2answers
223 views

Evaluate the contour integral $\int_{\gamma(0,1)}\frac{\sin(z)}{z^4}dz.$

Let $\gamma(z_0,R)$ denote the circular contour $z_0+Re^{it}$ for $0\leq t \leq 2\pi$. Evaluate $$\int_{\gamma(0,1)}\frac{\sin(z)}{z^4}dz.$$ I know that \begin{equation} ...
0
votes
2answers
70 views

$\int_{0}^{\infty}\frac{\cos2\pi x}{x^4+x^2+1}dx=-\frac{\pi}{2\sqrt{3}}\mathrm{e}^{-\pi\sqrt{3}}$

Can somebody help me out with the following integral? Prove that: $\int_{0}^{\infty}\frac{cos2\pi x}{x^4+x^2+1}dx=\frac{-\pi}{2\sqrt{3}}e^{-\pi\sqrt{3}}$ I have already determined the ...
4
votes
0answers
86 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 ...
3
votes
0answers
83 views

Contour Integration - Quantum field theory

I am a physics student. In calculating transition amplitude for Klein-Gordon real-scalar field, I encountered the integral, $$ \frac{-i}{2(2\pi)^2\Delta x} \int^{\infty}_{-\infty} \,dk ...
3
votes
1answer
125 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
1answer
139 views

Choice of branches for contour integration.

Suppose I have the following function of a complex variable $$f(z)=\log(z)(z^2+1)^{1/2}.$$ Wolfram Alpha tells me the branch cuts of $f(z)$ are $z\leq 0$ (presumably for the logarithmic term), and ...