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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1answer
48 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
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1answer
20 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
23 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 ...
1
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1answer
47 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
-1
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1answer
21 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
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1answer
39 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 ...
-1
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0answers
24 views

Can anyone prove this using residue theorem? [on hold]

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
21 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 ...
1
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1answer
35 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
63 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
50 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
21 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
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0answers
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 + ...
1
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1answer
64 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
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1answer
17 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 ...
1
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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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1answer
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 = ...
0
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1answer
13 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 ...
3
votes
1answer
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 ...
2
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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 ...
0
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1answer
26 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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1answer
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 ...
2
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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 ...
4
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1answer
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 ...
2
votes
1answer
56 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, ...
1
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1answer
28 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 ...
1
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1answer
32 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!
3
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1answer
52 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 ...
1
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1answer
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 ...
2
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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 ...
1
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1answer
29 views

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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0answers
29 views

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 $x^n$ ...
2
votes
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 ...
0
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1answer
39 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 ...
9
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2answers
137 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 ...
3
votes
2answers
92 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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0answers
23 views

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
49 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, ...
2
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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 ...
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2answers
53 views

Simple Question About Residues/Poles/Zeros/Singularities

I'm having a little bit of trouble with residues. If we have the $f(z)=\left(\frac{\cos(z)-1}{z}\right)^2$ at $z_0=0$, we have a zero of order 2 in the numerator and a zero of order 2 in the ...
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2answers
45 views

Compute the following integral, where $C$ is the circle $|z|=3$

Evaluate:$$\int_{C} (1 + z + z^2)(e^\frac{1}{z}+e^\frac{1}{z-1}+e^\frac{1}{z-2}) dz $$ where $ C$ is a circle $|z|=3$ and $z \ \epsilon \ \mathbb{C}$ The function that is being integrated has ...
3
votes
3answers
40 views

Evaluation of real trigonometric integrals using the Cauchy Residue Theorem

$I = \int^{2\pi}_0 \dfrac{d\theta}{2 - \cos \theta}$ This is straight from a book I'm reading, which suggests to convert $\cos\theta$ into $0.5(z+1/z)$ and then solve the integral on the unit circle. ...
2
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0answers
56 views

Computing an integral using residues

I am trying to find an integral: $$\int_{-\infty}^{+\infty}\frac{e^{-\sqrt{(x^2 + 1)}}}{(x^2 + 1)^2}\,\mathrm dx$$ I went about applying contour integral over a semicircle with diameter along $ x = ...
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vote
3answers
93 views

Finding $\int_0^\infty\frac{\sin^{2}x}{1+x^4}dx$

I am trying to evaluate $$\int_0^\infty\dfrac{\sin^{2}x}{1+x^4}dx$$ and I am stuck on how to start. I am thinking the first step would be to substitute $$\dfrac{(1-e^{2ix})+(1-e^{-2ix})}{4}$$ for ...
0
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0answers
42 views

Fourier transform of a tough composite function (sinc, sqrt, polynomial…)

Is it possible to compute the Fourier transform of $\mathrm{sinc}(\sqrt{1+x^4})$ in closed form? It appears the problem to be suited for contour integration, and I started to tackle the mere ...
1
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1answer
30 views

Finding $a_{-n}$ where $\cot (\pi z)=\sum_{n=-\infty } ^\infty a_nz^n$

The following is problem 5.11.2 of Berkeley Problems in Mathematics. Let $\cot (\pi z)=\sum_{-\infty} ^\infty a_nz^n$ be the Laurent expansion for $\cot (\pi z)$ on the annulus $1<\vert z \vert ...
0
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1answer
43 views

Integral of a two-valued function with two branch cuts

Is it possible to calculate in closed-form the integral $\int_{-\infty}^{+\infty}\mathrm{sinc}(\sqrt{1+x^4})\,dx$ (sinc being the cardinal sine, $\sin(x)/x$)? The function is everywhere defined (all ...
2
votes
1answer
52 views

Simple Residue calculation

$$\int_{\gamma(0;2)}\frac{e^{i\pi z/2}}{z^2-1} \, dz$$ Using the residue calculus i got $$-2\pi$$But the answer is $$=i$$ I must be wrong at this, but shouldn't the answer have $\pi$ at least since ...
0
votes
1answer
32 views

How to compute the following real integrals using the residue theorem?

How to compute the following real integrals using the residue theorem: $$\int_{-\infty}^{\infty} \frac{1}{(x^2+p^2)(x^2+q^2)} dx$$ $$\int_{0}^{2\pi} \frac{sin^2(\theta)}{5+4cos(\theta)} d\theta$$ ...