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

learn more… | top users | synonyms

2
votes
2answers
70 views

$\int_0^{2\pi} \log |1-w e^{i\theta}| \, d\theta$

Let $w$ be a complex number such that $|w| < 1$. Evaluate the integral $$\int_0^{2\pi} \log |1-w e^{i\theta}| \, d\theta.$$ I am having a hard time moving forward on this question. I tried ...
1
vote
0answers
35 views

Evaluating real trigonometric integral using contour, with pole order n

Use the residue theorem to compute the real integral: $$I = \int_{0}^{2\pi} \sin^{2n}\theta d\theta$$ I have considered a contour around a unit circle C, and used the substitutions: $sin\theta = \...
3
votes
2answers
72 views

Calculating residues of function with branch cut

Show that $$I= \int_{0}^{\infty} \frac{\ln x}{x^\frac{3}{4} (1+x)} dx = -\sqrt{2} \pi^2$$ I used a keyhole contour, with branch point at $z=0.$ Around $\Gamma$, $|zf(z)|$ tends to $0$ as $z$ tends ...
1
vote
1answer
91 views

Representing the function $f\left ( x \right )=\frac{1}{e^{2}e^{\cos\left ( x \right )}-1}$ in terms of Fourier series

The function is periodic with main period of $2\pi$, and it is even. So only the coefficients of the cosine terms remain. Wolfram alpha gives the result for $a_{0}$ as follows: I guess it is only ...
3
votes
1answer
72 views

Fourier series of $\frac{1}{5+4 \cos x}$ using contour integration

The function $$f(x)=\frac{1}{5+4 \cos x}$$ is periodic with the main period being $T=2\pi$. The graph is easily obtained, but here is a graph from Desmos as it looks better: The function is even, ...
0
votes
1answer
41 views

What is the definition of this symbol $ \int_{\sigma-i\infty}^{\sigma+i\infty} f(s) \, ds, \quad \sigma>0.$

What is the definition of this symbol $$ \int_{\sigma-i\infty}^{\sigma+i\infty} f(s) \, ds, \quad \sigma>0.$$ Thank you in advance
0
votes
0answers
47 views

how to evaluate this contour integral

how to evaluate $\int_0^\infty \! e^{ipx} \, \mathrm{d}x$, I know I can take the contour in the first quadrant, but why how does the integral over the arc vanish as R goes infinity, as it does not ...
5
votes
2answers
88 views

Computing alternating sum using contour integration

By considering the integral of: $$\left(\frac{\sin\alpha z}{\alpha z}\right)^2 \frac{\pi}{\sin \pi z},\quad \alpha<\frac{\pi}{2}$$ around a circle of large radius, prove that: $$\sum\...
6
votes
2answers
130 views

Computing $\sum\limits_{n=1}^\infty\frac{\sin n}{n}$ with residues

I'm running into some error in computing the sum. Since $\dfrac{\sin n}{n}$ is even, I'm considering the function $f(z)=\dfrac{\pi\sin z\cot\pi z}{z}$ and the contour integral $$\oint_\gamma \frac{\pi\...
1
vote
1answer
55 views

Express $\int_{\sin nx}^{\sin(n+1)x}\sin t^2dt$ in terms of $x$ and $n$

Please help me to express $$\int_{\sin nx}^{\sin(n+1)x}\sin t^2\,dt$$ in terms of $x$ and $n$. If it is not possible please help to establish bounds on the integral again in terms of $x$ and $n$. The ...
3
votes
3answers
89 views

Contour integral with a logarithm squared

The integral I'd like to evaluate is $\int_0^\infty \frac{\log^2 x \, dx}{(1+x)^2}$. I consider the function $f(z) = \frac{\text{Log}^2 z}{(1+z)^2}$, which has a pole of order 2 at $z=-1$ and has a ...
3
votes
1answer
80 views

Violating Cauchy's Integral Theorem

With regards to utilizing Cauchy's Integral Theorem for integration over closed contours: https://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem In particular the result that $\int_\gamma f(z)\,...
2
votes
2answers
238 views

Finding Cauchy Principal Value for $\int_0^\infty \frac{\ln^2x }{(1-x)^2\sqrt{x}(x-4)} dx$

I am trying to find Cauchy Principal value for $$\int_0^\infty \frac{\ln^2x }{(1-x)^2\sqrt{x}(x-4)} dx$$ Can you please suggest me where to start? Any help would be appreciated. Thanks!
3
votes
1answer
67 views

evaluate the path integral around a circle in complex plane

Let $a \in \mathbb{C}$ with $|a|>1$. I need to evaluate the path integral around the unit circle in $\mathbb{C}$: $$\int_{|z|=1}\frac{|dz|}{|az-1|^2}$$ where $|dz|$ represents integration with ...
2
votes
1answer
60 views

Laurent expansion of $\operatorname{sech}(z)$ centred at $\pi i/2$

I have found that the roots of the $\cosh(z)=0$ occur at $\frac{(2k+1)\pi i}{2}$ where $k \in \mathbb{N}\cup{0}$. But I want to find the order the poles of $\operatorname{sech}(z)$ so I'm trying to ...
16
votes
2answers
407 views

How to prove that $\int_0^\infty\frac{\left(x^2+x+\frac{1}{12}\right)e^{-x}}{\left(x^2+x+\frac{3}{4}\right)^3\sqrt{x}}\ dx=\frac{2\sqrt{\pi}}{9}$?

A friend gave me this integral as a challenge $$ \int_0^\infty\frac{\left(x^2+x+\frac{1}{12}\right)e^{-x}}{\left(x^2+x+\frac{3}{4}\right)^3\sqrt{x}}\ dx=\frac{2\sqrt{\pi}}{9}. $$ This integral can be ...
2
votes
3answers
51 views

Two Indefinite Integrals

Looking for some hints to evaluate the following integrals (with complex analysis or otherwise): $$\int_0^\infty\frac{x^{p-1}}{x+1}\,dx,\;\;\;\; 0<p<1,$$ $$\int_{-\infty}^\infty e^{-s^2+isz}\,ds,...
5
votes
2answers
112 views

How do I get $ \int_0^1 \frac{dz}{\sqrt{z(z - 1\,)(z+1\,)}} = \frac{\sqrt{\pi}}{2} \frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{9}{4})}$?

While reading physics papers I found a very interesting integral so I decided to write it down. Let $p(z) = z^ 3 - 3\Lambda^ 2 z$ where $\Lambda$ could be any number. If you want $\Lambda = 1$ and $...
2
votes
1answer
38 views

contour integration problem.. [closed]

how can we find $$\int_C e^{2z} 9^{z-2} dz,$$ where $C$ is the the contour from $z = 0$ to $z = 1 − i$
0
votes
0answers
32 views

Matsubara sum with general exponent

Matsubara sums of the form $$\sum_{i\omega}\frac{1}{(i\omega-\xi_1)^a}\frac{1}{(i\omega-\xi_2)^a} $$ have closed-form solutions for $a=1,2$. See Wikipedia. Are there also closed-form solutions for ...
1
vote
1answer
51 views

If $\lim\inf_{r\to 0}{r}\cdot \max_{|z|=r}|{f(z)}|=0$ then $0$ is removable singularity.

$\lim\inf_{r\to 0}{r}\cdot \max_{|z|=r}|{f(z)}|$ show $0$ is removable singularity, given $f$ is analytic in a punctured neighborhood of $z=0$. What makes it difficult for me is the fact that the ...
1
vote
1answer
146 views

Integrating $\sin(x)/x$, how to treat the pole at the origin? [duplicate]

I want to use residue theory to integrate $$\int_{-\infty}^{\infty}\frac{\sin(x)}{x}dx$$ What would be a good contour to use? I plan to take the imaginary part of this integral: $$\int \frac {e^{...
0
votes
0answers
43 views

Is the integral of any even complex function equal to $0$ on any contour?

Is it true that: $\oint _{C(5i+1,8\sqrt3)} \frac {z}{sh(z)} dz = \oint _{C(i,\sqrt{10})} \frac {z^2}{(1-cos2z)^4}dz = \oint _{C(\pi + i,4)} \frac {z}{tan(z)} dz = 0$ The problem is that i lost my ...
0
votes
1answer
74 views

Determine poles and residues of contour integral using Laurent series

I want to find the residues of the integral $F = \int_{-\infty}^{\infty} \dfrac{1}{x+(a-ib)} \dfrac{1}{\exp(-x/c)-1} dx$ I know that $x=-(a-ib)$ is a simple pole which contributes a non-zero residue....
4
votes
0answers
70 views

inverse Laplace transform by finding residues of essential singularities

I want to find the inverse Laplace transform of $$F(s)=\exp\Big(-\sqrt{2s}\tanh(\sqrt{2s})\Big).$$ Despite the square roots, $F$ doesn't have any branch points since $$\sqrt{2s}\tanh(\sqrt{2s})=\frac{\...
2
votes
2answers
72 views

Inverse Laplace transform of an exponential function

What is the inverse Laplace transform of $$\frac{e^{\frac{-2}{s}}}{s}$$ I have seen an answer using Maclaurin series expansion of this function. This function is not analytic at $0$, so, is such ...
1
vote
1answer
54 views

Complex Line Integral of absolute value of z

How do we proceed for the following complex line integral? $$\int\limits_\gamma |z|\:dz$$ where $\gamma$ is the half circular $|z|=1$, $0\leq \arg (z) \leq \pi$ taking $z=1$ as the initial point. ...
6
votes
1answer
146 views

Contour integral of $\sqrt{z^{2}+a^{2}}$

Suppose $a$ is real and nonnegative. Say we wanted to compute the above function (for whatever reason, be it to solve an improper real integral, or something else) along the curve $C$, as on the ...
3
votes
1answer
56 views

Contour Integration with $\cos (n\theta)$

How can I calculate this integral using contour integration? $\displaystyle\int_0^\pi \frac{3\cos(n\theta)}{5+4\cos(n\theta)}d\theta$ I know I can start by using that $\cos(n\theta) = Re (e^{in\theta}...
0
votes
0answers
18 views

finding the residues and evaluating contour integral

find the contour integral $$\oint _{c} \frac{\sinh z}{z-1} dz$$, where C is a square of side 3 centered at the origin I have problem both with finding the residues and doing the integral. I ...
0
votes
0answers
49 views

evaluation of fourier transform of electric potential

I would like to ask how to evaluate equation 7? I have spent hours and still have no idea how to get a(k).
0
votes
1answer
47 views

Evaluating $\int_C e^{-z^2} dz$ as radius goes to infinity

I was trying to calculate the integral $$\lim_{R \rightarrow \infty}\int_{C_R} e^{-z^2}dz$$ where $C_R$ is parameterized by $C(\theta) = Re^{i\theta}$ for $\theta \in [-\frac{\pi}{4}, 0]$. I tried ...
1
vote
1answer
74 views

Integration $\int_0^{2\pi} \frac{\cos^2 3\theta d\theta}{5-4\cos2\theta}$ by using residues

$$\int_0^{2\pi} \frac{\cos^2 3\theta d\theta}{5-4\cos2\theta}$$ By substituting $\cos m\theta$ to $\frac{z^m+z^{-m}}{2}$ and $d\theta$ to $\frac{-i}{z}dz$,I get $$\int_0^{2\pi} \frac{\cos^2 3\theta ...
1
vote
1answer
21 views

Contour integral question with 3 line segments

I am really unsure as to how to tackle this contour integral question, Can I get a comprehensive guide to tackling this question? $$H(\lambda)= \int e^{i\lambda z^2}/(z-2-i) \space dz$$ where the ...
0
votes
1answer
27 views

Calculating $\int \limits _{\gamma_r}\frac{e^{iz}-1}{z^2}dz$

I don't understand the following example. The second term on the right-hand side is $\pi$, since $$\lim \limits _{r \to 0} \int \limits _{\gamma _r} \frac {\Bbb e ^{\Bbb i z} - 1} {z^2} \Bbb d z = ...
0
votes
1answer
13 views

Splitting a complex controur integration in two. Figuring out the orientation.

Say I have an integration $$\int_{L_1} f(z)dz $$ that I want to write as a sum of $$\int_{L_2} f(z)dz \quad and \quad \int_{L_3} f(z)dz $$ $L_1,L_2$ are positively oriented. Suppose $L_3$ be ...
1
vote
1answer
33 views

Is there a shorter proof to show that this complex intergral is constant?

I have the integral, $$I(R) = \int_{C_R}\frac{1}{z(z-1)^2} dz$$ with the property that $$\left|\frac{1}{z(z-1)^2}\right| \leq \frac{1}{R(R-1)^2} \quad |z|=R>1$$ Where $C_r$ is the contour ...
0
votes
1answer
26 views

Evaluating contour integral of complex conjugate

This is part of a homework assignment. Any hints will be useful, I haven't made any progress. I need to evaluate: $\int_{|z-1|=1} \bar{z}^n dz, n \in \mathbb{Z}$
4
votes
2answers
61 views

Contour Integration with $\cos(n \theta)$

I need to compute the following real integral using complex numbers. I'm unsure how to handle the numerator so that the ensuing calculations do not become too unwieldily. $\int_{0}^{2\pi} \frac{ \...
0
votes
3answers
122 views

Very tricky complex integral, with poles on both sides of the real line,

I am trying to evaluate$$\int_{-\infty}^{\infty} \frac {x^2 -x^4}{1-x^6}\,dx,$$ which is an old exam problem. There is a special note on this problem that reads: Note: Your answer need not be a ...
1
vote
0answers
17 views

Evaluate the integral $(x+1)/(x^2+2)^2$ by choosing an appropriate contour in the upper half plane

How do you solve this question? Evaluate the integral $(x+1)/(x^2+2)^2$ by choosing an appropriate contour in the upper half plane How would the answer change if this question was evaluated with the ...
4
votes
2answers
79 views

Integrating $\int_{-1}^{1}\frac{dx}{(x-a)\sqrt{1-x^2}}$

I'm asked to find the value of $$\int_{-1}^{1}\frac{dx}{(x-a)\sqrt{1-x^2}}$$ where $a$ is complex and $a\not\in[-1, 1]$. I think I should use Cauchy's integration formula but don't know how to ...
1
vote
1answer
51 views

Inverse Laplace transform seems to be always vanishing but it couldn't!

Let's consider $x\in (0,1)$ and the distribution $p(x)=\lambda x^\lambda$, $\lambda>0$. I would like to find the pdf of the sum. The characteristic function of the $N$ sum reads: \begin{equation} \...
1
vote
1answer
66 views

Solve $\int_{-\infty}^{\infty}\frac{x^3sin(x)}{x^4+16}dx$ using contour integration

I have $$\int_{-\infty}^{\infty}\frac{x^3sin(x)}{x^4+16}dx = \pi e^{-\sqrt{2}}cos(\sqrt{2})$$ and have been asked to show this using contour integration. I have chosen the semicircular contour along ...
1
vote
1answer
69 views

contour integral branch cut

I need some help to solve the following integral by contour integration. $$\int_{0}^{1} x^a (1-x)^{1-a}\,\mathrm{d}x$$ I attached my ideas and a picture of the paths to fix the labels. Kind regards,...
4
votes
1answer
158 views

Evaluating the integral $ \int_{-1}^{1} \frac{1}{(1+x^{2})(1-x^{2})^{1/4}}dx$

I've been trying to find a way to integrate $\int_{-1}^{1}\frac{1}{(1+x^{2})(1-x^{2})^{1/4}}dx$ using contour integration, but I'm having a hard time coming up with a contour to use. Since I have a ...
0
votes
0answers
21 views

Determining if a contour integral is independent of path

So, I have a contour integral that goes from a to b, and I'm to determine if it is path independent. I'm curious if I'm even going about this the right way, and if I'm not, if someone could point me ...
1
vote
1answer
36 views

Double complex integral

So basically I want to integrate over two complex variables, so my integration will look something like this $\int uv\cdot e^{-uv}dudv$ where u and v are complex coordinates, in this case two ...
0
votes
0answers
42 views

logarithmic singularities in contour integration

How to evaluate the contour integral using the residue theorem if there is a logarithmic derivative? For example this: $$\int_C \log\zeta(s)\frac{x^s}{s} ds$$ or even this: $$ \int_C \frac{\log x}{x}...
1
vote
0answers
48 views

Calculate $\int_{-\infty}^{\infty}\frac{e^{2x}}{\cosh \left ( \pi x \right )}dx$ using contour integration

The contour for the complex integral is the rectangle with vertices at $\left ( R,0 \right ), \left ( R,1 \right ),\left ( -R,1 \right ), \left ( -R,0 \right )$ The closed contour integral is equal to ...