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

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6
votes
3answers
283 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
1answer
39 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
77 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
114 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
74 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
143 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
116 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
274 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
94 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
105 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
132 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
194 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
67 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
65 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
463 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
238 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
130 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
67 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
352 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} {\cos\left(\pi x^{2}\right)\over 1 + 2\cosh\left(\,2\,\pi\,x\,/\,\sqrt{\,3\,}\,\right)}\,{\rm d}x ...
0
votes
1answer
61 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
154 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
194 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
126 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
330 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 ...
18
votes
4answers
356 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
161 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 ...
4
votes
3answers
216 views

Show that $\int_0^{2\pi}\frac{R^2-r^2}{R^2 - 2Rr\cos (\varphi-\vartheta) + r^2}d\vartheta$ is independent of $R>r>0$, using only real numbers.

The poisson kernel is sometimes written as $$ \frac{1}{2\pi}\int_0^{2\pi} \frac{R^2-r^2}{R^2 - 2Rr\cos(\varphi-\vartheta) + r^2} \mathrm{d}\vartheta = 1 \ , \ \ R>r>0 $$ Where $\varphi$ is some ...
0
votes
0answers
92 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
81 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
259 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
72 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
92 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
114 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
152 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
164 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 ...
2
votes
1answer
135 views

Analytic continuation of zeta is meromorphic on $\mathbb{C}$ with simple pole at 1

We have the following identity: For some contour $\gamma$ and $\forall s \in \mathbb{C} $ Re $s > 1$: $$-2i\sin(\pi s) \Gamma(s)\zeta(s)= \Large\int_{\gamma} \frac{(-z)^{s-1}}{e^z-1}dz$$ The ...
0
votes
1answer
44 views

Finding the types of singularities of $\oint \frac{\sin(\pi \cdot z)}{(z-1)^2}$

I want to find the types of singularities of $$\oint \frac{\sin(\pi \cdot z)}{(z-1)^2}$$ the point is $z=1$ I know that: $$f(z)=\frac{p(z)}{q(z)},q(a)=0,p(a)\neq 0,p(z)$$ so $p(z)$ analytic in $a$ ...
8
votes
3answers
290 views

How to show $\int^{\infty}_{-\infty}\frac{\sin(ax)}{x(x^2+1)}dx=\pi(1-e^{-a})$? ($a\ge0$)

$$\int^{\infty}_{-\infty}\frac{\sin(ax)}{x(x^2+1)}dx=\pi(1-e^{-a}), \ a\ge0$$ I tried to solve but came up with $\pi(2-e^{-a}) $. Could you tell me where did I do the mistake? if $x=z$ then ...
11
votes
2answers
278 views

Integration method for $\int_0^\infty\frac{x}{(e^x-1)(x^2+(2\pi)^2)^2}dx=\frac{1}{96} - \frac{3}{32\pi^2}.$

The following definite integral is obtained directly from Hermite's integral representation of the Hurwitz zeta function. But is it possible to obtain the same result via the residue calculus or ...
3
votes
2answers
146 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 ...
2
votes
0answers
120 views

What is an example of an integral that CANNOT be done without contour integration ? If that exist.

What is an example of an indefinite integral that CANNOT be done without contour integration ? If that exist. Im talking about closed forms for integrals, not numerical methods. Note that there are ...
2
votes
1answer
131 views

Evaluating a trigonometric integral by means of contour $\int_0^{\pi} \frac{\cos(4\theta)}{1+\cos^2(\theta)} d\theta$

I am studying for a qualifying exam, and this contour integral is getting pretty messy: $\displaystyle I = \int_0^{\pi} \dfrac{\cos(4\theta)}{1+\cos^2(\theta)} d\theta $ I first notice that the ...
12
votes
3answers
408 views

Evaluate $\int_0^{\frac{\pi}{2}}\frac{x^2}{1+\cos^2 x}dx$

Evaluate the following integral $$\int_0^{\frac{\pi}{2}}\frac{x^2}{1+\cos^2 x}dx$$ This function does not have an elementary anti-derivative. How can we solve this?
2
votes
2answers
223 views

How to find $\int_{0}^{1}\frac{1}{x^{2}+2x+2}\mathrm dx$ with contour integration

I want to evaluate the following integral $$\int_{0}^{1}\frac{1}{x^{2}+2x+2}\mathrm dx$$ by contour integration; I have a problem with the choice of the contour/ branch cuts. Where can I find some ...
2
votes
0answers
235 views

Contour integral with branch point

As preparation for my exam I "invented" the following problem as an exercise to see whether I understand how to work with branch points. $f(z) = \frac{z}{\sqrt{z^2+1} (z^2 +a^2)}$ The goal is to ...
0
votes
2answers
128 views

A contour integral for Fourier transform

How does one show the following, preferably with contour integral on the complex plane? $$\frac{\Gamma(\alpha)}{2\pi}\int_{-\infty}^\infty (ik)^{-\alpha}e^{-ikx}dk = (-x)_+^{\alpha-1},$$ where $x$ is ...
3
votes
2answers
167 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 ...
2
votes
3answers
75 views

Help with a contour integration

I've been trying to derive the following formula $$\int_\mathbb{R} \! \frac{y \, dt}{|1 + (x + iy)t|^2} = \pi$$ for all $x \in \mathbb{R}, y > 0$. I was thinking that the residue formula is the ...
6
votes
3answers
113 views

calculation of $\int^{\frac{\pi}{2}}_{0}\cos^{n}x\cos (nx)\ dx $, where $n\in \mathbb{N}$

Calculation of $\displaystyle \int^{\frac{\pi}{2}}_{0}\cos^{n}x\cos (nx)\ dx $, where $n\in \mathbb{N}$ $\bf{My\; Try}::$ Using $\displaystyle \cos (x) = \frac{e^{ix}+e^{-ix}}{2}$, we get ...