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

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3
votes
1answer
69 views

How to evaluate the summation $S_b$

This question is from my notebook, not hw or else, only exercise to understand better. I tried by myself. However, since my trail are too trivial, I dont need to write here. i am confused a bit. I ...
2
votes
1answer
156 views

Inverse Laplace Transform via residues

I have $\frac{1}{2 \pi i} \int_{\infty-iT}^{\infty+iT} \frac{e^{-s(1-t)}-e^{st}}{-s+e^{-s}-1} ds$ and I am trying to solve it using a contour. So I could have t>0, or t<0. I have a pole at 0. For ...
1
vote
0answers
24 views

possible results of integral along closed path after defining branch of sqrt

Prove that one can define a branch of the function $\sqrt{1-z^2}$ in every region $D\subset \mathbb{C}$ such that the points $-1$ and 1 belong to the same connected component of the complement of $D.$ ...
0
votes
1answer
101 views

Complex integral using residue theorem

I have $$\int_{|z|=1} z^m \sin\left(\frac{1}{z}\right)~dz,$$ for $m = 0,1,2,\dots$ I know that there is a singularity at $z=0$, and this singularity is within the curve, thus the residue theorem ...
1
vote
1answer
53 views

Evaluating $\int^{\infty}_{-\infty}{\frac{\cos x}{x^2+a^2}}$

I am working on some complex analysis problems related to poles and residues. I would really appreciate if someone could work the below problem. Its not a HW problem just one I picked out of a book ...
0
votes
1answer
61 views

Determining all possible values of contour integral of $\int\exp{z^{-1}}dz$

In our class, we are asked to find all possible values of $\int_{\gamma}\exp{z^{-1}}dz$ where $\gamma$ is any closed curve not passing through $z=0$. I wanted to ask if I can rewrite this expression ...
4
votes
1answer
159 views

residue of $\frac{1}{z^{2n}} \pi \cot(\pi z)$ at $z=0$

how to calculate the residue of $$\frac{1}{z^{2n}} \pi \cot(\pi z)$$ at $z=0$ I know the answer is $$(2\pi i)^{2n} \frac{B_{2n}}{(2n)!}$$ but I dont know how I saw an answer using "the coefficient ...
-1
votes
1answer
251 views

Residues in singular points of complex function.

I am asked to get the residues in the singular points of $f(z) = \frac{z^2 + 1}{z^2(z + 2)}$ . The problem is that I cant find what a singular point is for a complex function and how to get the ...
0
votes
1answer
95 views

handwaving substitution in integral involving branch cut and derivative of sqrt + generalization

Want to compute $$ I = \int_0^i \mathrm{d}z \frac{z}{\sqrt{z^2-1}}$$ on the complex plane using complex methods. QUESTION: is the result $i \left( \sqrt{2}-1 \right)$ which one gets imposing ...
1
vote
1answer
106 views

integral involving square root using complex methods: what choice of path?

I'm asked to compute using complex methods the following integral: $$ I(a)= \int_0^1 \mathrm{d}x \frac{\sqrt{1-x^2}}{x^2-a^2},$$ where $a>1.$ What I know is the following: for $|z|<1,$ the ...
2
votes
1answer
144 views

Calculating residue

I really don't know too much about residue or how to calculate it, but just out of curiosity how would I find the residue of $e^{-1/z^2} $ at $ z=0$? Would it make sense to do so? And would this value ...
0
votes
3answers
87 views

Calculate residues for this function

I want to understand the residues.. Here is a simple example I have in my book. But i just can`t understand what author explains there. Is there anybody who can explain me this? I need to find ...
2
votes
2answers
100 views

Residue theorem, $\int_{-\infty}^{\infty} e^{-ikx}(1-ika^2)^{-m} dk$ with integer $m$

I am trying to solve this integral $\int e^{-ikx}(1-ika^2)^{-m} dk$ using the residue theorem, but I cannot find the residue of the function. $$\frac{1}{(1-ika^2)^{-m}}=\sum (ika^2)^n(-1)^n ...
3
votes
3answers
283 views

Find the coefficient $c_{-3}$ in the Laurent series $g(z) = \frac{e^{iz}-1}{\cos z-1}$

The function $\displaystyle g(z) = \frac{e^{iz}-1}{\cos z-1}$ has a Laurent expansion of the form $\sum_{n=-\infty}^{+\infty} c_{n}z^{n}$ in the region $2\pi<|z|<4\pi$. Find the coefficient ...
4
votes
1answer
155 views

Expressing an integral in terms of the Bernoulli numbers

In Ahlfors' Complex Analysis text, the Bernoulli numbers, $B_k$, are defined as the coefficients in a Laurent development: $$(e^z-1)^{-1}=\frac{1}{z}-\frac{1}{2}+ \sum_1^\infty (-1)^{k-1} ...
2
votes
2answers
225 views

Integrating $\int_{0}^{\infty} \frac{(\log x)^2}{x^2+x+1}$ using residue theorem [duplicate]

Just out of curiosity, how does one integrate something like this using residue theory? $$\int_{0}^{\infty}\frac{(\log x)^2}{x^2+x+1} dx$$ According to Wolfram Alpha, the answer is ...
0
votes
2answers
274 views

Compute complex integral

Integrate the function using the residue theorem $$\int_0^{2\pi}\frac{d\theta}{(2-\sin \theta)^2}$$ Using the formula $\sin \theta=1/2i(z-1/z)$ and $d\theta=dz/(iz)$
2
votes
2answers
85 views
7
votes
2answers
317 views

Integrating $\int_0^\infty \frac{\log x}{(1+x)^3}\,dx$ using residues

I am trying to use residues to compute $$\int_0^\infty\frac{\log x}{(1+x)^3}\,dx.$$My first attempt involved trying to take a circular contour with the branch cut being the positive real axis, but ...
3
votes
2answers
63 views

A question about determing a residue

Let $f(z)$ be analytic at $z=w$ and have a pole at $z=a$. How does one show that the residue of $\displaystyle\frac{f(z)}{w-z}$ at $z=a$ equals the singular/principal part of $f(z)$ evaluated at ...
2
votes
2answers
114 views

Finding a definite integral by residue integration?

I have a probability distribution of the form $$\frac{k_1}{(k_2 x^2+k_3 x+k_4)^n}$$ and I want to find the mean and variance—but am running into problems with that. The mean would be ...
1
vote
0answers
93 views

Using Residue theorem to evaluate integral with complex poles

I'm trying to evaluate the integral of a function $$\frac{\psi_m^T\cdot P\cdot\phi_m\cdot\exp(-\gamma\xi i)}{(\gamma_m-\gamma)\cdot B_m}$$ with respect to $\gamma$, where $\psi_m$ is a $1\times n$ ...
12
votes
2answers
185 views

$\int_0^\infty \frac{\cos(tx)}{(x^2 - 2x + 2)}\,\mathrm{d}x$ for $t$ real

This was a question on an old prelim exam in complex analysis: compute $$\int_0^\infty \frac{\cos(tx)}{x^2 - 2x + 2}\,\mathrm{d}x$$ for $t$ real. I've tried… Residue calculus—it's ...
4
votes
1answer
197 views

Residues at poles

What is the residue of $$f(x)=\frac{1}{(x^2+1)^a}$$ at $x^2=\pm i$, where $0<a<1$ ? My intuition tells me that there must be a non-zero residue, but my attempts to compute tells me the residue ...
3
votes
0answers
139 views

Turning a Line Integral into a Contour one

I'm trying to compute an integral appearing in the article "On Determinants of Laplacians on Riemann Surfaces" of D'Hoker and Phong (page 541). It is as following. Fix $B\in \mathbb{R}_+$ and let ...
1
vote
3answers
426 views

Residue/Laurent series of $\frac{z}{1+\sin(z)}$ at $z=-\pi/2$

For some reason, I just can't quite figure out how to easily calculate the Laurent series for the following function: $$ f(z)=\frac{z}{1+\sin(z)},\quad z_0=-\frac{\pi}{2} $$ I don't really need the ...
0
votes
1answer
36 views

The residue of $\frac{(1-zx)^{n}(1-zx)^{1/2}}{z^{m+1}(1-z)^{n+1}}$ at $z = 0$

How do I calculate the residue of $\dfrac{(1-zx)^{n}(1-zx)^{1/2}}{z^{m+1}(1-z)^{n+1}}$ at $z=0$? Here, $x$ is a fixed real number and $m, n$ are positive integers.
0
votes
1answer
496 views

Calculate residue at essential singularity

I know you can calculate a residue at an essential singularity by just writing down the Laurent series and look at the coefficient of the $z^{-1}$ term, but what can you do if this isn't so easy? For ...
5
votes
2answers
95 views

Proving $g(\omega)=\frac{1}{2\pi i}\int_{\gamma}\frac{zf'(z)}{f(z)-\omega}\, dz$ where $g$ is the inverse of $f$

I have the following exercise: Let $G$ be an open subset of $\mathbb{C}$ and let $f$ be a one to one function in $H(G)$ such that $f'(z)\neq0$ for all $z\in G$. For each $\omega\in f(G)$ ...
2
votes
1answer
374 views

Contour integration using the residue at infinity

I posted a similar problem a few months ago but got no responses. So I'm going to try again with a different problem. I want to evaluate $ \displaystyle I ...
0
votes
1answer
38 views

Does ${\rm Res}_{z=z_0}\frac{p(z)}{q(z)}=\frac{p(z_0)}{q^{(m)}(z_0)}$ for all $m\ge 1?$

I know the following result involving pole: If $p(z),q(z)$ be two functions analytic at $z=z_0$ and $p(z_0)\ne0$ and $q(z)$ has a zero of order $m$ at $z_0$ (i.e. $q(z)=(z-z_0)^mf(z)$ where ...
7
votes
4answers
325 views

How to calculate $ \int_{0}^{\infty} \frac{ x^2 \log(x) }{1 + x^4} $?

I would like to calculate $$\int_{0}^{\infty} \frac{ x^2 \log(x) }{1 + x^4}$$ by means of the Residue Theorem. This is what I tried so far: We can define a path $\alpha$ that consists of half a ...
3
votes
1answer
117 views

Roots of $z^{2n} + \alpha z^{2n -1} + \beta ^2$

I've been looking at a problem available here. The problem is: Let $n$ be a natural number, and $\alpha$, $\beta$ nonzero reals. Show that the number of roots of $p(z) = z^{2n} + \alpha z^{2n -1} + ...
3
votes
0answers
58 views

Hint to compute the following integral

Can someone give a hint on how to solve the following integral? $$\int_0^{2N\pi}\frac{(-R\cos t)(\xi t-r)+\xi R\sin t}{(R^2+(\xi t-r)^2)^{3/2}}dt$$ I've tried some substitutions. First I've splitted ...
8
votes
0answers
223 views

Help in calculating the following integral $\int_0^{2\pi}\! \frac{(1+2\cos x)^n \cos (nx)}{3+2\cos x} \, \mathrm{d}x. $

I was asked to calculate this: $$\int_0^{2\pi}\! \frac{(1+2\cos x)^n \cos (nx)}{3+2\cos x} \, \mathrm{d}x. $$ My idea was to change the integration limits to $|z|=1$ in the complex plane and to ...
0
votes
1answer
26 views

Validity of residue outside the domain

Using the identity theorem I can see that $f(z)=\dfrac{2}{3+z}$ and hence 1 is true and 4 is false. This far is easy. But for 2 and 3 I can see that $f$ is not defined at $z=3$ and $-3$ is not a ...
3
votes
2answers
230 views

Taking the limit of an integral using residues, why is this wrong?

I have the integral $\lim\limits_{R\to\infty}\int_{-R}^{R} \frac{\cos(x)}{x^2+a^2} dx$ where $a$ is a positive real number. The strategy was to evaluate the limit of the integral on the boundary of a ...
0
votes
1answer
102 views

Evaluate this integral without using the residue theorem

I want to evaluate this integral $$ I = \int_c \tan z + \frac{\csc z}{z} dz $$ $$ c :|z| = 1 $$ apparently tanz is analytic in this region so its integral equals to zero now $$ I = ...
2
votes
1answer
232 views

Inverse Laplace transformation using reside method of transfer function that contains time delay

I'm having a problem trying to inverse laplace transform the following equation $$ h_0 = K_p * \frac{1 - T s}{1 + T s} e ^ { - \tau s} $$ I've tried to solve this equation using the residue method ...
1
vote
1answer
133 views

Generating Laguerre polynomials using gamma functions

An exercise given by my complex analysis assistant goes as follows: For $n \in \mathbb{N}$ and $x>0$ we define $$P_n(x) = \frac{1}{2\pi i} \oint_\Sigma ...
2
votes
2answers
231 views

Contour Integrals and Residues

I'm trying to figure out what it is all about, but my mind is blowing up. First of all, I have turned back and looked at the general definitions of integrals. Then I have looked to line integrals. ...
0
votes
2answers
142 views

Computing real integrals using the Residue Theorem where singularities are on the real line

How would you compute, for $a>0$ the integral $$\int_0^\infty \frac{\sin x}{x(x^2 + a^2)} dx \, \, ?$$ I've computed the residues of the function $$f(z) = \frac{e^{iz}}{z(z^2 + a^2)} $$ which I ...
3
votes
3answers
177 views

Laurent Series and residue of $\frac{z}{(z-1)(z-3)}$ around z = 3

As mentionned in the title, I'd like to get the function's Laurent series and after its residue, I have tried to separate the two denominators to get a partial fraction but I still have a z at ...
3
votes
4answers
118 views

Computing $\int_0^{\pi\over2} \frac{dx}{1+\sin^2(x)}$?

How would you compute$$\int_0^{\pi\over2} \frac{dx}{1+\sin^2(x)}\, \, ?$$
2
votes
1answer
42 views

Residue of a 1-form in a Riemann Surface does not depend of the chart

Let's suppose that $X$ is a Riemann Surface, $\omega$ is a meromorphic 1-form in $X$ and let $p$ be a pole of $\omega$ of order $M$. I want to show that the residue of $\omega$ at $p$, defined by $$ ...
2
votes
0answers
79 views

Find the inverse laplace transform: $\frac{1}{{{{({s^2} + 1)}^3}}}$

Find the inverse Laplace transform: $$x(t) = {L^{ - 1}}\left[ {\frac{1}{{{{({s^2} + 1)}^3}}}} \right]$$ with $x(t=0)=0$. I did: $${\left[ {{\mathop{\rm R}\nolimits} {\rm{es}}\frac{{{e^{st}}{{(s - ...
5
votes
2answers
400 views

Calculating integral with branch cut.

I'm learning how to calculate integrals with branch points using branch cut. For example: $$I=a\int_{\xi_{1}}^{\xi_{2}}\frac{d\xi}{(1+\xi^{2})\sqrt{\frac{2}{m}\left(E-U_{0}\xi^{2}\right)}}$$ where ...
3
votes
1answer
130 views

is this trig integral doable using contour integration?.

Is it possible to evaluate $\displaystyle \int_{0}^{\pi}\frac{x\cos(x)}{1+\sin^{2}(x)}dx=\frac{-{\pi}^{2}}{4}+ln^{2}(\sqrt{2}-1)$ by using residues?. I attempted it by considering $\displaystyle ...
5
votes
1answer
103 views

Finding a generalization for $\int_{0}^{\infty}e^{- 3\pi x^{2} }\frac{\sinh(\pi x)}{\sinh(3\pi x)}dx$

$\;\;\;\;$I was reading the introduction of Paul J. Nain's book "Dr. Euler's fabulous formula" where he talks about the sense of beauty in mathematics and quotes the G.N.Watson as saying that a ...
1
vote
2answers
134 views

Integration using residues

For the following problem from Brown and Churchill's Complex Variables, 8ed., section 84 Show that $$ \int_0^\infty\frac{\cos(ax) - \cos(bx)}{x^2} \mathrm{d}x= \frac{\pi}{2}(b-a)$$ where $a$ and ...