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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6
votes
2answers
217 views

What is the integral of 1/(z-i) over the unit circle?

At present there is a simple pole on the closed contour, so the Residue Theorem appears to be inapplicable. But I want to claim that we can enlarge this circle to make sure that it encloses the ...
0
votes
0answers
39 views

Integration (Residue theorem )

$$ \phi (x,y)=\int_{0}^{\infty }\left ( \frac{1-\cos zx}{z} \right )\sin zx \,e^{-zy}dz. $$ I have solved this integration this way: $$ \frac{\partial \phi (x,y) }{\partial y}=\int_{0}^{\infty }(\cos ...
2
votes
1answer
41 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) ...
3
votes
1answer
73 views

Limit of a difference of integrals that both look almost identical,

Let $\gamma (t) = t+i(e^t-1)$ for $-1\le t \le 1$. find $$\lim_{\epsilon \to 0^+} \left[\int_{\gamma} \frac{\sin(z)}{(z-i\epsilon)^2} dz - \int_{\gamma} \frac{\sin(z)}{(z+i\epsilon)^2} dz\right]$$ ...
1
vote
1answer
40 views

Multivariate/multidimensional residues

My very specific question: Given $(z_1,z_2) \in \mathbb{C}^2 $ and $$ I = \frac{1}{z_1 z_2 +1} $$ a) Where are the poles of $I$? b) What are the residues of $I$? Note: $z_1$ and $z_2$ are not in ...
8
votes
3answers
681 views

Need a hint for this integral

I'm trying to evaluate the following integral $$\int_0^{\infty} \frac{1}{x^{\frac{3}{2}}+1}\,dx.$$ This is an old complex analysis exam question, so I plan to use the residue theorem. How can I ...
1
vote
0answers
33 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
61 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
44 views

Let $f(z) = \frac{z^{-2}}{\sin( \pi z )}$. What is the residue for $z \neq 0$?

Let $f(z) = \frac{z^{-2}}{\sin( \pi z )}$. What is the residue for $z \neq 0$? In a pdf online, it states we may calculate the residue using the "derivative trick" to get: $$ \mathrm{Res}(f,n) = ...
1
vote
0answers
36 views

On the dog-bone contour around [-1,1], what are the arguments of these two lines approaching the real axis from above and below?

I am using a dog-bone contour to integrate around the interval [-1,1]. (-1 and +1 are branch points of the integrand.) I am using the principal branch of log, so I am restricting its argument to ...
0
votes
0answers
32 views

Calculating the residues of trigonometric functions of functions

For a generic function f(x), with simple poles at $a_1, a_2,..., a_n,$ I want to know if there is any way you can calculate the residues at those poles for the function sin(f(x)). I assume that if ...
1
vote
1answer
26 views

Quick question on the roots and poles of a meromorphic function,

Does the degree of the polynomial in the numerator always equal the degree of the polynomial in the denominator? In other words, the number of zeros, counting multiplicity, equals the number of ...
6
votes
2answers
127 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 ...
3
votes
1answer
45 views

Residue theorem with pole on integration path

I have to calculate the inverse Laplace transform of $\dfrac{1}{s^2+1}$ (which I know is sin(x)) by residue theorem: $\int^{i \infty}_{-i \infty}exp(t\cdot s)\cdot \dfrac{1}{s^2+1}\mathrm{d}s$. ...
3
votes
3answers
68 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
2answers
95 views

How can a positive integrand integrate to 0? [duplicate]

I integrated $\dfrac{\log x}{1+x^2}$ from $0$ to infinity with residue calculus and got... $0$. This also agrees with Wolfram Alpha. How can this be? Is it due to the behavior of $\log(x)$ near ...
2
votes
2answers
230 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!
2
votes
1answer
52 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 ...
1
vote
1answer
36 views

Problem with a residue calculation

I have a problem with calculating a residue. I want to calculate Res($f,\frac{1}{2}$), where $$ f(z) = \frac{z^6 +1}{z^3(2z-1)(z-2)} = \frac{z^6 +1}{2z^5 -5z^4+2z^3}.$$ $$ $$ One method is to view ...
0
votes
1answer
43 views

Find the residues of $\frac {z^2} {(z^4+1)^2}$

I know that $f(z)=\frac{z^2}{(z^4+1)^2}$ has four poles at -1 .$z_1= e^{i\frac{\pi}{4}}$, $z_2= e^{i\frac{3\pi}{4}}$, $z_3= e^{i\frac{5\pi}{4}}$,$z_4= e^{i\frac{7\pi}{4}}$.But how do I find the ...
1
vote
1answer
135 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
1answer
42 views

residue theorem to evaluate an integral

I have encountered a problem; Use Residue theorem to evaluate $\displaystyle\int_{-\infty}^{\infty} \frac{\sin z}{(z^2+4)(z^2 -2z +2)} \, dz$ How is this done?
0
votes
1answer
71 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
62 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
63 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
59 views

how to integrate the definite integral using residue theorem? [duplicate]

How to evaluate $\int_{0}^{\infty}\dfrac{1}{x^a+1}dx$, where $a>1$. I don't know where to start since $x^a+1$ could have infinitely many roots, then it is impossible(?) to evaluate its residues. ...
2
votes
2answers
44 views

$\int_{C_N} \frac{dz}{z^2\sin(z)}$ complex integral, problem with residues

Let $C_n$ be the rectangle, positively oriented, which sides are in the lines $$x=\pm(N+\dfrac{1}{2})\pi~~~y=\pm(N+\dfrac{1}{2})\pi$$ with $N\in\mathbb{N}$. Prove that $$ ...
1
vote
1answer
67 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 ...
3
votes
1answer
53 views

Find $ \int_0^{2\pi} \frac{1}{a^2\cos^2 t+b^2 \sin^2 t} dt \;; a,b>0$.

Using Residue Theorem find $\displaystyle \int_0^{2\pi} \frac{1}{a^2\cos^2 t+b^2 \sin^2 t} dt \;; a,b>0$. My Try: So, I am going to use the ellipse $\Gamma = \{a\cos t+i b \sin t: 0\leq t\leq ...
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 = ...
1
vote
1answer
54 views

Singularities of $\sin(z)/(1-\cos(\sqrt{z}\,))$

$\displaystyle f(z) = \frac{\sin(z)}{1-\cos(\sqrt{z}\,)}$. The assignment is to find all the singularities of $f$, determine the type of them and the residue. It is clear that the singularities are ...
1
vote
0answers
30 views

How to compute this integral on contour

How to compute this following integral? $$\int \limits_{C}^{}\frac{e^{az^2}\,dz}{z^4+1}$$ Given $ a>0$ and $C:=\{z: |z+1| = 1\}$ is positively oriented. Where should I start? What would ...
0
votes
0answers
49 views

evaluate $\int_0^\infty \frac{(ln(x))^2 }{1+x^2}dx$ [duplicate]

I am attempting to evaluate the following integral: $$\int_0^\infty \frac{(ln(x))^2 }{1+x^2}dx$$ Using the substitution $x=e^u$ and $dx=e^u du$, I get: $$\int_{-\infty}^\infty \frac{u^2}{e^{-u} + ...
0
votes
3answers
65 views

Integrate $\oint\frac{z}{\cos z-1}dz$ with residue theorem

$$\oint\limits_{|z-3|=4}^{}\frac{z}{\cos z-1}dz$$ My attempt: $$\cos z=1$$ $$z=2\pi k$$ The set includes only $z=0$ and $z=2\pi$. What next?
-1
votes
1answer
24 views

the residue at the singular point

We need to find residue $\frac{1}{\cos^2z}$ $\cos z=0$ $z=\frac{\pi}{2}+\pi k$ - order 2 poles as the next?
0
votes
0answers
18 views

rational function of complex polynomials can be uniquely written as: $R(z)=P(z)+\sum\limits_{i=1}^n\sum\limits_{j=1}^{r_i}\frac{a_{ij}}{(z-z_i)^j},$

Let $R(z)$ be a rational function of complex polynomials, i.e. $R(z)=\frac{f(z)}{g(z)}$ with $f(z),g(z)\in\mathbb{C}[z]$. Claim: $R$ can be uniquely written ...
0
votes
0answers
22 views

Residue of $\frac{\text{cot}(\pi z)}{z^6}$ at $0$

I am trying to compute $\zeta(6)$ = $\sum_1^{\infty} \frac{1}{n^6}$; I generally know how to do this using a residue-based proof, but I am stuck at the last bit, namely calculating the residue of ...
0
votes
3answers
111 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
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 ...
1
vote
1answer
31 views

finding residues for poles

I'm struggling to find the residues of the equation $$ \frac{-z\ln(z)}{(z^2+a^2)(2-z)} $$ with poles at $z=\pm ai$ and $z=2$ I have the residue for $z=2$ as $$ \frac{-2\ln2}{4+a^2} $$ but I am ...
1
vote
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} ...
0
votes
1answer
17 views

Calculate $\int_{C}\frac{e^{z+\frac{1}{z}}}{1-z^2}$

Calculate $$\int_{C}\frac{e^{z+\frac{1}{z}}}{1-z^2}$$ Where $C=\{|z|=2\}$ Ok so if I write $f(z)=\frac{e^{z+\frac{1}{z}}}{1-z^2}=\frac{e^z}{1-z^2}\cdot e^{\frac{1}{z}}$ Then $f(z)$ has an ...
2
votes
0answers
31 views

Integral of the principal value of a hypergeometric function

I am looking to write the hypergeometric function $${}_2F_1\left(1,1,2+\epsilon, -\frac{\alpha}{\beta}\right) = \int_0^1\,dt\,\frac{(1-t)^{\epsilon}}{1-tz + i\delta},$$ where $z=-\alpha/\beta$ and ...
0
votes
1answer
32 views

Show that the polynomial $p(z)=z^5+7z-1$ (in $\Bbb{C}[z]$) has one real root with absolute value smaller than one.

Show that the polynomial $p(z)=z^5+7z-1$ (in $\Bbb{C}[z]$) has one real root with absolute value smaller than one and that the rest of the roots are in $\{1<|z|<2\}$. Ok, so this exercise ...
0
votes
1answer
36 views

finding residue for complex analysis

I am having a tough time finding the residue for a function, suppose my test function is $$\frac{z^2}{{(z^2+a^2)}^2}$$ while I could determine the poles to be $+-ai$ and I know the formula to find ...
0
votes
0answers
46 views

integral vs. residue at infinity

I have an issue with residues at infinity. I am computing the integral $\displaystyle{\int_{C_3^+(0)} \dfrac{e^{3z}}{z^2(z^2+2z+2)} dz} $ Since all three poles ($0$ of order 2, $1\pm i$ of order 1) ...
1
vote
1answer
61 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 ...
4
votes
1answer
145 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 ...
0
votes
2answers
34 views

Using the residue theorem

Is it possible to evaluate $$\int_{-\infty}^\infty \frac{x^2}{(x^2+1)^2} \, dx $$ using the residue theorem, as opposed to Calc 1 methods? How can I get started using the residue theorem? What ...
0
votes
1answer
48 views

how to evaluate this definite integral? [duplicate]

For $\int_{0}^{\infty}\frac{\sin^2(x)}{x^2}dx$. I considered using residue theorem. But since the function inside is holomorphic except for a removable singularity at the origin. So whatever contour I ...