For questions about integration methods that use results from complex analysis and their applications

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If $c_{n}$ coefficient of the expansion of $f$. Show that $\sum_{n=0}^{\infty}\left|c_{n}\right|^{2}r^{2n}$ is an expression determined by an integral

Let $U\subseteq \mathbb{C}$ be an open set, $z_{0}\in U$ and $R>0$ such that $\mathcal{B}_{R}(z_{0}) \subseteq U$. Let $f:U\rightarrow \mathbb{C}$ be a holomorphic function with Taylor's serie ...
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If $f(z)=\sum_{n=0}^{\infty}c_{n}(z-z_{0})^{n}$, then $c_{n}r^{n}=\frac{1}{2\pi}\int_{0}^{2\pi}f\left(z_{0}+re^{-2nt}\right)dt$

Let $U\subseteq \mathbb{C}$ be an open set, $z_{0}\in U$ and $R>0$ such that $\mathcal{B}_{R}(z_{0}) \subseteq U$. Let $f:U\rightarrow \mathbb{C}$ be a holomorphic function with Taylor's serie ...
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18 views

Complex integration by substitution

Integrate $ f(z) $ counterclockwise around the unit circle. $$ f(z) = 1/(4z-3) $$ My solution C(contour) : $ z(t) = \cos{t} + i\sin{t} = e^{it}, 0<t\leq 2\pi $ $$ \oint_C \frac{1}{4z-3} dz = ...
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1answer
15 views

Writing a function $f : [-\pi,\pi) \to \mathbb{R}$ as $\sum c_k e^{ikx}$ where $c_k$ is to be found

I have a function on $[-\pi, \pi)$ defined as: $$ f(x) = \begin{cases} -1 & \mbox{if} \;x \in [-\pi,0) \\ 1 & \mbox{if} \;x \in [0,\pi) \\ \end{cases} $$ And I have to write it in the form ...
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50 views

Complex integration with infinitely many poles on imaginary axis

I'm trying to integrate with a closed contour on the upper-half of the complex plane. $I = \displaystyle\int_{-\infty}^\infty \dfrac{z\,\mathrm{sech(z)}}{[(z-a)^2+b^2][(z+a)^2+b^2]} dz$ There are ...
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44 views

Integration over complex surface

How to evaluate the following integral? $G(\omega) = \frac{1}{\pi} \int_{|\lambda| \leq 1} d^2 \lambda \frac{1}{\omega - \lambda}$ I have tried to change to polar coordinates with $\omega = Re^{i ...
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47 views

Let $\gamma=\{z\in \Bbb C: \lvert z \rvert=2\}$ in anti-clockwise orientation. Then $I=\frac {1}{2\pi i} \int_{\gamma} z^7 \cos \frac 1{z^2} dz$=?

$$I=\frac {1}{2\pi i} \int_{\gamma} z^7 \cos \frac 1{z^2} dz=?$$ The function $\cos \frac 1{z^2}$ is neither analytic at $z=0$ and nor it has a pole at $z=0$. By Cauchy Integral Formula can I get ...
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32 views

Show that $g(z)=\frac{1}{n}\sum_{k=0}^{n-1} f \left(\xi^{k}\sqrt[n]{z}\right)$ is an entire function.

Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be an entire function and $\xi=e^{\frac{2\pi i}{n}}$ for some $n\in \mathbb{N}$. Suppose that $f(\xi z)=f(z)$ for all $z\in \mathbb{C}$ and consider the ...
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94 views

There is an entire function $g$ such that $f(z)=g\left(z^{n}\right)$.

Let $f$ be an entire function and $\xi=e^{\frac{2\pi i}{n}}$ for some $n\in \mathbb{N}$. Suppose that $f\left(\xi z\right)=f(z)$ for all $z\in \mathbb{C}$. Show that there is a entire function $g$ ...
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1answer
29 views

Show that entire function $f$ is a polynomial of degree at most $n$

Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be a entire function. Suppose that there are $M$, $r>0$ and $n\in \mathbb{N}$ such that $\left|f(z)\right|<M\left|z\right|^n$ for all $z \in \mathbb{C}$ ...
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1answer
30 views

Parametrize the given curve and compute the integral (complex numbers)

The integral I have to evaluate is $\int_Czdz$, where $C$ is the line from 0 to $1+i$, and then from $1+i$ to 2. My work: $z_1(t)=(1+i)t$ and $z_2(t)=(t+1)+i(1-t)=t(i-1)+(1+i)$, $t\in[0,1]$. ...
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1answer
41 views

Calculate $\int_{\left|z-1\right|=2}z^{n}\sin\left(z\right)dz$ for $n\in \mathbb{Z}$

Calculate $$\int_{\left|z-1\right|=2}z^{n}\sin\left(z\right)dz$$ for $n\in \mathbb{Z}$ My attempt: According to the following result which was presented at my course as Cauchy's integral formula for ...
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2answers
27 views

A function that satisfies $|f(z)-\bar z|<0.9$ is not analytic in the unit circle.

I've came accros this excersize: Suppose that $D=\{z:|z| \le 1\}\subset \mathbb C$ and $$f:D\rightarrow\mathbb C$$ suppose that for every $z\in D$ such that $|z|<1$ $$|f(z)-\bar z|<0.9$$ where ...
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3answers
36 views

Compute the integrals using the residue theorem

Compute the following integrals: $I:=\int_{|z|=2}\frac{1}{(z-3)(z^{13}-1)}dz$ $J:=\int_{|z|=10}\frac{z^3}{z^4-1}dz$ I do not know where to begin. I know I am supposed to use the substitution ...
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27 views

Solutions of this integral $\int_{\gamma}\frac{e^{z^2}-1}{z\sin(z)}dz=0$

I must integrate the function $f(z)=\frac{e^{z^2}-1}{z\sin(z)}$ long the circle $\gamma$ of radius $r=1/4 $ and center in the origin. I thing that $\int_{\gamma}\frac{e^{z^2}-1}{z\sin(z)}dz=0$, ...
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1answer
17 views

Integration with Jordan's Lemma

I'm trying to integrate the following: $$I = \int^\infty_0 \dfrac{\cos x}{x^2 + 1} dx$$ What I did was: $$I = \int^\infty_0 \dfrac{\cos x}{x^2 + 1} dx = \dfrac{1}{2}\int^\infty_{-\infty} ...
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2answers
22 views

Conjugate symmetry to prove inner product

We have to show that $$\langle p,q\rangle=\int_a^b \overline{p(t)}q(t)$$ is an inner Product. I (think) I know what to do: I have to prove linearity, conjugate symmetry and positive definiteness. I ...
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0answers
40 views

Integral of complex logarithm on a disk in the plane

Let $a$ be a complex number and $D$ the disk centered around $0$ and of radius $R$. I would like to compute the integral I=$\int_D \log(|z-a|)d^2z$. I am interested in particular in the case $R\gg ...
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27 views

$|\int\limits_a^b f(t)dt| \leq \int\limits_a^b |f(t)|dt$, where $f(t) \in \mathbb C$

Let $f: [a,b] \to \mathbb C, t \to f(t) = \text {Re } f(t) + i\text{ Im } f(t)$. Suppose $f$ is continuous. Let $\int\limits_a^b f(t)dt = \rho e^{i\theta}$, where $\rho \geq 0$ is the module. Then: ...
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1answer
57 views

Over an integral arising from Kepler's problem [also: generally useful integral]

I'm dealing with this integral in my spare time, since days and days, and it's really interesting. I'll provide to write what I tried until now, and I would really appreciate some help in ...
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12 views

numerical differentiation of holomorphic function that uses square region instead of circle

In usual numerical differentiation of holomorphic function, circle region in the complex plane is used. But I was wondering if I can use square region. Is there any link or paper that uses square ...
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1answer
36 views

Finding residue of double pole

I'm trying to find the residue of the pole $+ib$ of $$f(z) = \frac{e^{iaz}}{(z^2+b^2)^2}$$ (the pole $-ib$ lies outside of the contour). I'm trying to do this by $$Residue = ...
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72 views

How to calculate $\int_{-\infty}^{\infty}\frac{x^2}{\cosh(x)}\mathrm{d}x$ [duplicate]

I know the poles are $z=i\pi/2+i n\pi$ and therefor I got an rectangular contour for the integration which wasn't so useful. I also know with change of variables I can get to ...
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1answer
33 views

Integral along the boundary is zero, if function has a compact support?

I've just missed a point on complex analysis lectures. Namely, we did an integral representation formula using Green's formula: Suppose we have function $f$ continuously differentiable on open set ...
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3answers
99 views

How to show that $\int_0^{\infty} dx \frac{\log{x}}{1+x^2}$ is zero using complex analysis

I want to show this using contour integration, the appropriate contour is a keyhole I think.
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49 views

complex integral for Trigonometry integral solution

I have two integral that must solve with complex integrals I know how to solve it in normal way but my university professor told me to solve it in complex integral solution way. I know that it would ...
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1answer
13 views

Contour Integral Inequality and Normal Families

I am trying to prove a certain family of functions $\mathcal{F}$ is normal, and my proof got very stuck. I am trying to show that the family of analytic functions on $\mathbb{D}$, continuous on the ...
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1answer
41 views

Using Cauchy's integral formula to evaluate $\int_0^{2\pi} \frac{1}{a^2 \cdot {\cos}^2(t) + b^2 \cdot {\sin}^2(t)} dt$ [duplicate]

I have to solve this integral using Cauchy's integral formula. I tried to substitute it with several different attempts but without a solution. Can anyone help? $$\int_0^{2\pi} \frac{1}{a^2 \cdot ...
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Improper integral computation using complex analysis

$$pv\int_{-\infty}^{\infty} \frac{x}{(x^2+4)(x^2+2x+2)}dx$$ I get an answer of $-\frac{\pi}{5}$ but wolframalpha disagrees by a factor of $2$ ($-\frac{\pi}{10}$): ...
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1answer
38 views

Trigonometric contour integral

I cannot figure out what I'm doing wrong: $$\int_0^{2\pi} \frac{1}{a+b\sin\theta} d\theta\quad a>b>0$$ $$\int_{|z|=1} \frac{1}{a+\frac{b}{2i}(z-z^{-1})} \frac{dz}{iz}$$ $$\int_{|z|=1} ...
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1answer
51 views

Is this Complex Integration correct?

I want to integrate $\displaystyle \int_{-\infty}^\infty dx \, e^{iax}\frac{1-e^{-bx^2}}{x^2}$ for a>0. I am going to try and do this using the method of contour integration. I will choose a ...
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1answer
50 views

complex integral over $dz\,dz^*$

the aim is to calculate $$\int dz\,d\bar{z} e^{-z\bar{z}}$$ when interpreting $dz\,d\bar{z}$ term, I got confused: $$dz\,d\bar{z}=(dx+i\,dy)(dx-i\,dy)=(dx)^2-(dy)^2 $$ ...
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47 views

Laplace trasform

i am trying to do this exercise but i do not get it. The laplace trasform is: \begin{equation} T(f)(s)= \int_{0}^{\infty} f(t)e^{-st} dt \end{equation} The exercise is: a) If $f$ is the ...
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2answers
38 views

The prove of the Laurent series formula

I am trying to understand why at Laurent series we get that . So, I need to understand why this formula is correct: . Could someone help?
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Why is this integral finite?

I am looking at this integral $\displaystyle\int_{-\infty}^\infty dx \, \frac{e^{iax}}{\sinh^2{bx}}$. Now dividing the integral into real and complex parts, respectively $\int_{-\infty}^{\infty} dx ...
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Integrating Around Cuts in the Complex Plane

I want to consider a simple hyper-elliptic curve $y^{2} = f(x)$, where for simplicity, let's just say $\rm{deg}(f)=4$. Thus, there will be 4 branch points on the $x$-plane over which the curve is ...
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Integration (Residue theorem )

$$ \phi (x,y)=\int_{0}^{\infty }\left ( \frac{1-\cos zx}{z} \right )\sin zx \,e^{-zy}dz. $$ I have solved this integration this way: $$ \frac{\partial \phi (x,y) }{\partial y}=\int_{0}^{\infty }(\cos ...
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Is there a general algorithm to determine new contours for multivariable change of integration variables

Is there a general algorithm to determine the new region of integration upon a multivariable change of variables (where the old variables are a function of all the new variables). I have to do a ...
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1answer
28 views

Proof statements with Moivre formula

So i have this statement in my course that i have to prove by using the moivre: $\cos(4x)=8\cdot\cos^4(x)-8\cdot\cos^2(x)+1$ Could anyone help me with this? (P.S. sorry for the English)
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Computing alternating sum using contour integration

By considering the integral of: $$\left(\frac{\sin\alpha z}{\alpha z}\right)^2 \frac{\pi}{\sin \pi z},\quad \alpha<\frac{\pi}{2}$$ around a circle of large radius, prove that: ...
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2answers
56 views

how to deal with $| dz |$ in complex integral, with two examples [duplicate]

I met two integrals: $$\int_{| z |=1} \frac {|dz |} {z} $$ and $$\int_{| z |=1} |\frac {dz} {z}| $$. Actually I have no idea of how to deal with $|dz|$. Any good suggestions? Many thanks. Best ...
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1answer
34 views

Integrating complex function $f(z)=\sin(z)$ along a path

What we learned in complex analysis is $$\int_{\gamma}f(z)\,dz=\int_{\gamma}(u(x,y)+iv(x,y))(dx+i\,dy)$$ if we let $f(z)=u+iv$ and $dz=dx+i\,dy$. So how could I calculate $f(z)=\sin(z)$ ? What is ...
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1answer
51 views

complex integration, residues, inverse Laplace transform, calculus

Dear Mathematicians, I kindly ask your expertise on complex integration. The problem is the last step in the solution to a differential equation, using an inverse Laplace transform. I know that the ...
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1answer
22 views

Finding univalent function with $f(z_1)=f(z_2)$

Let $\Omega$ be a simply-connected domain. Let $z_1,z_2\in\Omega$. Prove that exists an univalent function such that $f(\Omega)=\Omega$ and $f(z_1)=z_2$. Since $\Omega$ is simply connected, one ...
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1answer
30 views

What is the value of $\oint _{C(5i+1,8\sqrt3)} \frac {z}{sinh(z)} dz$?

sinh(z) = 0 $\Rightarrow z = ik\pi$ and we have to find the distance between $5i+1$ and $i\pi$ then between $5i+1$ and $-i\pi$ then with $2i\pi, -2i\pi, 3i\pi, -3i\pi...$ to see which poles are ...
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32 views

Integrate $\int{ \frac{\sin(\pi z^2)+\cos(\pi z^2)}{\{(z-1)(z-2)\}^{4}} dz }$

I want to evaluate $$\int{ \frac{\sin(\pi z^2)+\cos(\pi z^2)}{\{(z-1)(z-2)\}^{4}} dz }.$$ This is the contour integration I came across. I know Cauchy's integral formula and Cauchy's integral ...
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2answers
39 views

$\int_{C_N} \frac{dz}{z^2\sin(z)}$ complex integral, problem with residues

Let $C_n$ be the rectangle, positively oriented, which sides are in the lines $$x=\pm(N+\dfrac{1}{2})\pi~~~y=\pm(N+\dfrac{1}{2})\pi$$ with $N\in\mathbb{N}$. Prove that $$ ...
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0answers
42 views

evaluation of fourier transform of electric potential

I would like to ask how to evaluate equation 7? I have spent hours and still have no idea how to get a(k).
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1answer
52 views

Integration over complex plane

I have a problem with the following integral $$\int_{-\infty}^{\infty}\frac {x\sin x}{x^4+1}$$ Can someone please help me with the way the solution goes? I would highly appreciate it Thanks in ...
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2answers
62 views

the integral a complex variable

$\int\limits_{0}^{2\pi}e^{\cos\varphi}(\cos\varphi-\sin\varphi)d\varphi$ I think $e^{i\varphi}=z$ $\to d\varphi=\frac{dz}{iz}$ $\cos\varphi=\frac{z^2+1}{2z}$ $\sin\varphi=\frac{z^2-1}{2iz}$ ...