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

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1answer
56 views

Cauchy's Integral Formula - clarification on permissible closed curves

The Cauchy Integral Formula that I am working with says: Suppose that $f:E \rightarrow \mathbb{C}$ is holomorphic, $E$ is an open subset of $\mathbb{C}$, and $z_0 \in E$. Pick $\rho > 0$ such ...
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1answer
40 views

Does the analytical form of the following integral exist?

I have an integral $$\int_0^{2\pi}d\theta\cos(2\theta)e^{-a[1-\cos(\theta-\theta_0)]}.$$ Is there any analytical form for the integral above?
2
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1answer
41 views

What is meant by the “contour of a function?”

Suppose that we have $f(x,y)=(x+y)^2.$ What is meant by the "contour of a function," and what is an analytic expression for it? All software, such as Matlab, Mathematica,.. gives just a function like ...
1
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1answer
26 views

Use a rectangular contour to evaluate the integral

$$\int_{-\infty}^{\infty} \frac{\cos(mx) dx}{e^{-x}+e^x} = \frac{\pi}{e^{m\pi /2}+e^{-m\pi /2}}$$ I need to evaluate the above integral specifically using a rectangluar contour and at some point ...
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2answers
21 views

Piecewise continuous contours with discontinuity only at end points

Let $w(t)=u(t)+iv(t)$ where $a \leq t \leq b$ be a complex valued function on real variable $t$. For integrating $w(t)$ from $a$ to $b$ we require that $u(t) $ and $v(t)$ must be piecewise continuous....
2
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1answer
72 views

Evaluating the integral $\int_{-\infty}^{\infty} \frac{(1-ix)^{n-s_1}}{(1+ix)^{n+s_2}} dx$.

How can one show that for $s_1,s_2 \in \mathbb{C}$ $$ \begin{aligned} \int_{-\infty}^{\infty} \frac{(1-ix)^{n-s_1}}{(1+ix)^{n+s_2}} dx = & 2\sin(\pi(n+s_2))2^{1-s_1-s_2} \Gamma(1-n-s_2)\Gamma(...
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1answer
28 views

Estimating integral $\int\limits_{C}\frac{z^3}{z^2-1}\text{d}z$

Using the estimation lemma show that $$\left|\int\limits_C \frac{z^3}{z^2-1}\text{d}z\right|\le \frac{9}{8}\pi$$ where $C:\{z:|z|=3,\Re(z)\ge 0\}$. The length of $C$ is $\pi$ and $\displaystyle \...
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2answers
53 views

Prove $\int_{-\infty}^{\infty} \frac{dx}{(1+x^2)^{n+1}} = \frac{(1)(3)(5)…(2n-1)}{(2)(4)(6)…(2n)} \pi \ \ \ \forall n \in \mathbb{N}$

My attempt starts with a contour integral in the half disk, I let the radius -> infinity and so the contour integral \begin{equation} \int_{\gamma} \frac{dz}{(1+z^2)^{n+1}} = 2 \pi i \ res_{z_0 = i} ...
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1answer
35 views

Confusion over complex integral along a path

Compute $$I:=\int_C\frac{z^9}{5}dz,$$ where $C$ is the curve $z(t)=\sin t+i\sin10t$, $0\leq t\leq\pi/2$. Would the answer be: $I=\int^{z(\pi/2)}_{z(0)}\frac{z^9}{5}dz$ where $z(\pi/2)=1$ and $z(0)=0$,...
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1answer
57 views

Using contour integration to solve this integral

We need to use contour integration to solve $$\int_{-\infty}^\infty {e^{ax}\over e^x+1}dx$$ given that $0<a<1$. My question is about what contour to use, knowing that the singularities are at $z=...
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1answer
44 views

How to compute this integral with contour integration?

Consider the function $$g(z)=\dfrac{e^{izt}\phi(z)}{z},$$ where $\phi$ is a $C^\infty$ function. I want to compute the integral $$I=\int_{-\infty}^{\infty}\dfrac{e^{ixt}\phi(x)}{x}dx,$$ where $t$ ...
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0answers
29 views

Contour and perimeter recognition in binary image

I need to detect contour (object) and find the perimeter of a detected object. For example, I have the following image: http://i.stack.imgur.com/40TTX.png All images are binary, so they consist of ...
0
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1answer
45 views

Derive the Fourier Transform

I have been asked to derive the Fourier Transform for $$f(x)=\frac{1}{x^2+a^2}$$ where $a>0$. I know the Fourier Transform is equal to $$\hat{f}(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\...
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0answers
56 views

What is Complex Analysis? Why is it accompanied by Linear Algebra?

I hope this doesn't extend to a lengthy question. I studied Linear Algebra recently in my first term at university. I came to the realization however that some institutions would teach that course ...
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2answers
56 views

Evaluate using complex integration: $\int_{-\infty}^\infty \frac{dx}{(x^2+1)(x^2+9)}$

Evaluate $$\int_{-\infty}^\infty \frac{dx}{(x^2+1)(x^2+9)}$$ Firsly I found the residues of this function: $$Res(i)=-i/16$$ $$Res(-i)=i/16$$ $$Res(3i)=i/48$$ $$Res(-3i)=-i/48$$ I then closed ...
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1answer
40 views

Complex analysis - path integrals

I need to evaluate the following. $$\int_\gamma f(z) \: \text{d}z = \int_\gamma z^3+\cosh z \: \text{d}z$$ where $\gamma(t)=t^2+2it$ for $0\leq t \leq 1$. At first, I used the standard approach: $$\...
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1answer
43 views

Use Cauchy integral formula to determine an integral

I am trying to use CIF to solve $$\int_\gamma (z^2-4)^{-1} dz$$ where $\gamma$ is the unit circle traversed once in the positive direction. If I let $f(z) = z^2-4$, then $f$ is not analytic at $\pm2$....
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3answers
54 views

Contour integral of $\frac{x^{p-1}}{1+x}$

I am trying to find the integral $$\int_0^\infty\frac{x^{p-1}}{1+x}\;\mathbb{d}x$$ I know that this is easily expressible in terms of beta function. But i need to prove that it's value is $\dfrac{\pi}...
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4answers
171 views

Finding the integral $I=\int_0^1{x^{-2/3}(1-x)^{-1/3}}dx$

I have to find the following integral using contour integration without using information obtained from the Beta function: $$I=\int_0^1{x^{\frac{-2}{3}}(1-x)^\frac{-1}{3}}dx$$ I can change this ...
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1answer
39 views

Contour integral - $\int_C \frac{\log z}{z-z_0} dz$ - Contradiction

Let the domain $O=\mathbb{C}-(-\infty,0)$, the point $z_0 \in O$ and the circle $\gamma=C(0,r<|z_0|)$ in the positive direction. Compute $\int_C \frac{\log z}{z-z_0} dz$. The answer that my ...
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1answer
30 views

Contour integral around square using a parameterization with symmetry

A question asks to solve the integral $\int_{\gamma} \frac{1}{z \bar{z}} dz$, where $\gamma \subset \mathbb{C}$ is a square centered at the origin with sides parallel to the axes. Solution: Since the ...
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1answer
26 views

Contour integral - Use an example to contredict an answer

The answer that my teacher gives us is $2πi \log z_0$. I know that this answer is false according to the question $\int_C \frac{\log z}{z-z_0} dz$ - Cauchy theorem with $z_0$ outside the interior of $\...
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2answers
55 views

Deriving Cauchy integral formula

Within in the proof of Cauchy Integral Formula there is this line $$f^{(k)}(z)=\frac{k!}{2 \pi i} \int_\Im \frac{f(\zeta)}{(\zeta - z)^{k+1}} d\zeta \quad (k=1,2,3,...)$$ My goal is to derive this ...
2
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1answer
63 views

Contour integral on the function $\frac{\log z}{z-z_0}$ [duplicate]

Let the domain $O=\mathbb{C}-(-\infty,0)$, the point $z_0 \in O$ and the circle $\gamma=C(0,r<|z_0|)$ in the positive direction. Compute $\int_C \frac{\log z}{z-z_0} dz$. So far we didn't see the ...
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1answer
60 views

Modified Bessel Function Integral representation proof $K_{\nu}(z)=\frac{z^{\nu}}{2^{\nu+1}}\int_{0}^{\infty}t^{-\nu-1}e^{-t-z^{2}/4t}dt $

How do I proof the following integral representation for the Modified Bessel function of the second kind (Macdonald Function). $K_{\nu}(z)=\frac{z^{\nu}}{2^{\nu+1}}\int_{0}^{\infty}t^{-\nu-1}e^{-t-z^{...
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1answer
36 views

Definition of the domain - Cauchy theorem - Contour integration

In general, in that kind of question, there isn't a domain defined (I think). If I ask you to tell me the answer of the contour integral $\int_{\gamma}\frac{1}{z-2} dz$ where $\gamma$ is simply the ...
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0answers
88 views

Proof of Sophomore's Dream using Contour Integration

Sophomore's dream is a relatively common identity, that states $$ \int _0^1 x^{-x} dx = \sum_{n = 1}^\infty n^{-n}$$ The common proof is found using the series expansion for $ e^{- x \log x} $ and ...
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0answers
19 views

System of equations - Contour integral

A problem ask to find the constants $a$, $b$ and $c$ if $f(z)=az^2+bz+c$ and $\int_{\gamma}\frac{f(z)}{z}dz=2 \pi i$, $\int_{\gamma}\frac{f(z)}{z+1}dz=4 \pi i$ and $\int_{\gamma}\frac{f(z)}{z-1}dz=8 \...
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3answers
22 views

Inequality - Contour integral

I would like to solve the inequality $|\int_{\gamma} \frac{1}{z^2}dz|\leq 2$ where $\gamma$ is the line $[i,2+i]$. I thought about using the Cauchy theorem in closing the path between $i$ and $2+i$, ...
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1answer
27 views

Contour integral over the unit circle

I have to evaluate the contour integral $\int_{\gamma} \frac{\sum_{k=0}^m a_k z^k dz}{z^{n+1}} = \int_{\gamma} \frac{f(z)dz}{z^{n+1}}$ with $n \geq 0$ and $\gamma = C(0,1)$ counterclockwise oriented. ...
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1answer
41 views

Contour integral - Unit circle

I have to compute the contour integral $\int_{\gamma} \frac{(z-1)}{z(z+1)(z-2)}dz$ where $\gamma$ is (a) the circle $C(0;1)$, (b) the circle $C(0,\frac{3}{2})$ and (c) the rectangle of vertex $-2-i$, $...
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1answer
33 views

$r(\theta) e^{i\theta}$ - Parametrization of the square $\gamma$

In the question Contour integral - Circle instead of a square achille hui explains that $r(\theta) e^{i\theta}$ with $r(\theta)\min\left(\frac{1}{|\cos\theta|}, \frac{1}{|\sin\theta|}\right)$ is a ...
0
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1answer
61 views

Contour integral - Circle instead of a square

I would like to solve the integral $\int_{\gamma} \frac{1}{z \bar{z}} dz$. Here $\gamma \subset \mathbb{C}$ is a square centered at the origin and where his vertices are parallel to the axes. Could I ...
3
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1answer
38 views

Cauchy's theorem for contour integration

I have to compute $\int_C(z+\frac{1}{z})^{2n}\frac{1}{z}dz$, where $n \in \mathbb{N}$, and $C$ is the unit circle with positive orientation. So let $z(t)=\cos (t) + i \sin (t)$, with $-\pi \leq t <...
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1answer
80 views

Certain type of integrals $ \int_0^{\pi}d\theta\sin\theta \frac{1}{x+i\epsilon - \sqrt{y+z\cos\theta}}, $

I would like to do the following integral $$ \int_0^{\pi}d\theta\sin\theta \frac{1}{x+i\epsilon - \sqrt{y+z\cos\theta}}, $$ where the $i\epsilon$ has been added to avoid some possible divergencies. ...
3
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1answer
91 views

Use of residues to find I=$\int_0^\infty \frac{\sin^2(x)}{1+x^4} dx$

I'm working on the problem $$I=\int_0^\infty \frac{\sin^2(x)}{1+x^4} dx$$ I found 4 singularities and i would like to use the singularities in the 1st and 2nd quadrants to solve this integral; i.e. $...
2
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1answer
25 views

Is this dual behavior allowed in integration in Complex Analysis?

The integral $$I_1 = \int_{C} \bar zdz=4\pi i ,$$ when $C$ is the right-hand half $z=2e^{iθ},\ (-\dfrac{π}{2}≤θ≤\dfrac{π}{2})$, of the circle $|z| =2$ from $z =-2i$ to $z = 2i$. And, one also can ...
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3answers
75 views

Improper integral $\int _{0+0}^{1-0}\frac{dx}{\left(4-3x\right)\sqrt{x-x^2}}\:dx$

How do I solve this? $$\int _{0+0}^{1-0}\frac{dx}{\left(4-3x\right)\sqrt{x-x^2}}\:dx$$ I know it's a type 3 improper integral, and I'm having issues with these. I think that I need to write it as a ...
3
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1answer
53 views

How to evaluate the integral $\int_0^{2\pi}\mathrm{d}\theta e^{ia\cos(\theta-\theta_1)}\cos^2(\theta-\theta_2)$

I have an integral: $$\int_0^{2\pi}\mathrm{d}\theta e^{ia\cos(\theta-\theta_1)}\cos^2(\theta-\theta_2),$$ where $a, \theta_1$ and $\theta_2$ are reals. Any idea on how to evaluate this integral.
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2answers
146 views

Problem over a definite integral, which surely needs contour integration

During my Master Thesis work I came up with an integral which I am going to consider as a hard challenge. I have been trying for days to crack it, but still nothing. The integral is the following $$\...
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1answer
28 views

Complex: evaluating integral with residues

Having a bit of trouble here. Having this integral $$ \int_{0}^{\infty} \frac{dx}{(x^{2}+1)(x^{2}+4)^{2}} $$ I can tell it's even, so it has symmetry. Thus, $$ \frac{1}{2} \int_{-\infty}^{\infty} \...
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1answer
52 views

Prove that the following function has a branch cut

I am given with a function $$\zeta(z)=\int_{-\infty}^\infty\mathrm{d}x \frac{f(x)}{z-x}.$$ Any idea to prove that $\zeta(z)$ is discontinuous across real axis for $f(x)\neq 0$?
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2answers
86 views

Integrate $\int_{-\infty}^\infty\frac{e^{-ik\sqrt{x^2+a^2}}}{\sqrt{x^2+a^2}}dx$

I'm trying to evaluate the integral below for my research related to sound radiation. Assume $a$ is a positive constant. $$\int_{-\infty}^\infty\frac{e^{-ik\sqrt{x^2+a^2}}}{\sqrt{x^2+a^2}}dx$$ First,...
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0answers
48 views

How to solve a contour integral?

I am trying to solve the contour integral $$\frac{1}{2\pi i}\int_C\frac{e^{t}}{t(2at+x^2)}{\rm{exp}}\left(\frac{ax^2}{2(2at+x^2)}\right)\,{\rm d}t $$ where the path of integration $C$ starts at $-\...
5
votes
2answers
121 views

How to evaluate integral $\int_0^{\infty} e^{-x^2} \frac{\sin(a x)}{\sin(b x)} dx$?

I came across the following integral: $$\int_0^{\infty} e^{-x^2} \frac{\sin(a x)}{\sin(b x)} dx$$ while trying to calculate the inverse Laplace transform $$ L_p^{-1} \left[ \frac{\sinh(\alpha\sqrt{...
2
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1answer
90 views

evaluate $\int _0 ^\infty \frac{1-\cos(ax)}{x^2}dx$

Im trying to evaluate for a given $a\in \mathbb R$$$\int _0 ^\infty \frac{1-\cos(ax)}{x^2}dx$$ I have noticed that since $1-\cos(ax)$ is analytic in $\mathbb C$, the integral $$\int _{C} \frac{1-\cos(...
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0answers
63 views

Evaluation of an improper integral with complex exponential

Are there any convenient ways to calculate an integral of the form $$ \int_{-\infty}^\infty\frac{a_1 e^{j\omega\alpha}+a_2e^{j\omega\beta}}{1 + a_1a_2e^{j\omega\gamma}}d\omega$$ where $a_1,a_2,\alpha$,...
0
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1answer
59 views

Compute, for all integers $m,n$, $\int_{|z|=2}z^n(1-z^m)dz$

Compute, for all integers $m,n$, $\int_{|z|=2}z^n(1-z^m)dz$. Isn't $z^n-z^{mn}$ analytic, so the value is 0? And if it isn't analytic, letting $z(t)=2e^{it},t\in[0,2\pi]$ gives $$\int_{0}^{2\pi}(2^ne^...
1
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1answer
38 views

Compute $\int_{|r|=2}\frac{1}{z^2+1}dz$ [duplicate]

Compute $\int_{|r|=2}\frac{1}{z^2+1}dz$. Is there a special trick to solving this? I tried letting $z(t)=2e^{it},t\in[0,2\pi]$, which gave me $$\int_{|z|=2}\frac{1}{z^2+1}dz=\int_0^{2\pi}\frac{2ie^{...
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0answers
28 views

Contour integral $\int_{\gamma}e^{-t\mu}(-\mu)^{-m}\log\sqrt{-\mu}d\mu$

Let $m\in\mathbb{N},t>0$ how to compute the integral $\int_{\gamma}e^{-t\mu}(-\mu)^{-m}\log\sqrt{-\mu}d\mu$, where $\gamma$ is contour $\{|\arg(\mu+1)|=\pi/4\}$ transversed upward? Here are my ...