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Questions tagged [residue-calculus]

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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3answers
58 views

Evaluate $\int_0^\infty \frac{x^2}{x^4+16}\,dx$

It's a very simple problem. I put it into wolfram alpha and it gave me a hugely complicated answer. This makes me think that I am supposed to be doing it differently. I believe I am supposed to use ...
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0answers
17 views

Residue and Laurent series for an integral involving exponential and square root.

In another post, I am trying to compute this integral: $$\int_{-1}^1 \frac{e^{bx}}{(x-a)^2}\sqrt{1-x^2}dx\quad b>0,\;a>1$$ I try to tackle it using the residue theorem. When I write it with ...
2
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1answer
48 views

Is there any special function corresponding to $\int_{-1}^1 \frac{1-e^{b(x-a)}}{(x-a)^2}\sqrt{1-x^2}dx$?

I try to get an expression for this difficult integral: $$\int_{-1}^1 \frac{1-e^{b(x-a)}}{(x-a)^2}\sqrt{1-x^2}dx\quad b>0,\;a>1$$ It could also be written in terms of trigonometric functions ...
2
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4answers
26 views

Residue of $\cot(z)/(z-\frac{\pi}{2})^2$ at $\frac{\pi}{2}$

I want to know what type of singularity has $f(z)=\cot(z)/(z-\frac{\pi}{2})^2$ at $\frac{\pi}{2}$ and what is the residue of $f(z)$ at $\frac{\pi}{2}$. I thought that $f$ has a pole of order $2$ at $\...
2
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2answers
30 views

Residue of $\cot(z)/(z-\frac{\pi}{2})$ at $\frac{\pi}{2}$ [closed]

I don't know what type of singularity has $f(z)=\cot(z)/(z-\frac{\pi}{2})$ at $\frac{\pi}{2}$ and how can I calculate the residue of $f(z)$ at $\frac{\pi}{2}$. Can you help me? Thanks in advance.
2
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0answers
25 views

De forming the contour and showing that the contributions of contours of infinitely small lengths go to zero

I am considering an integral around the path $ \Gamma = C_1 \cup C_{\varepsilon_{1}} \cup C_2 \cup C_{\varepsilon_{2}}$ of a function $f(z)$ that has a pole in every cross in the images below. In ...
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0answers
45 views

Evaluating $\sum_{n=-{\infty}}^{\infty} \frac{1}{n^4+1}$ [duplicate]

I recently learnt that $$\sum_{n=-{\infty}}^{\infty} \frac{1}{n^2+a^2} = \frac{\pi}{a}\coth(\pi a)$$ My question is that is there a way to evaluate the closed form for the series $$\sum_{n=-{\infty}}^...
2
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1answer
43 views

show that $\sum_{k=1}^n \dfrac{1}{\prod_{j\neq k}(a_k-a_j)}=0$

Let $f(z)$ be a complex polynomial of degree at least $2$ and $R$ be a positive number such that $f(z) \neq 0$ for all $|z| \geq R$. Show that $\int_{|z|=R} \frac{dz}{f(z)}=0$ and deduce that $\...
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0answers
48 views

Inverse Laplace transform of $\frac{\sqrt{s}}{\sinh(\sqrt{s})}$

I was just wondering if it is possible to determine the Inverse Laplace Transforms of the function $$F(s) = \frac{\sqrt{s}}{\sinh(\sqrt{s} )}$$ by utilising the Residue theorem. I am fine in finding ...
1
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2answers
34 views

Residue $f(z)=\frac{e^z}{e^z-1}$

Find the residue at each pole of the function $$f(z)=\dfrac{e^z}{e^z-1}$$ I wonder that $z=0$ is a pole of $f(z)$ ? And is it a simple pole? Can I use the formula when $z_0$ is a simple pole of $h$ ...
4
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2answers
71 views

Use of residue theorem to evaluate a real integral

I have seen this integral on youtube, I tried to solve it for 5 hours but I expect I need to use residue theorem which I don't really know $$\int_{0}^1\frac{\ln(x+1)}{x^2+1}\,dx$$, numerical ...
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2answers
18 views

If $ \gamma$ is a closed path and $\gamma\subseteq B_R(0) $ then $\mathbb{C}\setminus B_R(0)\subseteq \operatorname{Ext}_\gamma$

I want to prove that if $ \gamma$ is a closed path and $\gamma\subseteq B_R(0) $ then $\mathbb{C}\setminus B_R(0)\subseteq \operatorname{Ext}_\gamma$ where $ \operatorname{Ext}_\gamma=\{a\not \in \...
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0answers
34 views

Evaluate $\int_{|z|=3} \frac{cos(\pi z)}{(z-2)^2(z+5)(z+1)} \ dz$

I am trying to solve $$I=\int_{|z|=3} \frac{cos(\pi z)}{(z-2)^2(z+5)(z+1)} \ dz.$$ My attempt: Residue Theorem: Let $$f(z)=\frac{cos(\pi z)}{(z-2)^2(z+5)(z+1)}.$$ Now, $$\text{Res}(f,2)=\lim_{z\to ...
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3answers
34 views

How to find residue of the following function.

Wolfram Alpha tells me that the residue of $$ \frac {1}{e^z-1} $$ at the point z= $ 2i\pi $ is 1. Now i understand how the formula for residue works for simple poles , i just don't understand how ...
1
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1answer
46 views

Poles of integral $\oint_D\left(\frac{\cos(\pi z/2)}{(z^2+i)(\sin^{2}(\pi z)}\right)\,dz$

Which are the poles of this integral and what order do they have? $$\oint_D\left(\frac{\cos(\pi z/2)}{(z^2+i)(\sin^{2}(\pi z)}\right)\,dz$$ where $D$ is $|z|<3/2$. In my opinion the should be $z=...
1
vote
1answer
31 views

Solving $\int_{\Gamma}\frac{\cos(z)}{z^2(z^2+1)} \ dz$ by the Residue Theorem

I am tring to find $$\int_{\Gamma}\frac{\cos(z)}{z^2(z^2+1)} \ dz,$$ where $\Gamma$ is the circle $|z|=2$. My attempt: Let $f(z)=\frac{\cos(z)}{z^2(z^2+1)}.$ Now, $f$ has singularities at $z=0,\pm i$...
1
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1answer
25 views

Residue of $z_0=1$ for $f(z)=\frac{z^3+5}{z(z-1)^3}$

Consider $$f(z)=\frac{z^3+5}{z(z-1)^3}.$$ I am trying to find the residue of the pole of order $3$, $\ z_0=1$. I know from calculations that $$\text{Res}(f,1)=\frac{1}{2}\lim_{z\to 1}\frac{\partial^2}...
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0answers
42 views

Find the Residues of $f(z)=\frac{z(z-\pi)^2}{\sin^2(z)}$

I am trying to find the residues of the function, $$f(z)=\frac{z(z-\pi)^2}{\sin^2(z)}.$$ My attempt: I have considered three singularities: $z_0=0,\pi,k\pi \ (k\in\mathbb{Z}, \ k\neq 0,1).$ For $...
2
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2answers
45 views

Finding $\text{Res}(f,0)$ where $f(z)=\frac{1}{z^2\sin(z)}$

I am trying to determine the residue of $z=0$ where $f(z)=\frac{1}{z^2\sin(z)}$. I have determined that $z=0$ is a pole of order $3$. Hence to compute the residue, I use $$\text{Res}(f,0)=\frac{1}{2}\...
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0answers
59 views

Compute $\oint_{S^1}\frac{z}{\sin^{3}(z/2)}dz$ without using Laurent series

Let $D=\{z\in\mathbb{C}:|z|<1 \}$. How to compute the integral : $$\int_{+\partial D}\frac{z}{\sin^{3}(z/2)}dz$$ without using Laurent series? My trouble is that I cannot find the poles and the ...
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0answers
16 views

How can we calculate the imaginary part of a fraction that has a term i0+ in the denominator (Sokhotski–Plemelj theorem)?

I have recently started dealing with thermal field theory for fermions and I am faced with a paper that, at some point, tries to calculate the imaginary part of a fraction that looks like: $$\frac{1}{...
2
votes
1answer
48 views

The Fourier transform of $f(x)=(x+iy)^{-k}$. for $y>0, k>2$

My actual goal is to prove the Lipschitz formula $$\sum_{n\in \mathbb{Z}}\frac{1}{(z+n)^k}=\frac{(-2\pi i)^k}{(k-1)!}\sum_{r=1}^\infty r^{k-1}e^{2\pi i r z}$$ with the help of the Poisson summation ...
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3answers
66 views

Finding the integral of $\int_{0}^{2 \pi} \sin^n(x)$

$$\int_{0}^{2 \pi} \sin^n(x) = \, ?$$ The key step is to consider the complex integration $\int(z-\frac{1}{z})^n\frac{dz}{z}$ around the unit disk. Notice that $$(z-\frac{1}{z})^n\frac{1}{z} = \frac{...
1
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0answers
68 views

Integral $\int\frac{\sin(2zj)}{z(z^{2}+\frac{\pi^{2}}{4})^{2}}dz = 0$ (residues)

Hi guys I'm solving this integral : $$\int_{+\partial D}\frac{\sin(2zj)}{z(z^{2}+\frac{\pi^{2}}{4})^{2}}dz\,,$$ where $D=\{z\in\mathbb{C}:|z|<\pi \}$ I have found that for $z=0$ the residue is $0$ ...
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2answers
32 views

How can I find the residue of this removable singularity?

We have the following function : $$f(z)=\frac{z^2}{1-cosz}$$ where $z_0=0$ is a removable singularity since the limit as z goes to 0 is 2. In such cases, in order to find the residue I proceed by ...
4
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1answer
109 views

Evaluating complex integral $\int_{0}^{\pi} \frac {x \sin x}{1+a^2-2a(\cos x)} $ via different contour

I got an complex integral $\int_{0}^{\pi} \frac {x \sin > x}{1+a^2-2a(\cos x)} $ for $a \ge 1$ and my given contour is a rectangle such that $|Re(z)|\le \pi$ and $0 \le |Im(z)| \le h \to \infty$....
0
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1answer
35 views

Find residue of $\cos(\frac{z}{1-z})$ at z=1.

Is the residue of $\cos(\frac{z}{1-z})$ at z=1 : sin(1)? i.e $\frac{1}{2\pi i}\int_c\cos(\frac{z}{1-z})dz=Res(\frac{1}{z^2}f(\frac{1}{z}),0)$=$\frac{1}{2\pi i}\int_c\frac{\cos(\frac{1}{z-1})}{z^2}dz$ ...
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0answers
24 views

Residue of f(z) with pole of 2nd order [duplicate]

Generally finding a Residue of a function with $n^{th}$ order pole is done with \begin{equation}\label{eq:1} Res(f(z),z_0) = \dfrac{1}{(n-1)!} \lim_{z \to z_0} \dfrac{d^{n-1}}{dz^{n-1}} (z-z_0)^n f(z)...
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1answer
45 views

Integral of $e^\left(1/z^2\right)$ around $|z|=1$ in the complex plane [closed]

$e^\left(1/z^2\right)$ has an essential singularity at $0.$ Don't know how to do this integral.
1
vote
2answers
86 views

Contour Integral of irrational polynomial from -1 to 1

I've been stuck at htis contour integral problem for a few hours now, and seem to be hitting brick walls. $$ \int_{-1}^1 \frac{\sqrt{1-x^2}}{1+x^4}dx\,, $$ I tried a trig substitution $x=\cos{\theta}...
1
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2answers
100 views

Integral $\int_0^\infty dp \, \frac{p^5 \sin(p x) e^{-b p^2}}{p^4 + a^2}$: any clever ideas?

I am trying to solve the following integral, with $a>0,$ $b>0$: $I \equiv \int_0^\infty dp \, \frac{p^5 \sin(p x) e^{-b p^2}}{p^4 + a^2} $ By expanding the $\sin$, I get $I = \sum_{n=1}^\...
1
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0answers
40 views

Integral evaluation via the Residue Theorem (example)

$$\int_{\gamma}\frac{e^z-1}{\sin^2z}\,dz, \quad \,\gamma(t)=4e^{it},\,t\in[0,2\pi].$$ Let $$f(z)=\frac{e^z-1}{\sin^2z},\quad z\neq n\pi,\,n\in\mathbb{Z}$$ The curve contains only the singular ...
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2answers
20 views

pole and essential singularity in the same point

In this case in $0%$ i have an essential singularity by $\sin(\frac{1}{s})$ but i have a ""pole"" too ( the denominator of the fractions $\frac{(...)\sin(...)}{**S**}$) , so in this case is this a ...
0
votes
1answer
42 views

Why do we consider an infinite semicircular contour for the integral $\int_{-\infty}^{\infty}f(x) dx$

I have noticed that $\begin{align*}\int_{-\infty}^{\infty}f(x) dx \end{align*}$ can be solved using residue theorem. My question is that while the actual integral doesn't have a closed path, it's a ...
0
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1answer
46 views

Why can't I just find the residue of the function?

I was solving the contour integral $$\oint \frac{z\sin z}{z^{2}+4} \ dz$$ in the upper half of the complex plane using the residue theorem and I couldn't figure out why I needed to convert it to $$Im\...
0
votes
1answer
90 views

Black hole integral

What I call as ‘black hole’, has a formal name of ‘limit point of singularities’. Suppose $k$ is a black hole of the function $f$ (e.g. $0$ is the black hole of $\csc \frac1z$), then how to evaluate ...
1
vote
1answer
45 views

Calculate integral with $z^a$ using residue theorem

I am trying to solve the following: Find $\int_0^\infty \frac{x^a}{x^2+3x+2}dx$ for $0<a<1$ by using the residue theorem. I thought to let $f(z)=\frac{z^a}{z^2+3z+2}$ and by taking the ...
0
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1answer
29 views

Integration of digamma function

I was trying to perform the contour integral of the digamma function $\oint\limits_C \psi(z)\,dz$ on the neighborhood (a small circle $-k+re^{it}$, $k \in \mathbb{Z}$ ) of $k$, before actually ...
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votes
1answer
76 views

How important is the assumption $\gamma$ is positively oriented? (Residues, Cauchy's Thm from Cauchy's Integral Formula)

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 4.32, Cor 8.27 Question 1. Should the following 2 statements in the textbook have an ...
1
vote
1answer
121 views

Are these the Big and Little Picard Theorems?

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 9.3, 9.4 These seem to be the Big and Little Picard Theorems or at least related to them. ...
1
vote
2answers
125 views

Proof of Casorati-Weierstrass [closed]

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Ch9.2 I have questions on the proof of Casorati-Weierstrass Theorem (Thm 9.7) - If $z_0$ is ...
0
votes
0answers
31 views

Inverse Laplace, residue, simple or essential pole from bivaluated hyperbolic trigonometric function?

I would like to compute the following function: $$f(t)=\mathcal{L}^{-1}\Big[\frac{1}{s(e^{a+\text{arcosh}(s+\cosh a)}-1)}\Big](t)$$ However, it seems that there is no other pole than the pole of ...
-2
votes
1answer
114 views

Must a positively oriented path be simple, closed and piecewise smooth? [closed]

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Ch1, Ch4 Question 1: Do all paths have an orientation? The following is a quote from ...
-1
votes
1answer
212 views

Prove $ \lim_{z \rightarrow z_o} f(z)$ is finite if $\lim_{z \rightarrow z_o} (z-z_0) f(z) = 0$

Context: A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 9.1, 9.3 (asked about here) Question From (here): How do we have that $$\...
-1
votes
1answer
78 views

Show that a holomorphic function $f$ has a pole iff we can find $g$ s.t. $f(z) = \frac{g(z)}{(z-z_0)^m}$

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Ch 9.2 Cor 9.6 of Prop 9.5(*) Suppose $f$ is holomorphic in $\{0<|z-z_0| < R\}$. ...
0
votes
1answer
157 views

$f$ zero (essential singularity) $\implies \frac 1 f$ pole (essential singularity)

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 9.1, 9.3 How do I do these? (I converted attempts to an answer.) (Exer 9.1) Prove $f$ ...
1
vote
2answers
78 views

Calculating $\int_{-\infty}^{\infty}\left( \frac{\cos{\left (x \right )}}{x^{4} + 1} \right)dx$ via the Residue Theorem?

In the text, "Function Theory of One Complex Variable" by Robert E. Greene and Steven G. Krantz. I'm inquiring if my proof of $(1)$ is valid ? $\text{Proposition} \, \, \, (1) $ $$\int_{-\...
3
votes
2answers
64 views

Residue at infinity calculating integrals

I have the following problem which I want to evaluate at infinity: $$\oint \dfrac{(z+2)}{(z^2+9)}dz$$ I approach this problem by saying that $z=\dfrac{1}{t}$ and $dz=\dfrac{-1}{t^2}dt$. And I plug ...
0
votes
0answers
26 views

Integration of definite integrals with residue theorem $\int_0^{2\pi}\frac {\cos(2\theta)} {5+4\cos(\theta)}\, d\theta.$ [duplicate]

$$\int_{0}^{2\pi} \dfrac{\cos2\theta}{5+4\cos\theta}d\theta$$ I am trying to take this integral. Solution manual of the book I'm using suggesting that I should take $\oint \dfrac{z^2}{5+2(z+z^{-1})}...
-1
votes
1answer
55 views

Is $\int_{C[-2i,r]} \frac{dz}{z^2+1} = 0 , 1 < r < 3$? I got $-\pi$.

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 4.33 It is given that $r \ne 1,3$ and that the answer is $0 \ \forall r$. I got $0 \ \...