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
1answer
33 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
19 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
36 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 ...
1
vote
1answer
26 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
28 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
25 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 ...
0
votes
1answer
18 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
10 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 ...
3
votes
1answer
39 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
101 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, ...
0
votes
1answer
54 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
71 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
38 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
32 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
56 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 ...
2
votes
0answers
81 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
54 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
51 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
27 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
48 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
49 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 ...
3
votes
2answers
86 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
26 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
63 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
44 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
31 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
51 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
39 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
38 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
46 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
113 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
197 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
63 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
110 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 ...
1
vote
2answers
33 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 ] = ...
2
votes
2answers
52 views

How to compute the residue of $(z^2+2z+1)\sin\left(\frac{1}{1+z}\right)$

This was an example given in my notes but all it concluded was with something about an infinite principal part. How do we compute it? we have it equal to $ \left( z + 1 \right)^2 \cdot \sin \left( ...
0
votes
2answers
65 views

How to find the residues of $\frac{1}{(z^4+4)^2}$?

How to find the residues of this function? $$\frac{1}{(z^4+4)^2}$$ So far, I found the poles: $z_1=-1-i$, $z_2 = -1+i$, $z_3=1-i$, $z_4=1+i$. I know they are of the second order. But I have ...
1
vote
0answers
22 views

calculating the residue of a complex function in an integral

I wonder how to solve this integral: $$P(q, \omega) = \frac{-2i}{(2\pi)^4}\times\int\frac{\mathrm d\mathbf{k}~\mathrm d\omega' e^{i\omega'\eta/h}}{\left[\omega' - E(\mathbf k) + ...
2
votes
1answer
49 views

Complex Analysis Integrals

I'm unsure how to apply what I've learned in complex analysis to the following question types: $$ \int_{-\pi}^\pi \frac 1 {1 + \sin^2(\theta)}\,d\theta $$ and $$ \int_{-\pi}^\pi \frac ...
2
votes
0answers
33 views

Calculate a complex integral using residues

Let $f(z)= \frac{2(e^\frac{1}{z}-1)(\sin^2z)}{z^3}$. Calculate $\int\limits_{\partial B_+(O,1)} f(z)\operatorname{d}z$ Could someone confirm my solution? Solution? I try to calculate the ...
1
vote
2answers
30 views

Compute $\int_\Gamma \frac{e^\frac{1}{z}}{z-1}dz$, where $\Gamma$ is the circle $|z-1|\le\frac{3}{2}$, positively oriented.

Compute $\int_\Gamma \frac{e^\frac{1}{z}}{z-1}dz$, where $\Gamma$ is the circle $|z-1|\le\frac{3}{2}$, positively oriented. The numerator is not analytic in $\Gamma$ so we can't use Cauchy ...
0
votes
2answers
78 views

How do I calculate the residue of $\sin(z+1/z)$?

How do I do this about $\displaystyle z=0$ ?. I tried creating a Laurent expansion and extracting it from there but I wasn't sure how to isolate the $\displaystyle 1/z$ expression. $$ \mbox{I ...
4
votes
1answer
50 views

Contour integral $\int_{|z|=1}\frac{2z^2+z}{z^2-1}\, dz$ using residues

I am trying to evaluate the contour integral $$\int_{|z|=1}\frac{2z^2+z}{z^2-1}\, dz.$$ In this case the two singular points lie on the boundary (on the contour). So do I count the residues at this ...
1
vote
1answer
68 views

Evaluation of $\int_0^{2\pi} \frac{1}{1+8\cos^2(\theta)}d\theta$ with Cauchy's residue Theorem

I have to proof $$\int_0^{2\pi} \frac{1}{1+8\cos^2(\theta)}d\theta = \frac{2\pi}{3}$$ with Cauchy's residue Theorem. I have showed it, but in my solution, there comes $-\frac{2\pi}{3}$. I Show you ...
0
votes
1answer
101 views

integration, laurent series, residue therorem

Evaluate the integral $\int_\gamma f(z)dz,$ where $\gamma(t)=e^{it}$, and $0\leqslant t\leqslant2\pi$. For $f(z)$ equal to: $$\dfrac{e^z}{z^3},\quad\dfrac1{z^2\sin z},\quad\tanh ...
0
votes
1answer
45 views

Residue theorem with contour integrals

I want to evaluate the integral $$ \int_{\gamma} \frac{1}{z^{2}\sin(z)} dz$$ where $\gamma(t) = e^{it}$ and $ 0 \leq t \leq 2\pi$ using the Residue theorem. I've tried expanding sin(z) with Taylor ...
2
votes
3answers
141 views

Evaluate $\int_0^{2\pi} \frac{\sin^2\theta}{5+4\cos\theta}\,\mathrm d\theta$

Evaluate $$\int_0^{2\pi} \frac{\sin^2\theta}{5+4\cos(\theta)}\mathrm d\theta$$ This is the final question on my review for my final exam tomorrow, and I will be honest and say that I have no clue ...
2
votes
2answers
90 views

Evaluate the Cauchy Principal Value of $\int_{-\infty}^{\infty} \frac{\sin x}{x(x^2-2x+2)}dx$

Evaluate the Cauchy Principal Value of $\int_{-\infty}^\infty \frac{\sin x}{x(x^2-2x+2)}dx$ so far, i have deduced that there are poles at $z=0$ and $z=1+i$ if using the upper half plane. I am ...
1
vote
1answer
38 views

Generating function of the Laguerre Polynomials

The Laguerre Polynomials have the following integral representations $$L_{n}^{\alpha} (x) = x^{-\alpha} e^x \frac{1}{2\pi i } \oint_c \frac{e^{-z} z^{n+\alpha}}{(z-x)^{n+1}} dz$$ where $c$ is an ...
2
votes
1answer
56 views

Countour integral using residue theorem

Evaluate the integral $$ \int_{\gamma} \tanh(z) dz $$ where $\gamma(t) = e^{it}$ and $0 \leq t \leq 2\pi$. I want to do this using the residue theorem but I am unsure of how to work out the poles of ...