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

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2
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1answer
38 views

Integrating secans over the imaginary axis using the residue theorem

I am trying to integrate $\sec(z)$ over the whole imaginary axis using the residue theorem. i.e., I want to calculate the integral $$\int_{\Gamma} \frac{dz}{\cos{z}}$$ where $\Gamma$ is the (open) ...
0
votes
0answers
32 views

Indefinite integral - tricks for expressing solution concisely

Consider the following indefinite integral: $I_n (b) = \int \mathrm{d}x \frac{\sin(nx) \sin(x)}{\cos^2(x)+b^2}$ where $b$ is a constant and $n = 1, 2, 3...$ Is it possible to write the solution ...
0
votes
0answers
34 views

Can any contour integral be calculated by direct methods?

Or is there ever an integral that can only be evaluated using integral theorems? Does substituting the parameterized contour into the equation and evaluating it as a Riemann integral always work?
6
votes
2answers
162 views

Confusion about contour integration of constant function: intuition vs. Residue Theorem

Let's say we have the holomorphic function $$f(z) = 1.$$ Because $f(z)$ has no poles, according the Residue Theorem we have $$\oint_\gamma f(z)\,dz = 0$$ for any closed counterclockwise path $\gamma$. ...
2
votes
2answers
50 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 ...
1
vote
0answers
28 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
47 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 $z$ ...
1
vote
1answer
86 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
58 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
39 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
31 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 ...
4
votes
2answers
65 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: ...
6
votes
2answers
113 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 ...
1
vote
1answer
54 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
52 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
72 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 ...
2
votes
2answers
211 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
33 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
37 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
370 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
50 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 ...
5
votes
2answers
110 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
37 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
25 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
108 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
29 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
68 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
47 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
53 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
34 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
117 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
45 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
16 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
42 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).
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votes
0answers
20 views

evaluating contour integral with given radius

I know how to do the contour integral $f(z)=\oint_{C} \frac{dz}{z^2+1}$ when the radius is infinite, I got $f(z)= 2\pi i * \frac{1}{2i} = \pi$, but how do I evaluate the same integral when it defines ...
0
votes
1answer
42 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
48 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
17 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
11 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
29 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
18 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
50 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
100 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
15 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
72 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
48 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
59 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
53 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 ...