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Questions tagged [contour-integration]

Questions on the evaluation of integrals along a locus in the complex plane.

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13 views

Evaluating $\int_{-\infty}^{\infty}\frac{1-a\cosh(\alpha x)}{(\cosh(\alpha x)-a)^2}\cos(\beta x)\,dx$

I would like to solve the following improper integral: $$\int_{-\infty}^{\infty}\frac{1-a\cosh(\alpha x)}{(\cosh(\alpha x)-a)^2}\cos(\beta x)\,dx$$ where $a$, $\alpha$ and $\beta$ are real constants....
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3answers
107 views

evaluate $\int_{|z|=1} \frac{1}{e^z -1-2z}dz$ by using Cauchy's residue theorem

evalulate the integral by using Cauchy's residue theorem $$\int_{|z|=1} \frac{1}{e^z -1-2z}dz$$ MY attempt : $ f(z) =\frac{1}{e^z -1-2z}$, now put $z= 1$ we get $f(z)=-1$ so $$\int_{|z|=1} \frac{...
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0answers
17 views

Imaginary part of contour integral

I am attempting to find the imaginary part of the following integral, $$ I_1 = \Im\bigg{(} \int_{-\infty}^{\infty} dx \int_{x}^{\infty}dy \frac{ \exp(x^2 -2y)}{\omega - y} \bigg{)}, $$ where $\omega$ ...
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0answers
57 views

Cauchy principal value of $\int_{-\infty}^\infty\frac z{8-z^3} \mathrm dz$ [closed]

Let $f(z) = \frac{z}{8 - z^3}$. Then find the Cauchy principal value of $$\int_{-\infty}^{\infty} f(x)\, dx$$ Please explain it properly with basic things.
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2answers
34 views

$\int _\gamma \frac {sinz} {z^2+1}dz$ for circle of radius 2

My task is to compute $\int _\gamma \frac {sinz} {z^2+1}dz$, where $\gamma$ is the positvely oriented circle centered at the origin of radius 2. I tried an approach similar to Jacky Chong's solution ...
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1answer
45 views

Integrate $\exp (iz^2)$ using contour integration

Question: Show that $$\lim_{R\rightarrow \infty} \int_{-R}^Re^{iz^2}dz = \sqrt \pi e^{i\pi/4} + \mathcal O \bigg(\frac 1R \bigg)$$ by using contour integration Attempt: First I observed that $$\...
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1answer
28 views

evaluating contour integrals along a curve

A couple of exercises I'm working on: $1) $ Let $\gamma$ be a closed curve lying entirely in the set $\mathbb{C} \setminus\{z \mid \text{Re} z \leq 0\}$. Show that $\int_{\gamma}\frac{1}{z}dz = 0$. $...
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2answers
31 views

For what simple closed curves does the equation hold?

$\int_{\gamma} \frac{dz}{z^{2}+z+1} = 0$. I know that by Cauchy's Theorem, if $f$ is analytic everywhere on and inside a simple closed curve, then $\int_{\gamma} f(z)dz = 0$. So I thought that the ...
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1answer
47 views

How to determine $\gamma$ in Fox H-function

In the following Fox H-function the contour $L$ is either $L_{-\infty}$, $L_{+\infty}$ or $L_{i\gamma\infty}$. $$ H_{p,q}^{\,m,n} \!\left[ z \left| \begin{matrix} ( a_1 , A_1 ) & ( a_2 , A_2 ) &...
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24 views

Lebesgue integral and contour integration

Suppose $f \colon \mathbb{C} \to \mathbb{C}$ is holomorphic and $B = B(0,R)$ is the ball of radius $R > 0$ centered at $0$. Identifying $\mathbb{C}$ with $\mathbb{R}^2$ and using $\lambda$ to ...
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1answer
28 views

Integral of conjugate complex numbers [closed]

I know contour integral but I don't know what to do with this??? $$\int_{0}^{1+i} ({z^{*}}) ^{2}dz\\$$ along the line $$\\x=3y\\$$ Thanks in advance!
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0answers
35 views

Evaluating a contour integral along two paths

I'm working on the following problem: Evaluate $\int_{\gamma} \overline{z}^{2}dz$ along $\gamma$, where $\gamma$ is the straight-line joining $(0,0)$ to $(1,1)$. I defined $\gamma : [0,1] \...
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1answer
15 views

Substitution in complex integral and the Argument Principle.

Let's say $C$ is a simple closed curve in the complex plane and $f(z)$ is holomorphic and doesn't vanish on $C$. According to wikipedia, one can make the following change of variables: $$\omega = f\...
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2answers
41 views

A Suitable Function for Terrain, Mountain Modeling

On Google Maps and various other mapping programs, one can see contour lines that correspond to elevation. Sometimes these contour lines are concentric corresponding to a mountain. My question is ...
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5answers
111 views

Quick way of solving the contour integral $\oint \frac{1}{1+z^5} dz$

Consider the contour integral in the complex plane: $$\oint \frac{1}{1+z^5} dz$$ Here the contour is a circle with radius $3$ with centre in the origin. If we look at the poles, they need to satisfy $...
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1answer
28 views

Complex Contour Integrals Answer check [closed]

can someone check my answers for the following question: Evaluate the integrals 􏰣 $\ \int{z^2 dz} $ and $\ \int{|z|^2} dz $ along the following paths a) line from 1 to i b) quarter of the unit ...
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2answers
165 views

Solving or bounding the real part of the integral $\int_0^{2 \pi i m} \frac{e^{-t}}{t-a} dt$

I would be interested in finding a closed form or, at least, bounding (in terms of $m$ as it becomes larger) the real part of the following itnegral: $$f(m,a):=\int_0^{2 \pi i m} \frac{e^{-t}}{t-a} ...
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1answer
24 views

ML inequality on $|\int_{\gamma} e^{iaz^{2}}dz|$ such that $z=Re^{i\theta},\theta \in [0,\frac{\pi}{4}]$

On a contour defined by sector of radius $R$ making an angle of $\frac{\pi}{4}$, I applied the ML inequality on the curvy(angular) part ofcontour and this is what i found $|\int_{\gamma} e^{iaz^{2}}dz|...
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5answers
69 views

Evaluate $\int_{0}^{\pi/2}\cos^{2n+1}(x)dx$

How can I compute this Integral for integer $n$ from $0$ to $\pi/2$ $$\int_{0}^{\pi/2}\cos^{2n+1}(x)dx?$$
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0answers
36 views

How to calculate the arc part when using Residue theorem?

Suppose the function $f(z)$ has singularities and I want to calculate the integral $$\int_{-\infty}^{\infty}f(z)dz=?$$ I use Residue theorem $$\oint_cf(z)dz=2\pi i\sum\text{Res}(f,z_0)$$ And $$\...
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1answer
38 views

evaluate $\int_0^{\infty}\frac{x^a}{x^2+1}dx$ for $a\in(0,1)$

I am stuck on this complex analysis homework problem :( here is my attack so far: If I integrate around a keyhole contour, I can show that $$\int_0^{\infty}\frac{x^a}{x^2+1}dx=\frac{2\pi i}{1-e^{2a\...
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2answers
96 views

Calculate the value of $\int_0^\infty \frac{\sqrt{x}\cos(\ln(x))}{x^2+1}\,dx$

I'm asked to evaluate the integral $\displaystyle\int_0^\infty \frac{\sqrt{x}\cos(\ln(x))}{x^2+1}\,dx$. I tried defining a funcion $f(z)=\frac{e^{(1/2+i)\operatorname{Log}(z)}}{z^2+1}$, taking $\...
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1answer
61 views

Demonstrating $\int_{\pi}^{0}{ie^{-i2\pi s \epsilon e^{i\theta}}d\theta} = i\pi \space \mathrm{sgn}(s)$

In his derivation of the Fourier Transform of $\dfrac{1}{x}$, Bracewell starts with $$\mathscr{F}\left\{\dfrac{1}{x}\right\} = P.V. \int_{-\infty}^{\infty}{\dfrac{e^{-i2\pi sx}}{x}}dx$$ And goes on ...
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0answers
659 views

Proof without words of $\oint zdz = 0$ and $\oint dz/z = 2\pi i$

I found this visual "proof" of $\oint zdz = 0$ and $\oint dz/z = 2\pi i$ quite compelling and first want to share it with you. But I have a real question, too, which I will ask at the end of this post,...
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2answers
47 views

Can someone help me? I started but i have problem

$\int_Φ \frac{e^{2z} } {z^4(z-2)}dz$ where Φ:[0,2Pi] and $Φ(t)=3e^{it} $ so we have $z=3e^{it}$ and $dz=3ie^{it}dt$ I want to calculate $ \int_0^{2Pi} \frac{e^{6e^{it}}} {e^{4}(3e^{it}-2)} 3ie^{it}dt =...
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1answer
44 views

Connection between sums and integrals over closed paths and their areas

Is there a deeper connection between the following identies involving sums or integrals over a closed path (resp. circle) resp. the area enclosed – e.g. a general principle that is underlying ...
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0answers
91 views

Visualizing Cauchy's integral theorem (and complex integration in general)

(I edited the question due to a hint from Giuseppe Negro who pointed out that I forgot about $dz$.) Consider Cauchy's integral theorem, i.e. $$\oint_\gamma f(z)dz = 0 $$ for holomorphic functions $...
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0answers
40 views

Nested Cauchy type integral

I have a function which is in integral form: $$f_+(z)=\exp\bigg(\frac{1}{2\pi i}\oint_{C}\frac{f(\alpha)}{(z-\alpha)}\,d\alpha\bigg),$$ where $C$ is a unit circle inside an annulus in $z$ complex ...
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integration involving Gaussian PDF and CDF, with a scale and offset [duplicate]

Suppose $\phi, \Phi$ are PDF and CDF for a $1$-dimensional normal Gaussian distribution, and $a,b>0$ are arbitrary constants. Is there a way to compute this integral analytically? $$\int_{-\infty}^{...
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3answers
51 views

Contour integral calculation 2

Show that \begin{align*} \frac{1}{a}\int_0^{\infty} \frac{\sin x}{x+a} \; dx= \int_0^{\infty} \frac{e^{-x}}{x^2+a^2}\;dx \quad (a>0) \end{align*} My Attempt: I showed that \begin{align*} \...
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1answer
35 views

Can I think of a toy contour as any closed piecewise-smooth curve which is simple?

I feel Stein doesn't give the specific definition of toy contour, can I think of a toy contour as any closed piecewise-smooth curve which is simple?
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1answer
34 views

Prove that $F(z) = \int_a^b f(t,z) dt$ is analytic

I would like to prove that if the complex-valued function $f(t,z)$, defined for $a \leq t \leq b$ and $ z \in D$ (where $D$ is some domain), is continuous and analytic for each fixed $t$ (for $z \in D$...
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0answers
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Show $\int _ { 0 } ^ { \infty } \frac { x \sin x } { \left( x ^ { 2 } + 1 \right) ^ { 2 } } dx = \frac { \pi } { 4 e }$ with residue theorem [duplicate]

Problem: Compute $\int _ { 0 } ^ { \infty } \frac { x \sin x } { \left( x ^ { 2 } + 1 \right) ^ { 2 } } dx = \frac { \pi } { 4 e }$ First we can study $\int_{\Gamma_R}g(z)dz$ where $\Gamma_R$ is ...
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Leibniz rule when range of integration is defined by inequality

I would like to solve this $$\frac{\partial\int_{S(x)>S(\theta)}(S(x)-S(\theta))dF(x)}{\partial\theta}$$ where $S$ is a single-valued, differentiable, and strictly increasing function and $F$ is ...
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0answers
55 views

Verification for $\int_{-\infty}^\infty \frac{\cos(az)}{z^{2n}+1}dz$

I decided to tackle this integral by integrating over a semicircular contour in the upper-half plane and the real line, where the path is denoted as $\Gamma$. $$\int_{\Gamma} \frac{e^{iaz}}{z^{2n}+1}...
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1answer
126 views

Verify that $\left|\int_{\gamma} \exp(iz^2)dz\right| \leq \frac{\pi\big(1-\exp(-r^2)\big)}{4r}$ where $\gamma(t)=re^{it}$, for $0\leq t \leq \pi/4$.

Verify that $$\left|\int_{\gamma} \exp(iz^2)dz\right| \leq \frac{\pi\big(1-\exp(-r^2)\big)}{4r}$$ where $\gamma(t)=re^{it}$, for $0\leq t \leq \pi/4$ and $r > 0$. I'm stuck. here is my attempt: $|...
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1answer
57 views

$\int_{0}^{\pi}\frac{\sin^2(x)}{a+\cos(x)}\,dx$

I have been trying to solve the integral below using contour integration. $$\int_{0}^{\pi}\frac{\sin^2(x)}{a+\cos(x)}dx, \quad a>1.$$ I'd appreciate to see how you would solve it, since the fact ...
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1answer
92 views

Confusion with this inverse Laplace Transform

I have earlier posted this question in Physics StackExchange but I feel that it is more relevant here. The question is about a contour integral and I have written most of the equations needed. Please ...
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0answers
17 views

Contour Integral over Heaviside Function

I am stuck on how to do a problem where I am deriving the Green's function for a nuclear scattering system. I am currently starting with the expression $$G^+(\mathbf{0})=\frac{2m}{(2\pi)^2}\int d\...
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1answer
48 views

A question about the Residue of $h=fg$

Let $f$ and $g$ be two functions (not necessarily analytic) of the complex variable $z$ such that for some $\varepsilon >0$ : 1) $f$ is continuous on $0<\left\vert z\right\vert <\varepsilon $...
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3answers
111 views

Evaluating $\int_{-\infty}^{\infty}\frac{e^{ax}}{\cosh{x}}dx $ using contour integration

Let $a \in \mathbb{C}$ with $-1 <$ Re $a < 1$. By considering a rectangular contour with corners at $R, R + i\pi, -R+ i\pi, -R,$ show that $$\int_{-\infty}^{\infty}\frac{e^{ax}}{\cosh{x}}...
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47 views

Estimating $\left |\log(z) \right |$ to calculate an integral?

I'm doing the classical example of treating the integral $\int_{0}^{\infty }\frac{\log(x)}{\left (1+x^2 \right )^2}dx$ using residue theorem on the function $$f(z)=\frac{\log(z)}{\left (1+z^2 \right ...
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2answers
193 views

Real integrals with two poles in the complex plane

I'm studying the Cauchy Integral Theorem / Formula, but realised I have a misunderstanding. Consider an integral over the function $f: \mathbb{R} \to \mathbb{C}$ $$ I = \int^\infty_{-\infty} f(x) \, ...
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1answer
237 views

Contour for $\int_0^\infty \arctan(z) e^{-z^2}\,dz$ or some variant

I'm trying to practice my contour integration skills and got interested in the following integral: $$\int_0^\infty \arctan(z) e^{-z^2}\,dz$$ I know that the usual way to calculate integrals on $[0,\...
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0answers
39 views

How can I solve this integral equation with the inverse Laplace Transform?

This question is related to Solving an integral equation with inverse Laplace transform. Let $\alpha,\beta,\mu>0$ with $\alpha/\beta>\mu$ and $X\sim\operatorname{Gamma}(\alpha,\beta)$. I am ...
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0answers
19 views

Meaning of the principal part for an analytic limit (as opposed to an integral)?

I am slightly confused as to the meaning of 'the principle part' of a limit, specifically in relation to the Kramers-Kronig relations as derived here on page 61. The source uses the 'Principal part' ...
4
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2answers
113 views

How to simplify the result I obtain for the following integral: $\int_0^\infty \frac{\cos(a x)}{x^4+b^4}\,dx$

I have been trying to calculate the following integral using the Residue Theorem but end up with an answer that seems to contain an imaginary part while the integral should be purely real: $$I_1=\...
2
votes
2answers
102 views

Integration problem solving without contour integration

Can the following question be solved without using contour integration. $F:(0,\infty)\times (0,\infty)\to \Bbb R$ be given by $F(\alpha,\beta)=\displaystyle\int_0^\infty\frac{\cos(\alpha x)}{x^4+\...
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1answer
45 views

Sign of the Hankel representation of the Gamma function

I have a question about the Hankel path representation of the Gamma function. The path of integration is displayed on the imege. The branch cut is taken as the negative real axis. $$ \Gamma(z) = \...
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1answer
86 views

How to solve this improper integral $\int _{-\infty}^{\infty} e^{ax}/(e^x+1) dx$? [closed]

I need to solve the following integral. $$\int _{-\infty }^{\infty }\:\:\dfrac{e^{ax}}{e^x+1}dx$$ where $0<a<1$.