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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5
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0answers
71 views

Evaluate $\int_{-\infty}^{\infty}\frac{\sqrt{a+ix}}{a^2+x^2}\,dx$ using residue calculus

I'm asked to evaluate $$\int_{-\infty}^{\infty}\frac{\sqrt{a+ix}}{a^2+x^2}\,dx$$ $\mathbb{R}\ni a>0$, using residue calculus (where $\sqrt{\cdot}$ is the PV $\sqrt{}$). My approach is as follows: ...
1
vote
1answer
38 views

Evaluating residua and simplifying complex expressions.

My question is in two parts, so please forgive its long-winded nature. Lets say that I want to find the residua of the following complex function: $$f(w)=\frac{2w+1}{w(w^3-5)}$$ Let us, ...
2
votes
1answer
62 views

Find the poles/residues of $f(z)=\frac{\sin(z)}{z^4}$

I'm trying to calculate the residua of the following complex function but am encountering problems trying to determine its poles: $$f(z)=\frac{\sin(z)}{z^4}$$ Expanding the denominator shows that we ...
1
vote
1answer
68 views

Evaluation of Residua

Suppose that I have the following complex valued function, and want to evaluate its residua: $$h(z)=\frac {z^5}{(z-3)(z^4+2)}$$ For both parts of the denominator we will have simple poles. For our ...
0
votes
0answers
55 views

Integration using residue theorem

Can you find, using residue theorem, ($\epsilon >0$), the value of this integral ($I$)? \begin{equation} I=\lim_{\epsilon->0^{+}} \int_{-\infty}^{\infty} \frac{dw}{w+i\epsilon} \end{equation} ...
0
votes
1answer
35 views

Find the residue of $e^{1/z}\sin(z)$ at $z=0$

I am looking for the residues of $f(z)=e^{1/z}\sin(z)$ at its singular points. Found that only $z=0$ is an essential singularity, where $$a_{-1}=\sum\limits_{n=1}^{\infty} ...
1
vote
1answer
49 views

How to determine the Laurent expansion of $\tan{z}$ around $z=0$ that is convergent in $z=\pi$

I want to determine the Laurent expansion of $\tan{z}$ around $z=0$ that is convergent in $z=\pi$ (only the first couple of terms). Now I know that if $\sum_{n=-\infty}^{\infty}c_nz^n$ then ...
2
votes
2answers
44 views

Calculating residue $\int_C \frac{8-z}{z(4-z)}dz$

I want to calculate the following: $$\int_C \frac{8-z}{z(4-z)}dz$$ $C$ is a circle of radius $7$, centered at the origin,negative oriented. I want to do this via finding the residues at $z=0,4$. I ...
1
vote
1answer
31 views

Exponential function and residues

Show that singular point of function $f(z)=\frac{1-e^{2z}}{z^4}$ is a pole and find the order $m$ of that pole and the corresponding residue. Here my question begins with the singular point ...
2
votes
1answer
36 views

Singular points and residues

In each case write the principal part of the function at its isolated singular point and determine whether that point is a pole, a removable singular point, or an essential singular point. ...
3
votes
5answers
290 views

Evaluate the complex integral of function

Use the residue theorem to evaluate $\int_\gamma \frac{z^5}{1-z^3}dz$ where $\gamma$ is the circle $|z|=2$. I have that $z_0=1$ is a singularity point and taking $g(z)=z^5$ and $h(z)=1-z^3$ and ...
0
votes
1answer
51 views

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

I need to find residue of function $f(z) = \frac{z}{1-\cos(z)}$ at $z=2\pi k$, where $k\in \Bbb Z$. I know residue at $z=0$ from here. I got a hint that need to substitute $z=\hat z+2\pi k$, so $\hat ...
4
votes
3answers
114 views

Evaluating $\int_{-\infty}^{\infty}\frac{\cos x}{e^x + e^{-x}}$ using the Residue Theorem

I consider the complexification $$f(z)=\frac{e^{iz}}{e^z+e^{-z}}$$ Poles of $f$: $\text{Denominator}=e^{-z}(e^{2z}+1)=0\Rightarrow e^{2z}=-1=e^{i(\pi + 2\pi k)}\Rightarrow z=\frac{i\pi(1+ 2k)}{2}$, ...
0
votes
0answers
55 views

Supposing $h$ has a pole, order m, at $z_0$, show the existence of a neighbourhood of $z_0$ and a new complex differentiable function $g$.

Suppose that $h$ has a pole of order $m\in\mathbb{N}$ at the point $z_0$. Show that there exists a neighbourhood of $z_0$ and a complex differentiable function $g$, so that $g(z_0)=z_0$, where ...
2
votes
2answers
85 views

Calculate the residue of this function

Find the residue at $z=0$ of the function $f(z)=\frac{\cot z}{z^4}$ I know that $z_0=0$ is a pole of order $k=5$, and $$Res(f;z_0)=\frac{\phi(z_0)^{(k-1)}}{(k-1)!}$$ but I cannot get the right ...
6
votes
1answer
73 views

Evaluating $\int_0 ^{\infty}\frac{dx}{x^{1/3}(1+x)}$ using Complex Analysis

I am trying to use the residue theorem to evaluate $$I=\int_0 ^{\infty}\frac{dx}{x^{1/3}(1+x)}$$ I'll explain my difficulty in finding a contour, then I explain my difficulty in finding a new contour ...
1
vote
3answers
59 views

Complex Integration, residues

Evaluate the following integrals by the method of residues i)$\int_0^{\infty}\frac{\cos x}{x^2+a^2}dx$, a real ii)$\int_0^\infty \frac{x^\frac{1}{3}}{1+x^2}dx$ I'm a little lost to ...
1
vote
1answer
53 views

ML-inequality: How to show that $e^{i2x} = e^{i2z}$ when evaluating $\int_{-\infty}^\infty \frac {\cos^2 (x)}{x^2 + 1} dx$

I am to solve the following integral: $$\int_{-\infty}^\infty \frac {\cos^2 (x)}{x^2 + 1} dx$$ We use contour integration in combination with residue calculus, so for $R > 1$ ($R$ is the radius ...
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3answers
48 views

Higher order poles, how high?

Clarification: I claim, for function $g(z)$, analytic and nonzero at $z=0$, if I have the function $f(z)=g(z)/z^n$ there is no use in trying to find poles of order smaller than n. And I would be ...
0
votes
2answers
30 views

Residues theorem, calculate the integrates

Find the residue of $f(z)=\frac{(z-1)^3}{z(z+2)^3}$ at $z=\infty$ In my refference they say that $Res(f;\infty)=-Res(\frac{1}{z^2}F(z);0)$ where $F(z)=f(\frac{1}{z})$ $$-lim_{z\rightarrow ...
2
votes
1answer
57 views

Residues theorem, and integrate

Evaluate $\int_\gamma\frac{z}{z^2+2z+5}dz$ where $\gamma$ is the unit circle I did but I don't know if it's right $$z^2+2z+5=0\Leftrightarrow z=-1\pm 2i$$ this I have that $z_1=-1+2i$ and ...
1
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0answers
28 views

Proof of Residue Limit Formula

We have the following theorem in Stein's Complex Analysis, p. 76: If $f$ has a pole of order $n$ at $z_0$, then: $$ res_{z_0}f = \text{lim}_{z\rightarrow z_0}{1 \over (n-1)!}\bigg({d \over ...
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1answer
34 views

Residues and poles, proof with poles

Proof that statements i)If $f_1$ and $f_2$ have residues $r_1$ and $r_2$, show that the residue of $f_1+f_2$ at $z_0$ is $r_1+r_2$. ii)If $f_1$ and $f_2$ have simple poles at $z_0$ show ...
3
votes
2answers
55 views

Complex analysis, find the residue

Find the residue of $f(z)=\frac{1}{z^2\sin z}$ at $z_0=0$ What I tried Let $g(z)=1$ and $h(z)=z^2\sin z$, both are analytics but they have zeros of different orders then $f(z)$ don't have ...
2
votes
1answer
45 views

Complex analysis, residues of function

If $f(z)$ has residue $b_1$ at $z=z_0$, show by example that $[f(z)]^2$ need not to have residue $b_1^2$ at $z=z_0$ What I tried Suppose that $f$ is analytics in the neighborhood of $z_0$ and ...
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vote
3answers
65 views

Calculate the integral with residue theorem

Calculate with residue theorem: $\int_0^{\pi/2} \frac{1}{a+\sin^2(z)} dz$ I tried to use a contour as follow (without the blue circle) : http://i.stack.imgur.com/P48XL.png but it didn't work well. ...
3
votes
2answers
146 views

Residues and poles proof

Let the degree of the polynomials $P(z)=a_0+a_1z+a_2z^2+\cdots+a_nz^n$ $a_n\neq0$ and $Q(z)=b_0+b_1z+b_2z^2+\cdots+b_mz^m$ $b_m\neq 0$ be such that $m\geq n+2$. Show that if all the zeros of ...
1
vote
2answers
41 views

Complex analysis, residues and integrate

Let $C$ denote the circle $|z|=1$ oriented counterclockwise. Show that i)$\int_Cz^ne^{\frac{1}{z}}dz=\frac{2\pi i}{(n+1)!}$ for $n=0,1,2$ ii)$\int_C e^{z+\frac{1}{z}}dz=2\pi ...
1
vote
1answer
33 views

Complex integrate and residues

Evaluate the integral of that $f(z)=\frac{z+1}{z^2-2z}$ around the circle $|z|=3$ oriented counterclockwise First I found that singularity points are $z=0,z=2$ ...
0
votes
1answer
26 views

Complex analysis, residues

Find the residue at $z=0$ of $f(z)=\dfrac{\sinh z}{z^4(1-z^2)}$. I did \begin{align} \frac{\sinh z}{z^4(1-z^2)} & =\frac{1}{z^4}\left[\left(\sum_{n=0}^\infty ...
1
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1answer
41 views

Are there different ways to find (the) residue(s) for a function with one simple pole vs. a function with several simple poles?

Regarding evaluation of residuals for functions with simple poles. Let's say $m$ represents the order of the pole, then in order to find the residual at each pole/the pole (if only one pole) we have ...
0
votes
2answers
57 views

ML-inequality: Why does this hold $|e^{-3y+3ix}| = e^{-3y}$ during numerator estimation of $f(z) = \frac{e^{3iz}}{z^2 + 1}$ [duplicate]

Given the following related to an ML-inequality for $R > 1$: Estimation of the numerator from the function $f(z)$ is supposed to develop as follows: I'm wondering why and how exactly the ...
1
vote
1answer
23 views

Compute an integral with Cauchy's residue theorem

Good evening everyone, I want to know if my result is correct. So: I have to compute the following integral: $$\int_\gamma \frac{ze^{\pi z}}{z^2+1}dz,$$ while ...
2
votes
2answers
55 views

Troubles working with Residue Theorem

I am trying to compute the integral on the positively oriented circle $$\int_{\partial D(1,2)} \frac{z dz}{(z+2)(z^2 -2z + 2)}.$$ So I apply the Residue Theorem. First I compute the singularities ...
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votes
1answer
39 views

Calculating residues of multiple poles?

How would I calculate $$\mathrm{Res}\left(\frac{\pi}{\sin(\pi z)(2z+1)^3}\right)?$$ I understand it has singularities at $z=n$ and $z=-1/2$, I'm interested in the residue when $z=-1/2$. I know that ...
1
vote
1answer
44 views

Best way to calculate residues

Basically, what is the best method to calculate residues, specifically, something like this: \begin{equation*} f(z)=\frac{1+z}{1-\cos(z)}. \end{equation*} For simple poles, I can just use L'Hopital ...
1
vote
1answer
82 views

How find the poles/residues of $\int_{-\infty}^\infty \frac{x^2 \, dx}{1 + x^4}$

I'm trying to find the poles/residues of this integral: $$\int_{-\infty}^\infty \frac{x^2 \, dx}{1 + x^4}$$ I've been given this attempt for a solution, but I don't really understand the procedure ...
0
votes
1answer
26 views

Residue Calculus - Showing that the quotient of polynomials have integral $0$ in a simple closed contour in a special case.

I'm having difficulty understanding the solution to the following problem. In the solution below, I can't understand why since $b_m\neq 0$, the quotient of these polynomials is represented by a ...
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0answers
28 views

Counting poles that are shared between $f$ and $g$

Suppose I have a meromorphic function $f(z)$ with poles at $f_i$ and $\mathcal{Res}(f,f_i)=1$, and $g(z)$ with poles at $g_i$ and $\mathcal{Res}(g,g_i)=1$. I would like to construct a function ...
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1answer
52 views

How can I expand this

How can I expand $\dfrac{\pi \csc(z\pi)}{(2z+1)^3}$? so then I can find the residue ? thanks
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votes
1answer
28 views

Singularities and Residue

For part (a) the singularity is 1/root2 + i/root2 ? And it is a pole of order 1? I am having trouble calculating the residue So far I have: residue = limit (as z tends to 1/root2 + i/root2) of ...
1
vote
1answer
52 views

Residue of essential singularity

$$f(z)=\sin(z)e^{1/z}$$Find the residue of $f$ at $0$. I think there is an essential singularity at $z=0$ ? How do I compute the residue of this... I know how to compute the residue of poles but not ...
0
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1answer
28 views

Application of Residue theorem

Let f(z,w) be holomorphic in $\mathbb{C}^{n}$ and not identically zero on the w-axis. Let {$b_{j}$} be the set of singularities of f(z,w) in some disk of radius $|w| < r$. Why does the residue ...
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vote
1answer
59 views

Function poles and divergence of series

Yesterday I tried to calculate the residues of a function the way below, but soon I realized it won't work. Now I have a question about the poles of a function, and a series representing it. $$z\in ...
1
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1answer
68 views

Let $f$ be an odd meromorphic function , what can I deduce about $res (f,0)$

Let $f$ be an odd meromorphic function. What can I deduce about $res(f,0)$?
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0answers
72 views

how can I show that $\cot\pi z$ and $\csc \pi z$ have simple poles for every integer $n$? so then I can calculate residues at those poles?

how can I show that $\cot\pi$z and $\csc\pi$z have simple poles for every integer $n$? so then I can calculate residues at those poles?
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0answers
30 views

branch point in real and imaginary part of a complex function

Is it possible to have branch point in both the real and imaginary part of a complex function f(z)? An example might be \begin{align} f(z) = u(\theta) + iv(\theta) \end{align} where $u$ and $v$ are ...
2
votes
0answers
27 views

Finding the number of zeros in the right half plane of $\mathbb{C}$. [duplicate]

I am attempting to learn an exercise in the chapter on Rouche's Theorem in Ahlfors Complex Analysis. It comes from page 154, number 3: Find the number of zeros of $$f(z) = z^4 + 8z^3 + 3z^2 + 8z + ...
1
vote
1answer
73 views

Are functions with singularities, but no poles, manipulable by Residue Calclus?

Consider these two functions $$\int_0^{\infty} {{1} \over {(x^2+a)^{3/ 2}}} \ dx$$ And $$\int_0^{\infty} {{1} \over {x^2+\cos(x)}} \ dx$$ They have singularities, however, wolfram alpha says that ...
1
vote
1answer
21 views

Residue integral solution differs from wolframalpha

Im trying to solve the following integral using the residual theorem: $$ \int\limits_0^{2\pi} \frac{\mathrm{d}\phi}{3 + 2\cos \phi} $$ Using $2\cos \phi = z - \overline z$ and ...