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Questions tagged [complex-integration]

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

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2answers
55 views

Given $f(z)$ is entire and a bound on f at each $ z \in \mathbb{C}$, prove that the integral is 0

The Problem: It's given that $f(z)$ is an entire function on $\mathbb{C}$ and that there exist $M > 0$ and $A > 0$ such that $|f(x+iy)|\le\frac{Ae^{2\pi M|y|}}{1+x^2}$ for all $x,y \in \mathbb{...
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1answer
28 views

Evaluate $\int_c (z - z^2) dz$

In an exercise, it's given that $C$ is a circle such that $|z-2|=3$ Now I know that, a circle with radius $r$ can be represented by $z=re^{i\theta}$, hence, $$ z-2=3e^{i\theta} \implies z=2+3e^{i\...
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2answers
18 views

Expressing coefficients of a complex function using integrals

Let $f(z)$ be the function defined by: \begin{equation} f(z)=a_{-3}z^{-3}+a_{-2}z^{-2}+a_{-1}z^{-1}+a_0+a_1z+a_2z^2+a_3z^3 \end{equation} How can we express the coefficients $a_i$ using integrals? ...
2
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2answers
33 views

Complex Integral $\int_{C} \frac{e^{iz}}{z^3} dz$ using Cauchy Integral Formula of Derivatives

I am trying to find $\int_{C} \frac{e^{iz}}{z^3} dz$ on circle of $|z| = 2$ traversing once on positive direction. My approach was using Cauchy derivative formula $f^{(n)}(a) = \frac{n!}{2\pi i} \...
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29 views

Computing this integral [on hold]

could you give me a hint on how to compute the integral $$\int_0^s \dfrac{e^{icx}}{a+b\cos(tx)}\,dx$$ with $a,b,c,s,t$ constants (and are all but $c$ positive) such that $a > b?$
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23 views
+50

Identity with regards to error of sinc approximation

I have this issue that I'm kind of clueless about, it is peripheral to what I typically do. I will state all the assumptions meticulously, even though I suspect they are not all needed. It is problem ...
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2answers
44 views

Compute complex integral inside an open curve

I need to compute this complex integral: $$ \int_\gamma \frac{1}{(z-i)(z-2i)} dz $$ $\gamma$ is defined as: $$ \gamma (t) = t + i(3e^t\cos^2(t)) $$ The parameter t belongs to the following ...
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2answers
31 views

Find the integral of $\int_{\gamma} (z - z_0)^{n}dz$ [on hold]

Given an arbitrary point $z_0$, and a circle $\gamma$ of radius $r>0$ centered at $z_0$, oriented counterclockwise. Find the integral. \begin{equation*} \int_{\gamma} (z - z_0)^{n}dz \end{equation*}...
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18 views

How to solve this Complex Integral using poles?

I want to find the green's function of a free particle, which depends of the integral: $$ I = \frac{1}{4\pi ²ir} \int^{+\infty}_{-\infty} \frac{ke^{ikr}}{E-\frac{\hbar²k²}{2m}+i\eta} dk\,. $$ Then, ...
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0answers
7 views

Why can we use the principal value of the complex integral to solve real improper integrals

I've been taught that you have some real integral, $\int_{-\infty}^{\infty}f(t)dt$ $f(t)=\frac{n(t)}{d(t)}$ where the order of d(t) is 2 or more greater than the order of n(t) For example, $\int_{-\...
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1answer
26 views

Integral of the reciprocal of a complex polynomial [duplicate]

For a polynomial P(z) of degree $n\geq2$, show that there exists some $R_{1}>0$ such that for $R>R_{1}$ it holds that: $$\int_{C_{R}}\frac{1}{P(z)}dz=0$$ where $C_{R}$ is a circle of radius R ...
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1answer
29 views

Questions about Cauchy's thm. on complex integration

I have the following integral $$I = \int_{\gamma_{1}} \frac{e^{z^2}}{(z-1)^2}dz,$$ where $$\gamma_{1} : [0,2Pi] \rightarrow \mathbb{C}, \\ \quad t \quad \mapsto \quad 2e^{i t} .$$ I want to show that ...
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17 views

Integration of a (0,1)-form on the boundary of a Riemann surface

In Simon Donaldson's book, he says that for any (0,1)-form $\theta$ on a compact connected Riemann surface $X$, the integral of $\partial\theta$ over $X$ is zero by Stokes' theorem - but that seems ...
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2answers
44 views

A line integral equals zero implies a real integral also is zero

I'm asked to check that the following line integral is zero: $$\int_{C(0,r)} \frac {\log(1+z)}z dz=0$$ (where $C(0,r)$ is the circle of radius $r$ centered at $0$) and then to conclude that for ...
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1answer
49 views

a basic definite integration and its result used in evaluating limit

Question : $\mathbf\Omega(n)=\displaystyle\int _{0}^{2\pi}\log(n^2-2n\cos t+1)dt\ ,\ n\geq1 $ then , find : $\displaystyle \lim_{n \to \infty} \left(1+\dfrac{\mathbf \Omega(n)}{4\pi}\right)^{\log(n+...
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0answers
12 views

can solve this by contour integration?

how is done this integral: $$\int_{0}^{2 \pi} \ln(z-Re^{i \theta})e^{i \theta}d\theta$$ I try by substitution of $u=z-Re^{i \theta}$ and i get 0 but i don't know if is correct, perhaps by contour ...
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0answers
9 views

Possible values of winding numbers

Suppose that $D$ is a plane domain. Let $N$ be the set of all integers $n$ such that there is a closed, piecewise smooth curve $\gamma$ in $D$ whose winding number about the origin equals $0$: $$N:=\{...
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0answers
19 views

Interpretation of integrals

How to interpret integrals in the image ? Length of curve ? Or etc. (https://i.stack.imgur.com/tJcrX.jpg)
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0answers
36 views

Find the line integral of $\int\frac{e^{iz}}{z}\, dz$ [duplicate]

This is a homework problem, I need to calculate $\int\frac{e^{iz}}{z}\, dz$ over some curves, but I can find a useful parametrization of $z$ that make this calculations not so difficult. For example ...
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1answer
33 views

Find the complex integral

I have $$\int_{\gamma} {\frac{z}{\overline z}}dz$$ where $\gamma$ is the edge of $\{1 < |z| < 2\ $and $\Im (z) > 0\}$. I think the way to solve this is to calculate the integral for $|z|=1$ ...
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0answers
35 views

Integral of a complex function over contour (continued)

I'd like to double check that my method to evaluating the following integral over a contour is correct. I asked how to go about integrating over such a contour in Integral of a complex function over ...
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1answer
41 views

How to evaluate a complex integral? [closed]

Show that |$\int_{C} \frac{e^z}{\bar z + 1} dz$| ≤ $2\pi e^2$ where C is the circle |z-1| = 1. I think C can be parametrized by z = $1 + e^{it}$ and $z' = ie^{it}dt$ with 0 ≤ t ≤ $2 \pi$ How do i ...
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0answers
21 views

How to find the parametrization of Re(z)? [duplicate]

Let Γ be the simple closed contour obtained by moving from 0 to 1 along the real axis, then from 1 to i along the unit circle, and finally from i back to 0 along the imaginary axis. I know we can ...
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2answers
80 views

Integral of a complex function over contour

I'd like to double check that my method to evaluating the following integral(s) over a contour is correct: $\Gamma$ is a simple closed contour given by the path moving from 0 to 1 along the real axis,...
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2answers
37 views

Integral of complex number over a contour

I have $$\int_{-1}^1 |z|dz$$ I need to calculate the integral where the integration contour is the upper semi-circle with unit radius. I calculated the integral in $(-1; 1)$ section; the answer is 1, ...
2
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1answer
47 views

Prove $\int_{0}^\pi{e^{a\cos(x)}\cos(a\sin(x))}dx=\pi$

Prove: $$A=\int_{0}^\pi{e^{a\cos(x)}\cos(a\sin(x))}dx=\pi$$ (I think) that a suggestion was made to calculate and later use it: $B=\int\frac{{e^{az}}}{z}$, over the path gamma $\gamma=e^{it}, t\in[-\...
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1answer
34 views

Lineintegral of absolute value with path $\gamma: [0,1]\rightarrow\mathbb{C},t\mapsto i +\exp(i\pi t)$

Calculate: $\int_{\gamma} |z|dz$ with $\gamma: [0,1]\rightarrow\mathbb{C},t\mapsto i +\exp(i\pi t)$ I tried calculating it and actually made some progress, where one term vanished when splitting ...
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16 views

Inverse Laplace transform of Gamma with branch cut

In solving a particular physical problem I have had to perform inverse Laplace transforms of sum and products of Gamma functions. Since my actual problem is complicated, I will state a simple example. ...
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3answers
54 views

What are we really doing when we integrate a complex exponential?

Firstly, I have read this related question from this site, but it does not answer what I am asking here. What I like to know is why $$\int_{-n}^n e^{ix}dx \ne 0$$ for any finite $n \in \mathbb N$ My ...
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0answers
34 views

Ahlfors page 171 Poisson Integral

Tl;dr : compute the last integral with $z$ fixed. If $C_1$ and $C_2$ are complementary arcs on the unit circlw, set $U = 1$ on $C_1$ and $U=0$ on $C_2$. Let $0 \leq \theta_0 < \theta_1 \leq 2 \...
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1answer
44 views

Integration of $e^{ax}\cos^n bx$ and $e^{ax}\sin^n bx$

Q: Integration of $e^{ax}\cos^n bx$ and $e^{ax}\sin^n bx$ I know how to integration $e^{ax}\cos bx$ using $\cos bx=\frac{e^{ibx}+e^{-ibx}}{2}$.Using the same trick here I got $$\int e^{ax}\cos^n bx ~...
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0answers
22 views

Computation of complex valued integral

I would like to get a simple formula for the complex valued integral $$ \frac{1}{\sqrt{2\pi\sigma_Z^2}}\int_{-\infty}^\infty\exp\left(iux+\delta d(t)\vert x\vert\right)\exp\left(-\frac{(x-\mu_Z)^2}{2\...
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2answers
23 views

Using the Cauchy Integral Theorem for Derivatives to evaluate an integral

I ran into this question which hints me to use Cauchy's Integral Theorem for Derivatives, however I don't seem to be able to fit this integral into the form of the Integral Formula $$\displaystyle \...
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1answer
44 views

Evaluating the Integral of $\pi e^{\pi \overline z}$ with respect to $z$

I've been given a question as part of the homework on Complex Integration. I cannot seem to think how to integrate it especially with the presence of the conjugate of $z$ in the expression $$\...
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1answer
37 views

Does the specific contour matter when integrating over a closed loop

I've heard different people say yes and no, so I want to ask it here on math stack, but often, as in with cauchys integral theorem on wiki for instance, the contour is specified as a circle in the ...
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1answer
86 views

Contour integration - complex analysis

I am trying to solve the following integral using a contour (large semi-circle connected to smaller semi-circle in the upper-half plane): $$\int_0^{\infty} \frac{\log^4(x)}{1+x^2} dx.$$ I have ...
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1answer
37 views

Lineintegral $\int_{\gamma}|z|^2dz$ over ellipse

Let $a,b\in\mathbb{R}_{>0}$ and $\gamma: [0,2\pi]\rightarrow\mathbb{C},t\mapsto a\cos(t)+ib\sin(t)$ calculate the line integral $\int_{\gamma}|z|^2dz$ My calculation turns out to be really ugly. ...
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2answers
51 views

Why is integral not equal to zero even though the path is closed?

I have an integral $$\int{z^{i}}dz$$ the path is $e^{it}$ where $t$ is between $0$ and $2\pi$. Why is it not equal to zero even though the path is closed.
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0answers
32 views

equality in integral inequality for compelx functions

I am trying to understand why: $\left|\displaystyle{\int}_{\gamma}f\left(z\right)dz\right|=\displaystyle{\int}_{\gamma}\left|f\left(z\right)\right||dz|\iff\forall z$ on $\gamma$ it is $\text{arg}\...
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1answer
8 views

Is the role of the boxed condition $z'(t)\neq 0$ to avoid going back?

The role of the boxed condition $z'(t)\neq 0$ is to avoid going back, isn't it?
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1answer
41 views

Complex integration lemma: shorter proof?

The black line is the branch cut. Lemma $$\lim_{\Delta\to0^+}\left(\int_{\gamma_1}+\int_{\gamma_2}\right)f(z)\ln(z-s)dz=-2\pi i\int_{pe^{i\theta}}^{qe^{i\theta}}f(t)dt$$ where $\arg(z-s)\in[\theta,...
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1answer
10 views

Prove simple closed curves $f$'s exist, so $\Gamma = C-\sum_{i=1}^{k}{f_i}$ satisfies $ \int_{\Gamma}{\frac{z^3e^{1/z}}{(z^2 + z + 1)(z^2 + 1)}dz}=0$

Let $C$ be the circle $C(0,2)$ traversed one time counter-clockwise. Prove that there exist $k\in \mathbb {Z}_+$ and $C^1$ simple closed cuves $f_1, \dots ,f_k$ such that the cycle $\Gamma = C-\sum_{i=...
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1answer
21 views

Argument Principle-like complex integral involving logarithm

Let $g$ be a holomorphic function on $C$ and its interior, where $C$ is a circle with a radius $r$ around the origin. Furthermore let's assume that $g$ doesn't assume a zero value on $C$ and its ...
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1answer
39 views

$\int_\gamma \frac{e^z}{(z^2+1)^2} dz$ where $\gamma$ is $C(0,2)$ traversed twice counter-clockwise.

Evaluate $\int_\gamma \frac{e^z}{(z^2+1)^2} dz$ where $\gamma$ is $C(0,2)$ traversed twice counter-clockwise. This $\gamma$ should be represented as a cycle I believe. Hi all, I am wondering about ...
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0answers
36 views

Branch cut for $\sqrt{z^2-a^2}$ in the lower half of the complex plane (instead of $[-a,a]$) - How does this change the function?

I have given a function $\sqrt{z^2-a^2}$ with $a>0$. At first i have chosen the branch cut on the real axis at $-a<z<a$: $$f(z)=\sqrt{z-a}\sqrt{z+a}=\sqrt{|z-a||z+a|} \exp({i\frac{\theta_1+\...
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1answer
46 views

Computing a real integral using the Cauchy integration formula

Compute the real integral: $$\int_0^{2{\pi}} \frac{dx}{2+\sin(x)}.$$ The idea here is to convert this real integral into a complex integral, and then solve it using the Cauchy integration form. ...
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1answer
15 views

Prove a certain relation holds for a holomorphic function.

Suppose $f(z)$ is holomorphic on the disk $|z|<R$ and $|f(z)|\leq M$ for all $z$ with $|z|<R$. Let $0<r<R$. Prove that for any $z$ with $|z| < r$ it holds that $|f^{(n)}(z)| \leq \...
1
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1answer
32 views

Prove a certain holomorphic function does not exist.

Prove that there does not exist a holomorphic function f(z) on any open set containing 0 such that $f^{(n)}(0) = n^n\cdot n!$ I tried to use the Cauchy integral formula for higher derivatives and ...
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0answers
11 views

Computing complex contour integrals, can contours touch?

Suppose I wish to evaluate the following contour integral $\oint \dfrac{1}{z - z_o} dz$ where the contour is given by the rotated square in blue and $z_o$ is in the center of the square. Can I ...
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4answers
65 views

Methods to solve $\int_{0}^{\infty} x^{n}\cos(x)\:dx$

I've been playing around with the following integral and was wondering if it can be generalised to any Real $n$. Does anyone know of any methods to approach this one? $$ I = \int_{0}^{\infty} x^n \...