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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0
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1answer
68 views

Application of Residue Theorem and limits

I am trying the following problem from Fisher's Complex Variables book: If $f$ is analytic on a plane except at poles $\gamma_1, \cdots \gamma_N$ and none of them are integers and ...
0
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1answer
19 views

Look for the group members $\left ( (\mathbb Z/24\mathbb Z)^*,\underset{24}{\odot} \right )$ and calculate their orders.

Look for the group members $\left ( (\mathbb Z/24\mathbb Z)^*,\underset{24}{\odot} \right )$ and calculate their orders. The elements are $\overline1,\overline2,\ldots,\overline{23}$. Have the ...
3
votes
1answer
46 views

Residue of $\frac{1}{(1-z)^3}$ at $z=1$

I know there is a singularity of $z=1$ but I am a bit confused on how to find the residue at that point since if we have that $f(z)=\frac{g(z)}{h(z)}$ with $g(z)=1$ and $h(z)=(1-z)^3$ then $g(z)$ has ...
0
votes
1answer
25 views

Application of Residues

So in applying the residue theorem to solve improper real integrals, we agree to take our semicircles to be as large or as small as necessary such that all the poles we wish to work with lie inside ...
5
votes
2answers
111 views

Find the Fourier Transform of $\dfrac{x}{x^4+4}$

I have a problem here that becomes quite difficult to manage. I have to find the fourier transform of: $$f(x)=\frac{x}{x^4+4}$$ I'm sure there will be many ways to do this and I'll post my method ...
2
votes
2answers
90 views

Riemann zeta, why are the residues either zero or one?

One more question, probably equally simple to answer but I don't know how this is true either: Why is the residue of Riemann zeta zero - trivial or non-trivial: $$\text{residue}\left(\zeta ...
1
vote
0answers
33 views

obtaining inverse z-transform by different methods

how can i obtain the z-transform of $X(z) = \frac{z+1}{(z-1)(z+2)^2}$ by: 1) Partial fraction expansion, 2) residue theorem, and 3) direct division method any help is appreciated.
5
votes
1answer
32 views

Meromorphic and even

I would like to do the following exercise : Let $f$ be a meromorphic function and $\mathcal{P}$ the set of its poles. We also assume that $f$ is even ($\forall z \in \mathbb{C}, \; ...
1
vote
1answer
52 views

Deducing Laplace Formulas

I have to compute the followings integrals $\forall\; b\in \mathbb{C},\; \text{Re} \;b \gt0,p\gt 0$ $$ \int_{-\infty}^\infty \frac{e^{ipx}}{x-ib}$$ $$ \int_{-\infty}^\infty \frac{e^{ipx}}{x+ib}$$ ...
0
votes
1answer
146 views

Is f(z)=1/z truly an analytic function

For an analytic function $f(z)$, we have $$\frac{\partial f}{\partial \bar{z}}=0.$$ Consider the function $f(z)=\frac{1}{z}$, which, at first sight, is a bona fide analytic function. However, we can ...
2
votes
1answer
43 views

I want to compute $\int_0^\infty \frac{x^t}{1+x^2}dx \; \forall t \in (-1,1)$ using residue theroem.

I want to compute $$\int_0^\infty \frac{x^t}{1+x^2}dx \qquad \forall t \in (-1,1)$$ using residue theroem. I consider $$f(z) = \frac{z^t}{1+z^2}$$ I find two pole of order 1 in $z=i$ and $z=-i$ with ...
3
votes
0answers
92 views

Is there a simple and fast way of computing the residue at an essential singularity?

Is there a simple and fast way of computing the residue at an essential singularity ? I mean if we have a pole of order $n$ at $c$ we can use the formula : $$\mathrm{Res}(f,c) = \frac{1}{(n-1)!} ...
1
vote
1answer
64 views

Complex contour integral with sign function:$-i \int \limits_{-\infty}^\infty \frac{{\rm sgn}(x)^2 ~x~ e^{i x}}{1+ax^2} dp$

I am trying to evaluate the integral: $-i \int \limits_{-\infty}^\infty \frac{{\rm sgn}(x)^2 ~x~ e^{i x}}{1+ax^2} dx$ with sgn$(x)$ the sign function and $a$ positive real. Naively applying the ...
2
votes
2answers
64 views

compute the integral using residue theory

I am trying to compute an integral in an example in my complex analysis textbook: $$\int_{-\infty}^\infty {xsinx\over x^4+1}dx$$ The book gives some startup hints, but I don't quite follow, I set ...
0
votes
0answers
20 views

computing integral using residue theory [duplicate]

I want to compute the integral $\int_{-\infty}^\infty {x^4\over {1+x^8}}dx$ by using residue theory. I find the zero of $Q(x)$ is $i^{1/4}$. Do I have to factor the denominator into 8 different ...
2
votes
1answer
65 views

Find the Laurent series of the function and give the residue

I have $f(z)={z^2\over {z^2-1}}$. I want to find the Laurent series of $f$ and the residue at the point $z_0=1$. Can I say that $f$ has a pole of order 2 at $z=1?$ Or is that only used when the ...
3
votes
1answer
13 views

$a_{-1}$ term of a function on the punctured plane?

Our instructor said if f is analytic on the punctured complex plane (missing origin), then $$a_{-1}=\oint_{|z|=r} f(z) dz$$ This may be an obvious question, but what happened to the $\frac{1}{2\pi i}$ ...
2
votes
2answers
62 views

Find $\int_\Gamma\frac{2z+j}{z^3(z^2+1)}\mathrm{d}z$ where $Γ:|z-1-i| = 2$

pls, some ideas for integral solution (residue theory)? $$\int_\Gamma\dfrac{2z+j}{z^3(z^2+1)}\mathrm{d}z$$ Where $Γ:|z-1-i| = 2$ is positively oriented circle. Thx, for help!
0
votes
1answer
21 views

Residue Calculation Problem

I need to find the residue of the following equation at $z = 0$. $$ \frac{\cot(z)\cot(hz)}{z^3}$$ My attempt is as follows: The residue will be the coefficient of $1/z$ in the Laurent Series ...
0
votes
1answer
71 views

Residue theorem in evaluating complex integrals?

It's been a while since I used residue theorem to evaluate anything. I remember that whenever we have a real valued function, we can use residue theorem to evaluate its integral with associated ...
0
votes
2answers
117 views

Calculating residue of $z\sin{ \frac {z+1}{z-1}}$

Let $f=z\sin{ \frac {z+1}{z-1} }$. Calculate the residue of $f$ in $z=1$. I think $f$ has an essential singularity at $z=1$ so the only way I can proceed is with Laurent series. I've defined ...
1
vote
1answer
34 views

complex residue involving exponent of quotient of polynomials

I was trying to work out an integral and came to trying to find the complex residue of $$R = \exp\left(\frac{Ax^2 + Bix}{Dx + 1}\right)$$ at $x = -D^{-1}$. I used partial fractions to get: $$ = ...
1
vote
1answer
74 views

$\int e^{\cos(x)} \cos(nx)\ dx$ using the residue theorem

I am trying to evaluate the following integral using the residue theorem: $$\int_0^{2\pi} e^{ \cos(\theta)} \cos(n\theta) d\theta$$ I have already evaluated $\int_0^{2\pi} e^{e^{-i\theta}} e^{i ...
4
votes
1answer
118 views

Residue theorem with exponential and trig functions

The following integral should be doable using the residue theorum: $$\frac1{2\pi}\int_{0}^{2\pi}e^{\cos\theta}\cos(n\theta) \,d\theta$$
0
votes
0answers
30 views

Contour Integral of $I= \int_{-i\infty}^{i\infty}\frac{a^{z+1}}{1+z} dz$ [duplicate]

I'm trying to evaluate the following integral: $I= \int_{-i\infty}^{i\infty}\frac{a^{z+1}}{1+z} dz$ $0<a<1$ I've integrated from 0 to $ i\infty $ then from ...
4
votes
0answers
89 views

Ugly-nice double series

I'm trying to evaluate the following ugly double sum (presented in raw notation as used in my calculations): $\sum _{m=1}^{\infty } \sum _{n=1}^{\infty } \frac{4 m \cos \left(\frac{2 \pi m ...
1
vote
1answer
114 views

Compute $\int_0^{\pi/2} \frac{1}{(\sinh t)^{2}+(\sin{\theta)^{2}}} d\theta$ by residues

We want to compute $\int_0^{\pi/2} \frac{1}{(\sinh t)^{2}+(\sin{\theta)^{2}}} d\theta$ with $t>0$ using residues. The first thing I want to do is using $z=e^{i\theta}$ to transform the integral ...
1
vote
1answer
75 views

Calculation of a Residue

Does anyone know of a good way to calculate the residue at zero of the following function? I was able to calculate it with the higher order pole formula for residues and then used Mathematica to find ...
0
votes
0answers
94 views

Convergence of improper real integrals using the residue theorem

Let $f:\mathbb{R} \to \mathbb{R}$ be a funktion that satisfies $$ |xf(x)| \to 0 \qquad (|x| \to \infty) $$ If $f$ can be extended to a holomorphic function with finitely many singularities $S$ and the ...
1
vote
2answers
97 views

Infinite sums and integrals using residues

I have no idea how to solve these two, any help? $\mathtt{i)}$ $$\frac{1}{2\pi i}\int_{a-i\infty}^{a+i\infty}\frac{e^{tz}}{\sqrt{z+1}}dz$$ $$ a,t\gt0$$ $\mathtt{ii)}$ $$ \sum_{n=1}^\infty ...
2
votes
0answers
43 views

Integral on the circle

It's a standard fact that to calculate integrals of the form $$\int_{0}^{2\pi}\mathcal{R}(\cos(\theta),\sin(\theta)) \ d\theta$$ with $\mathcal{R(x,y)}$ a rational function in two variables without ...
1
vote
1answer
41 views

The definition of Residue

In Wikipedia the definition of a residue of a function $f$ in a point $a$ is a unique value $R$ such that $f(z)-\frac{R}{z-a}$ has an anti derivative in a punctured disk $0<|z-a|<\delta$. How is ...
3
votes
1answer
129 views

prove $\int_{0}^{2\pi}(1-\cos x)^n\cos nx dx=(-1)^n\frac{\pi}{2^{n-1}} $

prove $$\int_{0}^{2\pi}(1-\cos x)^n\cos nx dx=(-1)^n\frac{\pi}{2^{n-1}} $$ I tried with $2\cos x =z+\frac1z$ then use residue theorem but I faced some troubles my try is : $2\cos x =z+\frac1z$ ...
0
votes
0answers
33 views

Residue at Infinity Question

So, I'm trying to determine a residue of the following at infinity: $$f(z)=\frac{z^6}{(1+z)^3}$$ How would I even begin to approach this? Here are my thoughts; I'm mostly looking for a confirmation ...
0
votes
1answer
63 views

Residue integration

Evaluate\begin{align} \int_{-\infty}^{\infty} \frac{\sin(x)}{x^2+4x+5} \, dx \end{align} I set $f(z) = \frac{1}{z^2 + 4z + 5} = \frac{1}{(z-z_0)(z-\bar{z_o})}$, where $z_0 = -2+i$ and $\bar{z_0} ...
0
votes
3answers
48 views

How to calculate $I(x)=\int_{-1}^1 \frac{dt}{\sqrt{1-t^2}(t-x)}$ by Residue theorem [$|x|>1$]

How to calculate $\int_{-1}^1 \frac{dt}{\sqrt{1-t^2}(t-x)}$ for $|x|>1$ by Residue theorem? I could do is just as: $$I(x)=\int_{-\pi/2}^{\pi/2}\frac{d\theta}{\sin\theta-x}\\ ...
0
votes
1answer
75 views

Using Jordan's Lemma for residue integration

Evaluate \begin{align} \int_{0}^{\infty} \frac{\cos(ax)}{x^2+1} \, dx \end{align} So far, I set $f(z) = \frac{1}{z^2+1}$, and the singularity point are $z = \pm i$. But I am using the top ...
2
votes
3answers
460 views

Using residues to evaluate an improper integral

Use residues to evaluate the improper integral \begin{align} \int_{0}^{\infty} \frac{1}{(x^2+1)^2} \, dx \end{align} First, I said $f(z) = \frac{1}{(z^2+1)^2}$. My only question so far is how ...
1
vote
1answer
43 views

Finding a residue

I am given $f(z) = \frac{\text{Log} z}{(z^2 + 1)^2}$. The singularity point occurs at $z = i$, with a pole order of $m = 2$. How can I go about finding $\text{Res}_{z=i} f(z)$? The correct answer ...
0
votes
0answers
41 views

Polynomial Expansion in a Complex Improper Integral

I want to evaluate the integral below by using the Residue theorem. $$\int_{-\infty}^{\infty}\frac{\exp(-iwt)w}{(w-ic)\sqrt{w^2-aw+b}}dw$$ There are branch cut points due to the square rooted term ...
15
votes
3answers
276 views

Need help with $\int_0^\infty\frac{\log(1+x)}{\left(1+x^2\right)\,\left(1+x^3\right)}dx$

I need you help with this integral: $$\int_0^\infty\frac{\log(1+x)}{\left(1+x^2\right)\,\left(1+x^3\right)}dx.$$ Mathematica says it does not converge, which is apparently false.
0
votes
0answers
47 views

Integration by Residue Theorem - Is This Integral equal to Zero?

$$ \int_{-\infty}^{+\infty}\frac{\sin(x)}{2x^{5}-3jx^{3}+2x}dx=0 $$ ...
0
votes
2answers
57 views

Calculate the integral $\int_{\varGamma}\frac{3e^{z}}{1-e^{z}}dz$

I am looking to solve $$\int_{\varGamma}\frac{3e^{z}}{1-e^{z}}dz,$$ where $\varGamma$ is the contour $|z|=4\pi/3$. We have been asked first to consider $e^{z}=1$ and $e^{z}=-1$ which I get to be ...
0
votes
0answers
33 views

Integration of a complex function having square rooted denominator

How to evaluate this integral ? $$\int_{-\infty}^{\infty}\frac{1}{(w-ia)\sqrt{w^2-ibw+b^2c}}dw$$ where $a$, $b$ and $c$ are real and greater than zero. Does the residue theorem work on it? Please ...
7
votes
3answers
162 views

How prove this sum$\sum_{k=0}^{\infty}\frac{(-1)^{\frac{k(k+1)}{2}}}{(2k+1)^2}=\frac{\sqrt{2}\pi^2}{16}$

How show that $$\sum_{k=0}^{\infty}\dfrac{(-1)^{\frac{k(k+1)}{2}}}{(2k+1)^2}=\dfrac{\sqrt{2}\pi^2}{16}$$ My idea:I know how to prove the following sum ...
0
votes
1answer
46 views

Computing $\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t$

I'd like to calculate the following integral on the interval $[0,2\pi]$: $$ I=\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t = 2\pi. $$
3
votes
2answers
168 views

Integral $\int_{0}^{2\pi}\log|e^{i \theta}-1|d \theta$

Consider $$\int_{0}^{2\pi}\log|e^{i \theta}-1|d \theta$$ Is it equal to $0$ ? Why ? Any hint ?
4
votes
1answer
115 views

Using Cauchy integral formula to calculate $\int_\gamma \frac{\cos{z}}{z^n}$

Let $\gamma(\vartheta)=\mathrm{e}^{i\vartheta},\,\vartheta\in[0,2\pi]$, and consider the integral $$I(n)=\int_\gamma \frac{\cos{z}}{z^n},$$ where $n\in \{0,2,4,6,...\}$. Is there any way to prove ...
4
votes
0answers
59 views

Can we use a sum of residues to develop an asymptotic expansion for this unknown function?

In the course of solving a particular physical problem, I have derived a relationship between two unknown functions: $$ f(s) = \frac{s \sinh{\frac{\pi s}{2}}}{2 \pi i \beta} \int_{-c- i ...
0
votes
0answers
16 views

A question on Residues and a family of functions

I have the following: $$\alpha(t)=\int_{\left|z\right|=1}z^{k}\frac{f^{\prime}_{t}(z)}{f_{t}(z)}\,dz$$ where $f_{t}(z)$ is a family of entire functions depending on $t \in \Delta$ and $k \geq 0$. By ...