Questions on the evaluation of residues, on the evaluation of integrals using the method of residues or in the method's theory.

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4
votes
2answers
209 views

Using complex analysis to evaluate $\int_0^\infty\frac{(\ln x)^3}{1+x^2}d x$

Here is my attempt: Let $R>1>r$ and $C$ be the closed curve in $\mathbb{C}$ consists of the following pieces: $$C_1=\{Re^{it}: t\in(0,\pi)\},\quad C_2=[r,R],\quad C_3=\{re^{it}: ...
0
votes
0answers
55 views

Contour integral with two branch cuts

I'm trying to solve this integral: \begin{equation} \int_0^\infty d\omega \,\frac{\left(\left(\omega ^2+1\right) \cos (\delta )-2 \omega \right) \log ...
0
votes
0answers
35 views

Evaluating this contour integral.

I was reading a paper that had the following integral $\displaystyle\prod_{n=1}^L\oint_{C_n}\frac{dx_n}{2\pi i}\prod_{k<l}^L(x_k-x_l)\prod_{m=1}^L\frac{Q_w(x_m)}{Q^+_\theta(x_m)Q^-_\theta(x_m)}$ ...
0
votes
2answers
39 views

What are some books or online resources about the Sommerfeld-Watson method for series summation?

I'm looking for resource recommendations on the Sommerfeld-Watson summation method, i.e. the use of the residue theorem to obtain expressions like $$ \tag 1 \sum_{n \in \mathbb{Z}} g(n) = - \pi ...
0
votes
1answer
33 views

Residue theorem and Angle of modular function

Let $f$ be a meromorphic function on the region $Im(z)>0$, $v_p(f)$ be the order of $p$. (The number $n$ such that $\frac{f(z)}{(z-p)^n}$ is holomorphic and non-zero at $p$.) Moreover, assume $f$ ...
1
vote
1answer
107 views

Computing Complex Integral to Determine Analytic Continuation of $f(z) = \int_0^\infty {{e^{-zt}} \over {1 + t^2}} dt$

My question is the following: Find the analytic continuation of the function $f(z)$ defined by $$ f(z) = \int_0^\infty {{e^{-zt}} \over {1 + t^2}} dt, \ \ \vert \arg(z) \vert < {1 \over ...
4
votes
4answers
102 views

Can I get some guidance on solving $\int_{-\infty}^{\infty} \frac{\sin^2(x)}{x^2} \, dx$?

I am trying to evaluate: $$I = \int_{-\infty}^{\infty} \frac{\sin^2(x)}{x^2} \, dx.$$ Using a contour semi-circle (upper plane), I can get: $$ \oint_{C} f(z) \,dz = \oint_{C} \frac{1 - e^{2iz}}{z^2} ...
0
votes
1answer
26 views

Can inverse fourier transform be formulated in terms of residue?

Today I ran into a peculiar problem when trying to perform the inverse fourier transform of $\frac{1}{a+jw}$ where a is some number $$ \mathcal{F^{-1}}(\frac{1}{a+jw}) = ...
0
votes
1answer
221 views

Residue Theorem for Laplace Transform

I need to know what's the Residue Theorem for a Laplace Transform. Does anyone know the name or something, so I can search it? I couldn't find anything. For example, if I have this two equations: ...
0
votes
1answer
31 views

integration of an open curve about isolated singularities

I know if I integrate a circular arc of an angle $\theta < 2\pi$ about an isolated singularity of the complex funciton I would get a fraction $\frac{\theta}{2\pi}$ of the residue of that ...
1
vote
1answer
44 views

contour integral in a region where the function doesn't have any poles

What is the value of the following contour integral? The contour is a circle with radius $0.5$ around $z=i$ point: $|z-i|<\frac{1}{2}$ $$\oint_C\frac{dz}{2-\sin z}$$ I think it is $0$ because ...
0
votes
1answer
57 views

How can I calculate the integral $ \int_{\left| z \right| = r} \frac{dz}{(z-a)^n(z-b)^n} $ [closed]

How can I calculate the integral? $$ \int_{\left| z \right| = r} \frac{dz}{(z-a)^n(z-b)^n} $$ for $ \left| a \right| < r < \left| b \right|$ and $ m, n > 1$ I tried to use the cauchy ...
9
votes
2answers
189 views

Integrating around simple pole and semicircle

Let $f$ be a holomorphic function on $\mathbb{C}$ with simple pole at $z_0$. Then if $\Gamma$ is a circle around $z_0$ oriented counter-clockwise with radius $r$ and $r\rightarrow 0$, then ...
4
votes
1answer
75 views

Check my answer - complex analysis, using residue and rouche's theorem

I was asked the following questions and I am unsure of my solutions, any advice would be appreciated, maybe there is a better way of doing this. Question: We are given $f(z)=2z-\sinh (z)$ defined on ...
0
votes
1answer
20 views

Bounding quantities that appear after using the residue theorem

for an exercise using the residue theorem I need to prove that this term $$\left|\dfrac{e^{R+it}-e^{-R-it}}{\left(e^{R+it}+e^{-R-it}\right)^2}\right|$$ tends to zero as $R\to\infty$. It's clear that ...
0
votes
3answers
73 views

How can I compute the residue of $\frac{\sinh(z)\sin(\omega z)}{\cosh^2(z)}$ at $z=\frac{i\pi}{2}$?

I need to compute the residue of $$f(z)=\frac{\sinh(z)\sin(\omega z)}{\cosh^2(z)},$$ where $\omega\in\mathbb{R}$ is a parameter, at $z=\frac{i\pi}{2}$, but can I do it without computing the integral ...
2
votes
2answers
134 views

Pole on a contour. Problem with integration

I have a problem with calculation of the complex integral $$\int_{|z|=1}\frac{z^2+3z+2i}{(z+4)(z-1)}dz$$ Apparently integrand has a pole in $1$ lying on our circle. What can I do? I cant use Cauchy ...
1
vote
1answer
30 views

Finding the limit of a function with sines and cosines by using the taylor expansion

I need to find the residue of a second order pole $z=0$, the residue works out to the following: $$\lim_{z\to 0}\frac{2z\sin{z^2}-2z^3\cos{z^2}}{\text{sin}^2{z^2}}$$ My professor said it's ...
1
vote
0answers
32 views

Higher order residues using series

For example if we have any $f(z)$ with a singularity at $z=0$ but it is of order 12 Instead of using the limit definition, suppose $f(z)$ is in the form of: $$f(z) = a_0z + a_1z^2 + a_2z^3 + ... + ...
0
votes
1answer
29 views

Finding the Residue at $z=n$

Find the residue of: $$f(z) = \frac{(\psi(-z) + \gamma)}{(z+1)(z+2)^3} \space \text{at} \space z=n$$ Where $n$ is every positive integer because those $n$ are the poles of $f(z)$ This is a simple ...
1
vote
2answers
58 views

Residue Formula in complex analysis

I understand the residue formula but I just can't understand the cancelling down of $$ \operatorname{res}_{z=z_1} (f)= \lim \limits_{z \to z_1}(z-z_1) \frac {z^2}{z^4+1} = \frac {z_1^2}{4z_1^3}.$$ ...
0
votes
1answer
21 views

Classifying singularities and finding their residues

How would one find the residues of: $f(z)=z/cos(z)$ I believe that the singularities are $z=\pi/2 + 2k\pi$ where k is an integer, but I'm not sure how to go about classifying them and then finding ...
4
votes
1answer
46 views

Computing Fourier Transform of $\frac{1}{t^2+a^2}$

I know this should be relatively simple, but I'm not getting the complete answer correct when I check with Wolframalpha. Here is my attempt. Going straight from the definition, with $x,t,a \in ...
10
votes
1answer
119 views

Integrate $\int_0^\infty \frac{dx}{(x^2+2x+12)^2}$ using residues

I want to find the integral $$I=\int_0^\infty \frac{dx}{(x^2+2x+12)^2}$$ using contour integration; I am familiar with the trigonometric substitution in real analysis. There are no branch cuts, ...
1
vote
1answer
63 views

Compute the value of the complex integration

If $a\in \mathbb C$ with $|a|<1$ then find the value of the integration: $$\dfrac{1-|a|^{2}}{\pi}\int_{|z|=1} \dfrac{|dz|}{|z+a|^{2}}.$$ I can't proceed anyway, my main difficulty is for $|dz|.$ ...
1
vote
2answers
89 views

Explanation for applying Cauchy Integral Formula

I do not understand the last part. How do you get: $$\oint_{C_N} f(z) dz = \frac{-7\pi^3}{45} + 4\sum_{n=1}^{N} \frac{\coth(n\pi)}{n^3}$$ How do you derive this, and what part of cauchy's formula ...
0
votes
1answer
46 views

Find the poles of this function

$$F(z) = \frac{\pi\cot(\pi z)\coth(\pi z)}{z^3}$$ The book says this function has a pole of order 5 at $z=0$ Then the book says this has poles at (simple poles): $z = \pm 1, \pm 2, \pm 3, ....$ $z ...
2
votes
2answers
34 views

Complex integral $\oint_L \frac{\cos^2{z}}{z^2}dz$

Compute $$ \int_L \frac{\cos^2 z}{z^2}\,dz$$ where $L$ is the closed loop that goes counterclockwise around the square with vertices $-1$, $-i$, $1$ and $i$. I was trying to compute this ...
2
votes
0answers
67 views

Can this integral similar to the Fourier transformation of $\delta$ function be calculated analytically?

I want to calculate the following integral: $$\int_{-\infty}^{+\infty}dk\ \exp\left[i\big(kx-\sqrt{k(k-b)}\big)\right]$$ where $x$ and $b$ are both real. If $b=0$, the integral reduces to the Fourier ...
1
vote
1answer
40 views

Residue of function with different branch points

I'm wondering what would one do when one wishes to find the residue of a function $$\text{Res}_{z\to z_0} f(z)$$ where $f(z)$ has multiple branch points, for instance $f(z)$ may be a function such as ...
2
votes
0answers
94 views

What is the value of the integral $ \int_{-\infty}^{\infty}\frac{\sin(2x)}{x^3}dx$?

I tried to evaluate the integral $$ \int\limits_{-\infty}^{\infty}\frac{\sin(2x)}{x^3}dx$$ using residues but the answer comes out to be a negative value, $-2 \pi$, which seems strange. Any help on ...
1
vote
2answers
60 views

Compute $\oint \Bigl[ z e^{3/z} + \frac{\cos z}{z^2 (z - \pi )^3} \Bigr] \, dz$ [closed]

Compute $$\oint \left[ z e^{3/z} + \frac{\cos z}{z^2 (z - \pi )^3} \right] \, dz$$ $$|z| = 5$$ My question is how to do residue at $$\oint ze^{3/z} \, dz $$
0
votes
1answer
62 views

Evaluate $\int_c {{{\tan z} \over z}dz}$ using residue theorem

Using residue theorem, evaluate the following; $C:\left| {z - 1} \right| = 2$ $$\int_c {{{\tan z} \over z}dz}$$ I want you guys to check my answer.Is it correct? $$\displaylines{ {\mathop{\rm ...
0
votes
0answers
30 views

Showing that an integral of a curve in $\mathbb{C}$ vanishes when the parameter approaches infinity

I'm trying to solve a problem where you have to use the residue theorem in order to get the value of a certain integral, but I cannot go on from this point: I need to show that $\int_{0}^\pi ...
3
votes
1answer
54 views

Cauchy P.V. Of an improper inegral

the poles are $x=+1,-1,i,-i$ we should take only the upper have of axis so we should take residue of $1$ and $i$? right in this problem the book took only $x= i$. I don't know why !! please help
2
votes
1answer
58 views

Sum of Residues of $\psi^2(-z)$

Compute the Sum of residues of $f(z) = \psi^2(-z)$, where $\psi(z)$ is the digamma function. There are singularities for $z= 1, 2, 3, \ldots$, i.e. for all natural numbers. But how do I compute the ...
4
votes
2answers
100 views

Calculating Harmonic Sums with residues.

Evaluate: $$\sum_{n=1}^{\infty} \frac{H_n}{(n+1)^2}$$ A user stated: "most of the time sum up the residues of $(\gamma+\psi(z))^2\cdot r(z)$. To determine the residues, just expand the digamma ...
0
votes
0answers
31 views

Residues of a digamma based function.

I was wondering how we can find the residues of the digamma function, From: Integral Calculation Find the residues of: $$f(z) = \frac{(\gamma + \psi(-z))^2}{(z+1)(z+2)^3}$$ The answer is in the ...
4
votes
1answer
74 views

Integrate using residue theorem

This was a question on my complex analysis take home final. Since the semester is over and grades have been posted I believe I can post it now. Let $a > 0$ and $b > 0$. Verify that ...
1
vote
1answer
69 views

solve integral with residue theorem [duplicate]

I want to show that for positive $a$ $$\int_{-\infty}^{\infty}{\frac{\cos(x)}{x^2+a^2}} dx = \frac{\pi e^{-a}}{a}$$ I'm not even sure how to define a smart contour… I guess it can't be a half ...
0
votes
1answer
36 views

Finding the residue of the improper integral $\frac{1}{z^4+4}$

$$f(z) = \frac{1}{z^4+4}$$ the roots of this are: $z^2=\pm i\sqrt{2} \implies z=\pm\sqrt{i\sqrt{2}}$ and $z=\pm i\sqrt{i\sqrt{2}}$ i.e. $$f(z) = \frac{1}{(z\pm\sqrt{i\sqrt{2}})(\pm ...
2
votes
1answer
57 views

Functional equation for the $\zeta$-function, bounding a contour

In one of my textbook the following problem is written: Proving the functional equation for the $\zeta$-function: $$\zeta(z) = 2^z\pi^{z-1}\sin\frac{\pi z}{2} \Gamma(1-z)\zeta(1-z) \qquad ...
2
votes
2answers
50 views

Prove $\operatorname*{res}_{z=z_0} f(z)g'(z) = - \operatorname*{res}_{z=z_0} f'(z)g(z) $

If $f$ has an isolated singularity at $z_0$ show that: $$\operatorname*{res}_{z=z_0} f(z)g'(z) = - \operatorname*{res}_{z=z_0} f'(z)g(z)$$ Here is my proof using partial integration: Proof ...
1
vote
1answer
41 views

How to calculate the residue of $\frac{z^{2n}+1}{z^n[iaz^2+(1+a^2)z-ia]}$ at $z=0$

Could someone give me some pointers how to calculate: $$\operatorname*{res}_{z=0} \frac{z^{2n}+1}{z^n[iaz^2+(1+a^2)z-ia]}$$ I don't think it's possible using the limit formula, but I'm having ...
3
votes
3answers
58 views

Small questions regarding residue of $\frac{e^z}{\sin^2 z}$ at $z=k\pi$

Could someone check the correctness following and answer the small questions? Calculate the residue of $$f(z) = \frac{e^z}{\sin^2 z}$$ at $z=k\pi \;(k \in \mathbb{Z})$ I classify $z=k\pi$ as a ...
4
votes
3answers
166 views

Residue Theorem for Gamma Function

I am kinda stuck and not sure what to do at this point of the calculation where: $$\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left(\,x^{-1}\sigma\,\sqrt{\, 2\,}\,\,\right)^{s}\Gamma\left(\,{s \over ...
2
votes
1answer
219 views

How to do contour integral on a REAL function?

Suppose we are given the problem: Evaluate: $$\int_{0}^{\infty} \frac{1}{x^6 + 1} dx$$ Where $x$ is a real variable. A real variable function (no complex variables). I was reading Schaum's ...
0
votes
2answers
114 views

pole on the contour using the residu theorem, what is this formula of Plemelj?

I've tried solving the following problem but I get stuck at the very end... $f(z)$ is defined as $$f(z)=\frac{1}{(z-\alpha)^2(z-1)}$$ with $\alpha \in \mathbb{C}$ and $\operatorname{Im}(\alpha) ...
6
votes
2answers
136 views

Evaluating sums using residues $(-1)^n/n^2$ [duplicate]

I am an alien towards compelx analysis, with very little know I am posing a question, who someone may want to help with. Evaluate: $$\frac{1}{4}\cdot \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$$ In ...
2
votes
3answers
51 views

How is $ \lim_{z \to z_o} (z-z_o)\frac{f(z)}{g(z)} = \lim_{z \to z_o} \frac{f(z)}{g(z)-g(z_o)/(z-z_o)}= \frac{f(z_o)}{g'(z_o)}$?

I was reading this proof in Gamelin Complex Analysis (page 196): If $ f(z) $ and $ g(z) $ are analytic at $ z_o $ and if $ g(z) $ has a simple zero at $ z_o $ $$ Res[ \frac{f(z)}{g(z)},z_o ] = ...