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

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2
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2answers
25 views

Contour integral qualitative behavior

consider a holomorphic function $f(z)$ and the paths $\gamma_1:(0,\pi)\rightarrow \mathbb{C}, t\mapsto r\cdot i\cdot e^{i t}$, $\gamma_2:(0,\pi)\rightarrow \mathbb{C}, t\mapsto r\cdot i\cdot e^{i (-t)}...
0
votes
0answers
27 views

Bounding the integral of the reciprocal of a complex polynomial

So, I would like to bound $\int_{C_R} \frac{1}{P(z)} dz$ where $C_R$ is the circle radius R centred at the origin, and $P(z)$ is a polynomial of degree $N=0,1,...,n$ i also want to deduce for what ...
1
vote
1answer
39 views

Developing a Process for Contour Integration

I am working on developing a personal process to follow (i.e. generalize, as much as is possible) for contour integration. I have seen the following things happen in worked examples and I am not sure ...
2
votes
1answer
66 views

Show that the Hankel type contour integral $\int_{-\infty}^{0^+}\frac{\mathrm{Log}(t)}{e^{-t}-1}dt=0$

Show that $\int_{-\infty}^{0^+}\frac{\mathrm{Log}(t)}{e^{-t}-1}dt=0$ where the integral is over a contour of the Hankel-type. What I mean is that the contour looks like this but reflected across the ...
0
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0answers
29 views

Calculating complex integral over contour

I'd like to calculate $\int_{\phi}^{} \frac{z^2}{z^3-1}dz$ where $$\phi(0)=i,\phi(1)=-1, \phi: [0,1] \rightarrow C$$ and $\phi((0,1)) \subset \left\{ z:|z|<1\right\}$ It seems pretty easy and ...
1
vote
0answers
21 views

Interchanging Limit and Integral sign

I'm reading a book on composition operators, and the author makes the following claim: Given a self-map of the unit disc, and a $H^2$ function $f$, where $H^2$ is the Hardy space, if we fix a radius $...
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0answers
21 views

complex line integral for function of several variables

In complex analysis of one variable, we studied a property of line integral that: let $f$ be holomorphic on open set $U\subset\mathbb{C}$ then for $z,z_0\in U$ we have $$f(z)-f(z_0)=\int_0^1 \frac{\...
0
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0answers
30 views

Generalization of a usual complex analysis fact

Let $f$ be a continuous function on $\mathbb{C}$ and assume that $\lim_{z\to \infty} zf(z) = \lambda.$ Let us note for all natural $n$ $$C_n = \{z \in \mathbb{C} : |z|=n\}.$$ Then, a usual fact of ...
1
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0answers
34 views

Definite integral with exponential and algebraic functions

I came across definte integral: $I(a, b) = \int_{a-b}^{a+b} \frac{1}{e^x -1} \frac{1}{\sqrt{1-(x-a)^2/b^2}} ~\mathrm{d}x $ Mathematica was not able to guide a closed form solution, but I am hoping ...
1
vote
1answer
36 views

Does contour integral over $\mathbb{R}$ give a step function?

I have the following integral $$ \int \frac{dx}{2\pi i} \frac{1}{(x+ia)^2+b^2} $$ where $x$ is a real variable and we integrate in the real axis from $-\infty$ to $+\infty$. I am also given the fact ...
0
votes
1answer
37 views

Computing using residue $\int_{0}^{\infty}e^{-x^{2}}\cos(x^{2})\mathrm{d}x$ (but not Gaussian way)

I am wondering if there is a residue-trick for computing $\int_{0}^{\infty}e^{-x^{2}}\cos(x^{2})\mathrm{d}x$ without having to go through computing the Gaussian residue integral. For practice here ...
0
votes
0answers
26 views

Complex integration with varying degrees

So I'm studying for an exam and going over past exams and one problem is causing me a little difficulty. Any help would be appreciated. The problem is: Let $0 \leq p < n \in \mathbb{Z}$. Calculate ...
0
votes
1answer
30 views

contour integrals complex

Hi I'm having trouble with the following integral $$ \int_C\left( \frac{sinz}{z+3-i}+\frac{e^z + z^2 - 1}{(z+1)^2} \right)dz $$ Where $ C: |z| = 2$ This is what I have so far. $$ z + 3 - i = 0$$ $$...
1
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1answer
32 views

Evaluate Contour Integral

I have provided my solution below, a confirmation on my solution would be appreciated, thanks in advance.
1
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1answer
68 views

Definite integral with trigononmetric functions

I have arrived at definite integral with trigonometric functions $I(a, b) = \int_0^{2 \pi} \frac{1 - a \sin(\theta) - b \cos(\theta)}{(1 +a^2 +b^2 - 2 a \sin(\theta) - 2 b \cos(\theta) )^{3/2}} \...
-2
votes
1answer
38 views

Integrating along a branch cut

What contour would one use to integrate the following equation? $\int_{0}^{\infty}\frac{x^a}{(x^2+1)^2}dx$ where $-1 < a <3 $ and $x^a= e^{aln(x)}$
2
votes
1answer
50 views

Contour integral of $f(z) = \frac{1}{z^2+iz+6} $

Need help evaluating a certain contour integral. $f(z) = \frac{1}{z^2+iz+6} $ Steps so far: Poles: $ z^2+iz+6 \rightarrow \frac{-i \pm \sqrt{-1-24}}{2}=0 \rightarrow z_0 = +2i, -3i $ Residues: ...
1
vote
0answers
20 views

Counting zeros of a function

I have defined the following function: $f(z)=\sin^2(π\frac{\Gamma(x)+1}{x})+\sin^2(πx)$ I want to count its zeros along the positive real axis up to a point, call it $x=A$. By the Cauchy ...
2
votes
0answers
139 views

complex contour integration

I have gotten stuck on this question: $$f(l,q)=\int_{-\pi}^{\pi} \frac{e^{-i l \theta}}{1-q \cos \theta} d\theta$$ where l is an integer and q is a complex number with |q|<1 I am supposed to ...
0
votes
1answer
43 views

Finding the path over which to compute a complex line integral when converted from a real integral

I have the integral $$\int_0^{2\pi}\frac{1}{5+3\cos t}dt.$$ And I want to convert this to a complex line integral. My idea was to use $\cos t=(e^{it}+e^{-it})/2$, but what is the path $a$ over which ...
1
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0answers
35 views

general procedure for contour integration of $\int_{0}^{\infty} \mathrm{Ai}(x)^{n} dx$

In Richard Crandall's On The Quantum Zeta Function, following eq. 4.11: $$ \int_{0}^{\infty}\mathrm{Ai}(x)^{2}dx=\frac{\Gamma(\tfrac{5}{6})}{2\pi^{5/6}12^{1/6}} $$ “again derivable by contour ...
1
vote
2answers
53 views

Evaluate the integral using the theory of residues: $\int_{-\infty}^{\infty} \frac{dx}{(x^2+1)^2(x^2+16)}$

$$\int_{-\infty}^{\infty} \frac{dx}{(x^2+1)^2(x^2+16)}$$ My attempt: We are integrating over the real axis, which is the real part of a set of complex numbers, so $$=\int_{-\infty}^{\infty} \frac{dz}...
5
votes
1answer
73 views

Evaluate the integral $\int_0^{2\pi} \frac{d \theta}{5-3 \cos \theta}$

$$\int_0^{2\pi} \frac{d \theta}{5-3 \cos \theta}$$ My attempt: Let $z=e^{i\theta}$ which gives $d\theta = \frac{dz}{iz}$ Thus, $$\oint_C \frac{1}{5-3(\frac{z+z^{-1}}{2})}\frac{dz}{iz}$$ $$=\frac{...
2
votes
1answer
86 views

Evaluate the integral using the theory of residues: $\int_0^{2\pi} \frac{(\cos \theta)^2 d \theta}{3-\sin \theta}$

$$\int_0^{2\pi} \frac{(\cos \theta)^2 d \theta}{3-\sin \theta}$$ I''m having trouble simpliyfing this into a form that will allow me to use Residue Theorem. I got it to the point where the integrand ...
0
votes
1answer
48 views

Cauchy's Integral Formula for calculating complex line integrals

I have the integral $$\int_a\frac{1}{(z-1)(z-3)}dz$$ with the paths $a:[0,2\pi]\rightarrow\mathbb{C},x\mapsto2e^{it}$, $a:[0,2\pi]\rightarrow\mathbb{C},x\mapsto2005+73e^{it}$. My ...
1
vote
1answer
17 views

Estimating derivative for an entire function

Suppose that $f(z)$ is an entire function such that $f(z)/z^n$ is bounded for $|z|\geq R$. SHow that $f(z)$ is a polynomial of degree at most $n$. As $f(z)$ is an entire function i thought of ...
0
votes
1answer
30 views

Parametrizing a complex path

So I have the set $a=\{x+iy|y=x^3-3x^2+4x-1\}$ that connects $1+i$ and $2+3i$. How do I parametrize a complex path of this? Eventually I want to find $\int_a(12z^2-4iz)dz$ and it seemed to me ...
0
votes
1answer
55 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 ...
1
vote
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
votes
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
vote
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 ...
1
vote
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
votes
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(...
0
votes
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 \...
1
vote
2answers
52 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} ...
1
vote
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 $z(0)=0$,...
0
votes
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=...
0
votes
1answer
43 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 $t$ ...
0
votes
0answers
24 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
votes
1answer
43 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
54 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 ...
0
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2answers
50 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 ...
2
votes
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: $$\...
0
votes
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 $\pm2$....
1
vote
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}...
8
votes
4answers
169 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 ...
2
votes
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 ...
0
votes
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 ...
0
votes
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 $\...
0
votes
2answers
54 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 ...