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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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 ...
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25 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 ...
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55 views
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37 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 ...
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18 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 ...
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integrate $\Gamma(z)\zeta(z)$

I know that the poles of $\Gamma(z)$ are all negative integers with residue $(−1)^n\over n!$, and $\zeta(z)$ has a pole at $z = 1$ wich residue is one. How can I integrate the following integral ...
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2answers
42 views

Troubles working with Residue Theorem

I try 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 try working with the Residum Theorem. First I compute the singularities ...
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1answer
28 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 ...
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36 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 ...
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41 views

The Coin-Exchange Problem (Application of the Residue Theorem) [on hold]

These day, I have met a problem about application of the Residue Theorem, see section 10.4 of enter link description here.Could anybody help me solve it? (The Coin-Exchange Problem) Suppose $a$ and ...
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71 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 ...
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22 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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25 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
51 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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26 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 ...
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1answer
41 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 ...
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25 views

Can anyone prove this using residue theorem? [closed]

Can anyone prove this using residue theorem? $$\sum\limits_{k=0}^\infty \frac{(-1)^k}{(2k+1)^3}=1-\frac{1}{27}+\frac{1}{125}-\dots=\frac{\pi^3}{32}$$
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1answer
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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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40 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 ...
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1answer
65 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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56 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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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 ...
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25 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 + ...
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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 ...
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18 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 ...
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1answer
31 views

On the determination of residue

I need your help on the following. (1)First we are to find the residue of $\frac{x^s}{s}$ at $s=0$. Since $s=0$ is the pole of order 1, so we get Res$(\frac{x^s}{s},s=0)=\frac{1}{2\pi ...
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29 views

Residue class ring $\mathbb{Z}[x]$/I and $\mathbb{Z}[x]$/J

$I = \left\lbrace \sum_{i=1}^{n} a_ix^i : n \in \mathbb{N}, a_1, ..., a_n \in \mathbb{Z} \right\rbrace$ beeing an ideal of $\mathbb{Z}[x]$ with polynomials without a constant term and $J = ...
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1answer
14 views

Question about how to calculate a certain residue

I want to calculate the residue at $z=i$ for $f(z)=\frac{1}{(z^2+1)(z^2+9)(z^4+4)}$. I've calculated the residue correctly by using the formula $Res(f;i)=\lim_{z\to i}{(z-i)f(z)}$. I also know that ...
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57 views

Prove that the integral of $x\cos(x)/(x-2)(x-1)$ from negative to positive infinity is $\pi(\sin1-2\sin2)$. Use an indented contour

To do this I used the Residue Thm but the main issue here is that I cannot get the sine term to appear. Perhaps I'm ignoring something here. We know that the singularity is $x=1,2$ so we should just ...
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3answers
69 views

Prove that the integral of $\sin^2(x)/(5+3\cos(x))$ from $0$ to $2\pi$ is $2\pi/9$

I'm not really unsure of how to approach this problem. I was thinking of reparametrizing the sin and the cos to its exponential form but I realize that it becomes even messier and leads sort of ...
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1answer
27 views

Evaluate the integral of $e^{x}/(x+1)^4$ on $\rho$, which denotes the entire imaginary axis

I'm not entirely sure if my intuition is correct but the singularity for this equation is -1 but -1 does not exist on the imaginary axis, so does this integral equal 0? If not, what am I missing and ...
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37 views

Prove that the integral of $sin(z)/(z^2+4z+5)$ from negative to positive infinity is $-\pi sin(2)/e$

I think I've made the problem a lot nastier than it supposed to look. Here's what I have so far. First notice that $(z^2+4z+5)$ is equivalent to $(z^2+4z+4)+1$ so our singularities are -2-i and ...
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2answers
70 views

Prove that $\int_0^{\infty} \frac{x^2}{x^4+5x^2+4}dx = \frac{\pi}{6}$

Prove that $\int_0^{\infty} \frac{x^2}{x^4+5x^2+4}dx = \frac{\pi}{6}$ Obviously you would use Residue Theorem to tackle this problem. The correct answer to this is $\frac{\pi}{6}$ however I'm ...
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2answers
44 views

How to calculate $\oint_{\left | z \right |=3}^{ } \frac{dz}{\left ( z^{2}-4 \right )^{200}\left ( z-15 \right )^{2}}$

How to calculate $$\oint_{\left | z \right |=3}^{ } \frac{dz}{\left ( z^{2}-4 \right )^{200} \left ( z-15 \right )^{2}}$$ using Cauchy Integral theorem, formula or Residue theorem. Edit: $z$ is ...
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52 views

Why does the Residue Theorem still hold, when I let my contour get infinitely large?

The theorem (as I know it) only allows for a finite set of isolated singularities. I integrated, along a square box, a function that has simple poles at all the non-zero integers -- and a triple pole ...
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1answer
57 views

Help with this indefinite integral using residues?

Question: How to evaluate this integral using residues$$\int_{0}^{\infty} \frac{x \sin x}{1 + x^2} dx$$ I integrate over the entire real axis and dividing it by 1/2 since the integrand is even, ...
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1answer
29 views

Finding pole order and calculating residue

Part of the the problem I'm trying to do involves finding the residue of $\frac{1}{z-\tan z} - \frac{1}{z}$ at z=0 I am not sure of the order of the pole Here's what I did to find the order of the ...
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1answer
34 views

Residue theorem, double pole, sinh.

how can I use the residue theorem to calculate $$\int_{-\infty}^\infty dx\, \frac{e^{-i x}}{(\sinh x)^2}$$ Im confused about how to tackle the double pole at $x=in\pi$. Thanks!
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1answer
57 views

residue of a contour integral with a branch point on the boundary

I am considering a problem where I would like to find the contour integral given by \begin{align} \oint_C f(z) dz \end{align} where $f = u+iv$. $C$ is the wedge shaped contour where $0 \leq r \leq ...
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42 views

Find the residue of the function $g(z)=f(z^2)$ at a given point.

Let $f(z)$ be analytic in $0<|z|<R$. Find the residue of the function $g(z)=f(z^2)$ at $z_0=0$. I am looking for a solution to this problem. My thoughts: I know in order to find the residue ...
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1answer
32 views

Identify singularities and classify them. Find the residue of the function at a given point.

Identify singularities of the function $f(z)=\frac{1}{\cos{z^2}}$ and classify them. Find the residue of the function that the point $z_0=\sqrt{\frac{\pi}{2}i}$ . I am hoping to find a clear ...
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Complex Integral

I am stuck computing the following complex integral $$\int_{|z| = 1}\frac{z^2}{4e^z -z}dz$$ I do not even know if the integrating function has a pole and then using residue calculus. Using the ...
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Evaluating the inverse Laplace transform of $1/(s^2-\sum_{n=1}^\infty{n!s^{3-n}x^n})$

I want to evaluate at $t=1$ the inverse Laplace Transform $\mathcal{L}^{-1}\{F(s)\}\vert_{t=1}$ of $$ F(s) = \frac{1}{s^2-\sum\limits_{n=1}^\infty{n!s^{3-n}x^n}} $$ and find out the $x^n$ ...
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3answers
69 views

Evaluate the improper integral.

Evaluate the integral below. $ \int^{+\infty}_{-\infty} \frac{x^2}{{(x^2-8x+20)}^2} \, dx $ I feel that I know how to do this problem, but I'm getting caught up in all the calculations. I've been ...
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1answer
40 views

Gaussian Integral using contour integration with a parallelogram contour

I'm having trouble figuring out how to use contour integration to compute the Gaussian integral. The contour I'm using is a parallelogram with function, $f(z) = \Large \frac{ e^{i \pi z^2}}{sin(\pi ...
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2answers
148 views

Dog Bone Contour Integral

Would someone please help me understand how to integrate $$ \ \int_0^1 (x^2-1)^{-1/2}dx\, ? $$ This is a homework problem from Marsden Basic Complex Analysis. The text book suggested using a "dog ...
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2answers
93 views

Sum of series with binomial

How to calculate $$\sum_{n=0}^{\infty}\binom{2n}{n}\frac{2n}{2^{2n}(2n-1)}$$ ? I tried to use residues, generating function, combinatorics formulas, but unsuccessfully.
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Inverse $z$ transform - contour integration

Here is my task: Find inverse $z$ transform of $$X(z)=\frac{1}{2-3z}$$if $$|z|>\frac{2}{3}$$ using definition formula. I found that $$x(n)=\dfrac{1}{3}\left (\dfrac{2}{3}\right ...
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1answer
50 views

Calculating the residue of a function

Let $f(z) = \frac{1+z}{1-\cos(z)}$ I wish to calculate the residue of $f$ at $0$, $2\pi$ and $-2\pi$. I believe this can be done by the following since $f$ has simple poles at these points $Res(f, ...
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3answers
81 views

Compute the integral $\int_{0}^{\infty} \frac{(1 + x + x^2)}{(1+x^4)} dx $ with a residue on suitable contour.

I believe that I could try to compute the same integral with limits from $-\infty$ to $\infty$ using residue on a half circle and then let the radius tend off to infinity, and once I have that value I ...