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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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
52 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
170 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
222 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
120 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
143 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 ] = ...
2
votes
2answers
59 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
80 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
30 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
54 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
44 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
43 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
84 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
70 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
88 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
117 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
61 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
179 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
126 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
52 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
63 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 ...
0
votes
1answer
47 views

Evaluating $\int^{\infty }_{-\infty}\frac {z^3\sin az}{z^4+4}dz$

I'd like to evaluate following integral with contour integration $$\int^{\infty }_{-\infty}\dfrac {z^3\sin az}{z^4+4}dz$$ and I think the best way to solve is to recognize it is equal to the ...
3
votes
4answers
137 views

Evaluate $\int_{-\infty}^\infty \frac{1}{(x^2+1)^3} dx$

Evaluate $\int_{-\infty}^\infty \frac{1}{(x^2+1)^3} dx$ I wasnt exactly sure how to approach this. I saw some similar examples that used Cauchy's theorem.
2
votes
2answers
56 views

Poles of $\frac{1}{1+x^4}$ [duplicate]

The integral I'd like to solve with contour integration is $\int^{\infty }_{0}\dfrac {dx}{x^{4}+1}$ and I believe the simplest way to do it is using the residue theorem. I know the integrand has four ...
0
votes
0answers
28 views

Proving susceptibility in Lorentz Model satisy Kramers-Kronig relations

My instructor asked me to prove that the real and imaginary parts of the electric susceptibility derived from Lorentz Model satisfy the Kramers-Kronig relations using the residue theorem. The problem ...
0
votes
1answer
64 views

residue of this function at infinity

How do I calculate the residue of $\frac{\sin(z)}{z}$ at infinity ? I tried to use wikipedia definition for the case, $\lim_{|z|\rightarrow\infty}f(z)=0$ then ...
0
votes
1answer
59 views

If $f(z)$ is a polynomial function of degree $n \ge 2$, prove that the sum of the residues of $\frac{1}{f(z)}$ is zero

Let $f(z)=a_nz^n +a_{n-1} z^{n-1} +...+a_1z+a_0$ be a polynomial of degree $n \ge 2$. Prove that the sum of the residues of $\frac{1}{f(z)}$ is zero. Ok, so here is my thinking process so far: At ...
23
votes
2answers
515 views

How to solve $\int_0^{\infty}\frac{\cos{ax}}{x^3+1}dx$?

QUESTION. It is looked for a closed solution for following real integrals $\displaystyle\int_0^{\infty}\displaystyle\frac{\cos{ax}}{x^3+1}dx$ and ...
2
votes
2answers
129 views

Find the contour integral around unit circle.

Evaluate the below integral by turning it into a contour integral around a unit circle: $$\int_{0}^{\pi}\frac{\cos2\phi}{1-2a \cos\phi + a^2} d\phi$$ $where\;a\neq \pm1$
0
votes
0answers
72 views

Calculate $\int_0^3 \frac{x^{\frac{3}{4}} (3-x)^{\frac{1}{4}}}{5-x}\,dx$ using principal branch

I would like to calculate the following integral $$ I = \int_0^3 \frac{x^{\frac{3}{4}} (3-x)^{\frac{1}{4}}}{5-x}\,dx $$ using contour integration but using principal branch of the function, i.e. ...
5
votes
1answer
120 views

Residue of $\frac{\cos(\frac{\pi}{z-1})}{z^2 \sin z}$ at $z=1$

Residue of $$\frac{1}{z^2 \sin z}\cos\left(\frac{\pi}{z-1}\right)$$ at $z=1$. More importantly, I don't even know whether it exists or not. The one who creates this question has made questions that ...
1
vote
1answer
45 views

How can I calculate this complex integral?

The integral is the following: $$\int_{|z|=r} \frac{z+1}{z(z^2+4)} dz , r>0, r \neq 2 $$ I'm a little bit lost, I know that its partial fraction expansion is $$ \frac{z+1}{z(z^2+4)} = ...
4
votes
5answers
176 views

Finding $\sum_{n=1}^{\infty }\frac{243}{16(n\pi )^5}\sin(2n\pi /3)$

The WolfarmAlpha couldn't give me the sum of $$\sum_{n=1}^{\infty }\frac{243}{16(n\pi )^5}\sin(2n\pi /3)$$ therefore I thought that this problem is difficult so I used my calculator to get $(1/24)$ ...
4
votes
2answers
123 views

Complex integral using cauchy residue formula

I want to compute $ \displaystyle \int_{0}^{+\infty} \frac{dx}{x^n-1} $ I've proved that $ \displaystyle \int_{0}^{+\infty} \frac{dx}{x^n+1} = \frac{\pi}{n\sin\left(\frac{\pi}{n}\right)}$ in a ...
6
votes
3answers
184 views

Evaluating an Integral by Residue Theorem

Its been awhile since I have taken complex analysis and I am wondering how to solve the following integral when $a>0, \ a=0,$ and $a<0$ for $$\int^{\infty}_0\frac{\cos ax+x\sin ax}{1+x^2}dx.$$ ...
2
votes
1answer
41 views

How to compute contour integral?

Use Residue theorem to compute contour integral $$\int_C \frac{4e^z}{\sin z} dz$$ I need help figuring out singularities that are within the circle $|z|= 4$. I am stuck at that part. Thanks in ...
1
vote
1answer
89 views

Evaluate $\int_0^{2\pi} \frac{d\theta}{\left(1+\beta \cos (\theta )\right)^2}$

I am trying to evaluate the integral $$\int_0^{2\pi} \frac{d\theta}{\left(1+\beta \cos \left(\theta\right)\right)^2}$$ via change of variables and applying Cauchy's Residue Theorem. Here is how I'm ...
2
votes
1answer
48 views

Complex integral and Laurent series

Could you help with solving this complex integral: $$\int_C z^3\exp{\left(\dfrac{-1}{z^2}\right)} dz$$ where $C$ is $|z|=5$. I am expecting that the Residue Theorem will be needed. The answer should ...
3
votes
2answers
98 views

Find the Fourier transform of $u(x) = \frac{x \cos(2x)}{(1+x^2)^2}$

Find the Fourier transform of $$u(x) = \frac{x \cos(2x)}{(1+x^2)^2}$$ My work Okay so we want $$\int_\mathbb R \frac{e^{-ixt}x\cos(2x)}{(1+x^2)^2}dx$$ Of course we want to apply the residue ...
2
votes
1answer
89 views

Using complex analysis to find the Inverse Laplace transform

I have been reviewing for my comprehensive graduation exam where I have been solving the Inverse Laplace transform via complex analysis. Consider $$ H(s) = \frac{s^2 - s + 1}{(s + 1)^2} $$ Then we ...
3
votes
1answer
123 views

Evaluating $\int_{-\infty}^{\infty}\frac{1}{(x^2+b^2)^2}dx$

Find $$\int_{-\infty}^{\infty}\frac{1}{(x^2+b^2)^2}dx$$ We see that the only poles are at $x=\pm bi$. Integrating over the semicircular contour implies that it is equal to $2\pi i*Res_{(+bi)}$ ...
4
votes
2answers
96 views

Help finding the residue of $1/(z^8+1)$

Help finding the residue of $1/(z^8+1)$ I'm integrating over $\{ Re^{it} | 0 \leq t \leq \pi \}$, and I found 4 simple poles at $z_0=e^{in\pi/8}$ where $n = 0,...,3$ and I'm trying to calculate ...
5
votes
4answers
301 views

Evaluate $\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$ using complex analysis

How do I compute $$\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$$ What I am doing is take $$f(z)=\frac{(\log z)^2}{1+z^2}$$ and calculating $\text{Res}(f,z=i) = \dfrac{d}{dz} \dfrac{(\log ...
9
votes
6answers
248 views

Evaluate $\int_0^{\infty} \frac{\log x }{(x-1)\sqrt{x}}dx$ (solution verification)

I tried to find the integral $$I=\int_0^{\infty} \frac{\log x }{(x-1)\sqrt{x}}dx \tag1$$ I substituted $x=t^2, 2tdt=dx$ and chose $\log x$ and $\sqrt{x}$ to be principal values. We have ...
2
votes
0answers
40 views

Solution of gaussian integral with hyperbolic cotangent

I was wondering if the integral $$I=\int_{-\infty}^{\infty}d\omega \omega e^{-(\omega/a)^2}\coth(\frac{b\omega}{2})\cos(\omega c)$$ where $a,b,c>0$ can be solved using complex countour ...
5
votes
2answers
117 views

Prove that $\zeta(4)=\pi^4/90$

I am asked to "use the calculus of residues" to prove that $$\displaystyle\sum\limits_{n=1}^{\infty} \frac{1}{n^4}=\frac{\pi^4}{90}$$ I think I can do this given the Laurent series for $\cot z$ ...
1
vote
1answer
77 views

Counting number of roots inside a circle, using Rouche's theorem,

Using the Argument Principle and applying Rouche's Theorem, I know that there are 6 zeroes of the polynomial $$z^{10}-6z^6+3z^4-1$$ inside the unit circle $|z|=1$, no zeroes inside $|z|=1/2$, but I'm ...