Questions tagged [contour-integration]

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

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3
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0answers
30 views

Using a contour integral of $f(z)=x^{4n+3}e^{-z}$ to show $\int_0^\infty x^{4n+3}e^{-x}\cos x\;dx=(-1)^{n+1}(4n+3)!/2^{2n+2}$ [closed]

I am pretty close to finding the right-hand side of the desired proof (see below) but not able to finish it. I have done this far but, it seems too long to continue.
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26 views

Evaluating a Hankel contour of zeta-like function

I was reading a 'paper' about the zeta function (I think the main part is fatally flawed, but there is still some valid work). At one point, the author deduces for $\Re(s)>0$ $$ (s-1)\zeta(s)=\frac{...
2
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1answer
30 views

Solve $\int_{|z|=2}\frac{e^{3z}}{(z-1)^3}dz$ using residue.

I'm trying to evaluate $$\int_{|z|=2}\frac{e^{3z}}{(z-1)^3}dz$$ using the residue theorem. I get a pole of order $3$ at 1 with a residue of $\frac{9}{2}e^3$. But since the absolute value of the ...
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1answer
30 views

Integrate $\int_0^{2\pi} e^{\cos\theta} \sin(\sin\theta-n\theta)d\theta$ [closed]

How to integrate $\int_0^{2\pi} e^{\cos\theta} \sin(\sin\theta-n\theta)d\theta$ $\int_0^{2\pi} e^{-\cos\theta} \cos(\sin\theta+n\theta)d\theta$
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1answer
30 views

Closed-contour integral of a derivative

Suppose a function $f(z)$ is single-valued everywhere and holomorphic inside a closed contour $C$, except for one pole. The derivative $ g(z) = \frac{ \partial f}{\partial z}$ also has the same pole. ...
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1answer
21 views

Computing Definite Integrals using Complex Variables and Contour Integration

I was trying to prove the definite integral $$\displaystyle{\int_{0}^{2\pi}}\frac{d\theta}{a-b\cos{\theta}}=\frac{2\pi}{\sqrt{a^2-b^2}}\quad(a>b\geq0)$$ by using contour integration and the Residue ...
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2answers
52 views

Computing Definite Integrals using Complex Variables and the Residue Therorem

I was trying to prove the definite integral $$\displaystyle{\int_{-\infty}^{\infty}}\frac{dx\,x\sin{x}}{x^2+a^2}={\pi}e^{-a}$$ by using contour integration and the Residue Theorem. I checked the ...
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23 views

Contour Integral $\int_0^{2\pi} e^{\cos\theta} \sin(\sin\theta-n\theta)d\theta$ [closed]

How to evaluate the integral $\int_0^{2\pi} e^{\cos\theta} \sin(\sin\theta-n\theta)d\theta$
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41 views

Contour integrals, residues, and changes of variables

Maybe this is a little contrived; I'm trying to understand how a change of variables will change how to compute a residue or contour integral. Anyway, suppose we have complex contour integral $$\frac{...
3
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1answer
44 views

Asymptotics of $\int xdx ~ f(x) J_\nu(x) J_\nu(\alpha x)$ type integral for $\nu \to \infty$

I am interested in obtaining the asymptotic expansion of integrals of the form $$ I_\nu(\alpha) = \int_0^\infty xdx ~ f(x) J_\nu(x) J_\nu(\alpha x),$$ for $\nu \to \infty$ and some fixed, real, $\...
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Regarding the contouring integral, what will be the final results? [closed]

Good evening, could you explain me step by step how can I solve this integral? I see that first I have to parametrize $∫\frac{\ln⁡(z+1)}{z+1}dz$, from $1$ to $i$, along the arch of circumference $|z|=...
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Contour integration with fractional powers in denominator

I wish to integrate the following $$\int_0^{\infty} \frac{\sin(k r)}{ r^{1/2}(r-2m)^{1/2}} dr$$ I believe contour integration is the best way to proceed. I see that there are branch points at $0$ and $...
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1answer
36 views

What will be the result of calculating the contour integral [closed]

Good evening, how can I solve this integral? $\int_{-1}^{i} \frac{\cos z}{\sqrt{\sin z}} d z$ along the line segment that joins the points $z_{1}=-1, z_{2}=i,$ the branch of function $\sqrt{\sin z}$...
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0answers
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how can I solve this contour integral [closed]

How can I solve this integral, $\int\limits_C \cos z\ dz$, where $C$ is the line segment that joins the points $z_1= \pi/2$ and $z_2= \pi + i$?
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Evaluating $f(z) = \frac{ze^{iz}}{z^2+1}$ over a real segment using contours and Residue theorem.

My task is to evaluate this integral $$I = \int\limits_{l_R}\frac{ze^{iz}}{z^2+1}$$ where $l_R$ is the real line from $0$ to $R$ or $l_R:= t$ with $t\in[0,R]$. My first try was using a contour ...
2
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2answers
69 views

What is a method for solving principal value integral of $\frac{1}{\pi}\int_{-B}^{B} \frac{x \sqrt{B^2-x^2}}{x-y}\mathrm{d} x$?

Question: I am trying to solve a principal value integral involving a square root. Using Mathematica I can get an answer but I would like to know a general approach to obtain them by hand. To be ...
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1answer
31 views

Value of a contour Integral

If we have the contour integral $\frac{1}{2\pi i} \oint_y \frac{f(C)}{(C-z)^2}dC$, where $f: \mathbb{C} \to \mathbb{C}$ is holomorphic and $y$ a closed circle with centre at $z\in \mathbb{C}$, which ...
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Evaluation of a real integral using complex contour

Evaluate $\displaystyle \int_0^\infty \frac {\tan^{-1}(ax)}{x(1+x^2)}\, dx$ for $\displaystyle a>0, a \ne 1, $. Need to do this using contour integral. I know how to do it using Leibnitz rule. I ...
2
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1answer
85 views

calculate $\int_{|z|=2}\sqrt{z^2-1}\,dz$ [duplicate]

Given: Calculate $\int_{|z|=2}\sqrt{z^2-1} dz$ Hint:$\sqrt{z^2-1}=z\sqrt{1-\frac{1}{z^2}}:=z\exp(\frac{1}{2}\log(1-\frac{1}{z^2}))$ I tried it for several hours and didn't manage to get ...
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1answer
36 views

zeros of a function in a simple closed contour

i would really appreciate if you can help me with the following demonstration please: Let $C$ be a closed simple contour such that $| f (z) | = cte$ ($f$ is analytic inside $C$ and is not an ...
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47 views

How to compute this contour integral with a modulus?

Evaluate the integral: $$\int_{-\infty}^{\infty} \frac {\omega e^{-2i\omega t}}{(\omega_0^2-\omega^2-\frac{\gamma}{m}i\omega)^{2}}d\omega$$ With $\frac{\gamma^2}{4m^2}>\omega_0^2$. I solved the ...
3
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1answer
47 views

contour integration of $\int_0^\infty \frac{\ln(x)}{x^2-1}dx$

I asked for a problem for few days ago, regarding integration of $$I=\int_0^\infty \frac{\ln(x)}{x^2-1}dx$$ I know that I could do the substitution $x=it$ and do the integral where the denominater is ...
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32 views

Integral with two branch cuts

I need help calculating the following integral $$\int_0^\infty\!\mathrm{d}u\frac{u^{-s}(u+i\omega_n)^s}{u-w+i\omega_n+i\omega_m}$$ where $0<s<1$. What I have thought so far is to do contour ...
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1answer
60 views

Contour integral $\oint_C \frac{p^2(z)}{|p(z)|^2} \frac{dz}{iz}$ over the unit circle $C$

I'm trying to find the contour integral over the unit circle $C$: $$\oint_C \frac{p^2(z)}{|p(z)|^2} \frac{dz}{iz}$$ where $$p(z) = a - z \sum_{n=-N}^{N} c_n z^n \\a, c_n, z \in \mathbb{C}$$ ...
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How to evaluate contour integral to show analyticity

Let $f(z)$ be defined on the complex plane such that $zf(z) \to 0$ as $|z| \to \infty$ and $f(z)$ is analytic on an open set containing $Im(z) > −c$, where c is a positive real constant. Let C1 be ...
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21 views

Construction of a certain weight for a functional to satisfy given condition:

Consider the following function : $$F(z) = \omega(z)\frac{\sin^2\left(\frac{c\Gamma^2(z)}{z}\right)}{\sin^2\left(\frac{c}{z}\right)}$$ Here, $\omega(z)$ is a weight we have to construct and $c$ is a ...
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1answer
28 views

Evaluating $\int_C \frac{1}{(z+i)^2\cos z}\,\mathrm dz$

I need to calculate the following integral $$\int_{|z| = 4} \frac{1}{(z+i)^2\cos z}\,\mathrm dz$$ I would imagine you are suppose to use the residue theorem in this problem. I know that the ...
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1answer
23 views

The value of the integral $\oint_C(2z+1)dz$ on the contour C What is the value of the integral

The value of the integral $\oint_C(2z+1)dz$ on the contour C, comprised of line segments C1, C2, . . . , C11 shown in Figure What is the value of the integral? I think we can say (2z+1) is ...
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1answer
22 views

Evaluate the given integral along the indicated closed contour $\oint_C\frac{sinz}{(z^2)+(\pi)^2}dz; |z-2i|=2$

Evaluate the given integral along the indicated closed contour $\oint_C\frac{sinz}{(z^2)+(\pi)^2}dz; |z-2i|=2$ This should be solve by using Cauchy's formula but i couldnt find.( z=$\pi$i is in C) ...
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0answers
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How to prove two identities between contour integrals and hypergeometric functions by choosing the right contour

I need to understand how to establish two identities. The first is $$ \int_{C} z^{-1-q}(1-z)^{-1-\lambda } dz=\frac{2 \pi \Gamma (q+\lambda +1)}{\Gamma (\lambda +1) \Gamma (q+1)}, q\geq ...
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1answer
51 views

Evaluating $\int_{|z|=1}\frac{z^{11}}{12z^{12}-4z^9+2z^6-4z^3+1}dz$ using Rouche's theorem

I am trying to evaluate $$\int_{|z|=1}\frac{z^{11}}{12z^{12}-4z^9+2z^6-4z^3+1}dz$$ Using Rouche's theorem, I know that all zeros of $12z^{12}-4z^9+2z^6-4z^3+1$ are inside $|z|=1$, but computing the ...
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0answers
29 views

$\oint_C\frac{1+e^z}{z}dz$ with |z|=1 evaluate the given integral along the indicated closed contour.

Consider $\oint_C\frac{1+e^z}{z}dz$ with |z|=1. evaluate the given integral along the indicated closed contour. I think this can be solve by using Cauchy's formula. But i'm not sure. If I take$ f(...
2
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3answers
155 views

How do you integrate $\int_{0}^\infty \frac{\log(x)^2}{(1-x^2)^2}$ using contour integration?

I have tried using the standard keyhole integral, and looking at$\ \log(x)^3 $, but because the poles lie on the real axis, when I expand the integrand $\ \frac{(\log(x) + 2\pi i)^3}{(1-x^2)^2} $ I ...
2
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1answer
26 views

Hankel integral modulated by cosine

This interesting integral arises in the calculation of wave reverberation in air ducts, and I believe it evaluates to: $$ \int_{-\infty}^\infty \frac{e^{ik\sqrt{x^2+w^2}}}{\sqrt{x^2+w^2}}\cos(\alpha x)...
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0answers
12 views

Find the work of vector field

Find the work of vector field $F=(x,y,z)$ along the contour specified by the conditions: $z=1 -x^2 -y^2, z-x= 1, y>=0. and oriented by the direction from a point (0, 0, 1) to a point (-1, 0, 0). I ...
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1answer
45 views

$\oint_C\text{Re}(z)dz$ is independent of the path $C$ between $z_0 = 0$ and $z_1 = 1+i$.

My way to prove that$\oint_C\text{Re}(z)dz$ is independent of the path $C$ between $z_0 = 0$ and $z_1 = 1+i$. This statement is false. I tried to take $z(t)=t+it$ and then I found this; $\oint_C〖(...
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1answer
30 views

Morera's Theorem and simply connected sets

my lecture notes contain the following version of Morera's Theorem: Let $f$ be a continuous complex-valued function on an open simply connected set $\Omega$. If $\int_\gamma f(w)dw=0$ for any ...
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1answer
47 views

Complex contour integral and Laurent series of $\frac{\sin z}{z^3-z}$

I'm new with these complex contour integrals and I wanted to know if my approach to this one is correct. The integral is $$\oint_\gamma \frac{\sin z}{z^3-z}\,\mathrm{d}z$$ where $\gamma(t)=2e^{it}, t\...
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0answers
38 views

How to evaluate the following integral? [Integrand is a fraction under square root with each numerator and denominator being a 3rd degree polynomial]

I am stuck with a integration that is related to a pendulum problem. $$ I = \frac{2\sqrt{2}}{\pi} \int_{x_1}^{x_2} \frac{\sqrt{-x^3+(1+E)x-\bar{\beta}}}{\sqrt{x} \sqrt{2-x^2}}dx\, $$ Note that, we ...
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0answers
24 views

An improper integral related to logarithm function

I was interested in solving for $\eta'(1)$ where $\eta(s)$ is Dirichlet eta function, and I am now able to expand it like this based on the fact that $\displaystyle\Gamma(s)\eta(s)=\int_0^\infty{t^{s-...
1
vote
1answer
48 views

Importance of a region being convex

So I have been reading and solving problems from complex analysis ( A.R Shastri's book), many times we talk of certain results in the case of convex regions. Is there something special about convex ...
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2answers
24 views

Contour Integral and Removable Singularity

Could someone check if what I have done is correct? To evaluate $\int_{C}\dfrac{z^3}{2z-i}$ where $C$ is the unit circle. My solution is as follows: Let $f(z):=\dfrac{z^3}{2z-i}$. There seems to be ...
0
votes
2answers
45 views

Evaluate $\int\frac{e^z-1}{z}\mathrm dz$ along the unit circle

How do I evaluate the following integral? $$\oint_{C}\frac{e^z-1}{z}\mathrm dz$$ where $C$ is the unit circle (counter-clockwise). I have just learned Cauchy-Goursat's Theorem, but I cannot apply it ...
3
votes
1answer
98 views

Fourier transform of $H(x)\tanh(x)$

I would like to compute the Fourier transform of the product of $\tanh(x)$ and the Heaviside step function $H(x)$, i.e. $$\int_{-\infty}^{\infty} H(x)\tanh(x)e^{-ikx}dx = \int_{0}^{\infty} \tanh(x)e^{...
0
votes
2answers
27 views

Contour Integral from Laurent Series

Disclaimer: I'm a Physics student, not a Maths student. I'm working on some problems involving Laurent series and the Residue theorem, and I've come across something I can't quite get my head around. ...
0
votes
1answer
26 views

Complex Integral $\int_{C(0,2)} \frac{e^z}{i\pi -2z}$

Evaluate: $\int_{C(0,2)} \frac{e^z}{i \pi -2z}dz$ So using the Cauchy Integral Formula: $\int_{C} \frac{f(z)}{z-z_0} = 2\pi i f(z_0)$ If I define $f(z)= \frac{e^z}{-2}$ then the $z-z_0=\frac{-i\pi}{...
1
vote
1answer
35 views

Computing integral with contour

I am trying to evaluate the integral $\int_0^{\pi/2}\frac{1}{\sin^2 t+(\sin t)^{-2}}dt$. The first step I took was using symmetry to get $$\int_{0}^{\pi/2}\frac{1}{\sin^2 t+(\sin t)^{-2}}dt=\frac{1}{4}...
0
votes
0answers
14 views

What is the difference between the trace operator and the contour integral operator?

I noticed that the trace operator can be used to trace the boundary of a subspace of a spectrum. Some methods also use the contour integral in spectral analysis, however I am not sure if this is ...
1
vote
1answer
68 views

Show that $\int_{0}^{\infty}\frac{t^{\tau - 1}}{1+t}= \frac{\pi}{\sin(\pi \tau)}$, where $0<Re(\tau)<1$

I need to show that Mellin Transform of the function $\frac{1}{1+t}$ is $\frac{\pi}{\sin(\pi \tau)}$. So, by definition $(Mf)(\tau)=\int_{0}^{\infty}f(t)t^{\tau -1}dt=\int_{0}^{\infty}\frac{t^{\tau - ...
1
vote
1answer
45 views

Confusion using a complex line integral to find the area of a circle

I'm trying to understand the complex line integral equation as given in these notes. It's given as $$\int_{\gamma} f(z) dz = \int_a^b f(\gamma(t)) \gamma'(t) dt$$ where $\gamma(t)$ is a ...

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