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

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5
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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
28 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
135 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 ...
0
votes
0answers
37 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
71 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 ...
4
votes
0answers
62 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 ...
2
votes
2answers
65 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
48 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
142 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
52 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 ...
0
votes
0answers
17 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
46 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
68 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 ...
1
vote
1answer
20 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
26 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
32 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
24 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
60 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
113 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
64 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
63 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 ...
4
votes
1answer
145 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
33 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
40 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 ...
1
vote
0answers
42 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 ...
0
votes
1answer
32 views

What's wrong with this integral calculation?

I want to calculate the integral $$I = \int_0^{2 \pi} \sin^2 \theta\ \cos^4 \theta\ d \theta$$ by converting it into a complex integral around the unit circle. I use the identities $$\cos \theta = ...
1
vote
1answer
60 views

Evaluate the following integral $\int_{-1/2}^{1/2}\big(\frac{\sin(n\pi f)}{\sin(\pi f)}\big)^4 df$

There are similar questions out there, but I was hoping someone could show how to would evaluate the following integral $$\int_{-1/2}^{1/2}\bigg(\frac{\sin(n\pi f)}{\sin(\pi f)}\bigg)^4 df$$ I've ...
1
vote
1answer
46 views

Would a keyhole contour be advisable to use for this integration?

The integral is $$\int_0^{\infty}\frac {1}{\sqrt{x}(1+x^2)}dx$$ which is to be evaluated by contour integration. So, the integrand clearly has simple poles at $+/- i$. But what kind of pole ...
0
votes
0answers
42 views

Complex integral with roots

Integral on $C$ of $\int g(z)\,\mathrm dz$ $C$ is: (those points are $1$ and $e$) $$\begin{align} g(z) &= z^{1/4}\\ g(1) &= i\\ \end{align}$$ How do I evaluate this using Anti-derivative ...
0
votes
1answer
44 views

Fundamental Theorem of Calculus for complex line integrals

I am supposed to calculate $\int_{\gamma}\sin(2z)dz $ where $\gamma$ is the line segment joining $i+1$ to $-i$ Can we apply the fundamental theorem of calculus (because I think we are within the ...
1
vote
1answer
92 views

Trying to establish the following identity involving sums using residues

If $2z - 1 $ is not an integer, then $$ \frac{1}{\cos( \pi z) } = 1 + \frac{4}{\pi} \sum_{n=1}^{\infty} \left[ \frac{ 2z -1}{(2z-1)^2-4n^2} + \frac{4}{1-4n^2} \right]$$ ...
0
votes
1answer
130 views

Evaluating $\int_{0}^{\infty}\frac{\sin(ax)}{\sinh(x)}dx$ with a rectangular contour

I need to try to evaluate $\int_{0}^{\infty}\frac{\sin(ax)}{\sinh(x)}dx$ and it seems like this is supposed to be done using some sort of rectangular contour based on looking at other questions. My ...
0
votes
2answers
33 views

Integral on a contour curve

Find the line integral along curve $C$ of $[f(z)]^2=z$ where $f(1)=1$. Here is curve c: https://imgur.com/uOSLwdt (Sorry for the blur, the points are $1$ and $e$) How can I solve this? I am lost. Is ...
0
votes
1answer
51 views

Evaluate the complex integral: $\int_{|z|=1}xdz$

$\int_{|z|=1}xdz$ I ended up with $2\pi$ as my final answer, can anyone confirm and/or give me a shorter way to do it? Mine involved lots of sines & cosines.
-4
votes
1answer
56 views

Complex Integrals (No Residue allowed)

Complete the integrals along curve $C$ a) $\displaystyle\int_C\frac1z\ \mathrm dz$ b) $\displaystyle\int_Cf(z)\ \mathrm dz;\quad[f(a)]^2=z\ \&\ f(1)=1$ c) ...
2
votes
3answers
100 views

Evaluate the integral of function involving $\cosh$

Evaluate the integral $$ \int_0^{\infty} \frac{\cosh(ax)}{\cosh(x)}\,dx, $$ where $|a|<1$. Consider the closed loop integral of $\displaystyle\frac{e^{az}}{\cosh(z)}$ where the contour $C$ is ...
0
votes
1answer
54 views

Laurent series for $\exp(-x)$ centered at infinity

I want to expand $\exp(-x)$ in a series centered at infinity, i.e. , $\exp(-x)=\sum_{i=-\infty}^{\infty}b_n (x-\infty)^n$ Obviously, this does not make sense, so what I did is: We define $z=1/x$ ...
1
vote
4answers
69 views

Use Complex Integrals/ Residue to evaluate $\int_0^\infty \frac{dx}{(x+1)^3 + 1}$

Use Complex Integrals/ Residue to evaluate $\int_0^\infty \frac{dx}{(x+1)^3 + 1}$ I'm not sure how to do this integration. It looks like partial fractions but I'm unsure.
3
votes
2answers
104 views

Convolution of half-circle with inverse

I am trying to compute the function: $$f(\lambda)\equiv\int_{-1}^{1}\frac{\sqrt{1-x^2}}{\lambda-x}dx.$$ It arises as the convolution of the semi-circle density with the inverse function. When ...
1
vote
1answer
73 views

What happens to poles lying on branch cuts in contour integration?

Inverse the Laplace Transform $$\frac{1}{\sqrt{s}}\cdot\frac{1}{1 + s}$$ back to time domain requires evaluation of Bromwich integration: $$\frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty} ...
0
votes
1answer
32 views

Reversing the direction of a contour integral

If $$\int_{C} f(z) dz$$ is some contour integral over a closed curve $C$, and $-C$ is the contour taken in the opposite direction, can $$ \int_{-C} f(z) dz$$ be treated as a closed curve around the ...
0
votes
0answers
31 views

Contour integral with trigonometric functions

Does the following integral, where $n$ is an integer, have an analytic solution? $I_1 = \int_{0}^{2 \pi} \frac{\sin(n x) \sin(x) \cos(x)}{\cos(x)^2 + a} \mathrm{d}x $ I tried writing $\sin(n x)$ as ...