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

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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 ...
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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: $\int_{|z-1|=1} \bar{z}^n dz, n \in \mathbb{Z}$
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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{ ...
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3answers
119 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 ...
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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
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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 ...
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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} ...
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1answer
65 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
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1answer
67 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 ...
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1answer
153 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 ...
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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 ...
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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 ...
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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 ...
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0answers
47 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 ...
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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 = ...
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1answer
61 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 ...
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1answer
47 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 ...
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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 ...
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1answer
45 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 ...
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1answer
92 views

Trying to establish the following identity involving sums using residues

If $2z - 1 $ is not an integer, then $$ \frac{1}{\cos( \pi z) } = 1 + \frac{4}{\pi} \sum_{n=1}^{\infty} \left[ \frac{ 2z -1}{(2z-1)^2-4n^2} + \frac{4}{1-4n^2} \right]$$ ...
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1answer
141 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 ...
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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 ...
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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.
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1answer
61 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) ...
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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
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1answer
59 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 $z=1/x$ ...
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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
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2answers
105 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 ...
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1answer
83 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} ...
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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 ...
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0answers
32 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 ...
2
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1answer
89 views

Evaluate the integral $\int_0^\infty \frac {x^{1/2} dx}{x^2 + 1}$ using method of residues

I am trying to evaluate the integral $\int_0^\infty \frac {x^{1/2} dx}{x^2 + 1}$ using method of residues. I can solve this very easily without the $x^{1/2}$ on top, but I do not know what to do when ...
3
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1answer
64 views

Show that $f(0) + f'(0)=\frac{1}{2\pi i} \int_C\frac{f(w)e^w}{w^2}dw$

Let f be holomorphic on a region containing the closed unit disk $D(0, 1)$ and C be the unit circle traversed anticlockwise. Let n be a positive integer. Show that $$f(0) + f'(0)=\frac{1}{2\pi i} ...
2
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1answer
81 views

Calculating convolution integral analytically

How can i compute convolution integral analytically, without using graphs. I hate using graphs, shiftings which are error prone. If this is possible can you explain what way i must follow? For ...
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2answers
80 views

Application of Green's Theorem leading to different solutions of area integral

I have an area integral problem over an irregular convex polygon. I use Green's Theorem to convert the area integral to a contour integral, and solve using standard methods. Green's Theorem says I ...
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votes
3answers
109 views

Show that $\int_{0}^{+\infty} \frac{\sin(x)}{x(x^2+1)} dx = \frac{\pi}{2}\left(1-\frac{1}{e}\right) $

I'm trying to show that $$\int_{0}^{+\infty} \frac{\sin(x)}{x(x^2+1)} dx = \frac{\pi}{2}\left(1-\frac{1}{e}\right) $$ using Jordan's lemma and contour integration. MY ATTEMPT: The function in ...
2
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1answer
97 views

Show $\int_{0}^{\infty} \frac{x^{-z}}{(1 + x)^{2}} ~ \mathrm{d}{x} = \frac{\pi z}{sin(\pi z)}$

I need to solve the following integral: $$ I = \int_{0}^{\infty} \frac{x^{-z}}{(1 + x)^{2}} ~ \mathrm{d}{x}. $$ Wolfram Alpha gives the answer as $ \frac{\pi z}{sin(\pi z)}$, or equivalently, $\pi ...
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1answer
29 views

Limit of contour integrals over semi-circles

For each $r>0$ let $I(r)=\int_{\gamma} \frac{e^{iz}}{z}dz$ where $\gamma(t)=re^{it}, t \in [0,\pi]$. Show that $\lim_{r \to \infty} I(r)=0$ I used $\int_{\gamma} ...
2
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3answers
64 views

Computing the integral using cauchy's theorem

I need to integrate $$\oint _{|z|=1} \frac {\sin z}{z}\, dz$$ I write $\sin z=z-\frac{z^3}{3!}+\frac{z^5}{5!}-...$, then I get by dividing by $x$, the series $1-\frac{z^2}{3!}+\frac{z^4}{5!}-...$. I ...
4
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3answers
194 views

Evaluate improper integral $\int_0^\infty \frac{x\sin x}{x^2+1}dx$

How to prove that $$\int_0^\infty \frac{x\sin x}{x^2+1}dx=\frac{\pi}{2e}$$ I've tried several basic approaches like substitution and IBP but can't move forward.
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1answer
56 views

Determine the complex contour integral $\oint \limits_{C} \frac{2}{z^3+z}dz$ without using Residue Theorems

Without residue theory, determine $$\oint \limits_{C} \frac{2}{z^3+z}dz$$ if $C: \big|~z~-~\frac{i}{2}~\big|=1$ is positively oriented. We first find that our integrand has three distinct ...
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1answer
43 views

Show $\frac {1}{2\pi i}\int_\gamma\frac {f'(z)}{f(z)}$ is sum of poles and zeroes times their order

Let $f: B(a,r) \to {\Bbb C} \cup \{\infty\}$ be a meromorphic function. For $p$ a zero or pole of $f$, let $\mu(f,p)$ denote its order. Let $\gamma$ be a closed curve in $B(a,r)$. Then $$\frac {1}{2 ...
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1answer
71 views

Schwarz Function of an Ellipse

I want to find the Schwarz function of the ellipse define by $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad a > b > 0. $$ To do so, substitute $$ x = \frac{z+\bar{z}}{2}, \quad y = \frac{z - ...
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3answers
74 views

Integrate: $\int_0^{2\pi} \frac{d\theta}{1+\cos^2\theta}$

I'm trying to integrate this here fella: $\int_0^{2\pi} \frac{d\theta}{1+\cos^2\theta}$ from examples in Ablowitz I know that for $|A|^2>|B|^2$ and $A>0$, $\int_0^{2\pi} ...
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1answer
46 views

Complex Laurent Series and Contour Integral

Let $f(z) = \sin{(\frac{1}{z})}$, where $z \neq 0$. Find a Laurent Series expansion of $f$ around the annulus $D: 1< |z|<3$. Use the result to find $$\oint \limits_C ...
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1answer
68 views

Complex Integral Over an Ellipse

Suppose we had a function defined as $$ S(\zeta) = \zeta - \sqrt{\zeta^2 - c^2} $$ and we wish to evaluate $$ \oint_\gamma \frac{S(\zeta)}{z-\zeta}\, d\zeta, $$ where $\gamma$ is the (positively ...
1
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1answer
62 views

2 ways for $\int_{|z|=r} x^2dz$

As the title says, I have to compute $$\int_{|z|=r} x^2dz$$ for the circle traversed anti-clockwise in 2 different ways. If I use a parametrisation I quickly get to 0 - which might be wrong though ...
1
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1answer
30 views

An inequality from complex contour integral on page 81 of Stein and Shakarchi's Complex Analysis

On page 81 of Stein and Shakarchi's Complex Analysis, there is an inequality, \begin{equation} \left\lvert \int_{A_{R}} f \right\rvert \leq \int_{0}^{2\pi}\left\lvert\frac{e^{a(R+it)}}{1+e^{R+it}} ...
1
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1answer
56 views

How to Solve the Contour Integral $\int_{|z|=r}x^{2}dz$

I have been asked to compute the integral $$\int_{|z|=r}x^2dz$$ for the circle traversed anticlockwise, without parametrisation, by observing that $$x = ...
0
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1answer
35 views

Almost Gamma function with imaginary exponential: substitution/contour trick

In a physics course, this integral showed up: $\int_0^\infty dp\, p^2\, e^{ibp}\; , \quad b \in \mathbb{R}$ My prof proceeded with the seemingly insane substitution $x=-ib$ to yield $\Gamma(3)x^{-3} ...