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

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42 views

Integrating $\int_{-\infty}^{\infty}\frac{e^{ax}}{1+e^x}dx$ for $0<a<1$.

I'm supposed to find the value of $\int_{-\infty}^{\infty}\frac{e^{ax}}{1+e^x}dx$ for $0<a<1$. I wanted to integrate over the upper semicircle of radius $R$, and take the limit as $R\to\infty$. ...
3
votes
3answers
270 views

Evaluating an improper integral that involves $\exp(-|x|)$

I am trying to prove that the function $f:\mathbb C\setminus\mathbb R\rightarrow\mathbb C$ defined by $$ f(z) := \frac{1}{2\pi i}\int_{-\infty}^\infty\frac{\exp(-|x|)}{x-z}dx $$ is holomorphic. I ...
6
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1answer
158 views
+50

Another beta integral due to Cauchy.

I have the following identity which I want to prove: $$C(x,y):= \int_{-\infty}^{\infty} \frac{dt}{(1+it)^x(1-it)^y} = \frac{\pi \cdot 2^{2-x-y}\Gamma(x+y-1)}{\Gamma(x)\Gamma(y)}$$ where ...
0
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0answers
7 views

Deformation of Gamma function integral contour

Terence Tao has described the gamma function as the inner product of a multiplicative and an additive character with respect to the Haar measure on $\Bbb R^+$. The gamma function is defined as ...
0
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0answers
30 views

Contour integral of continuous but not holomorphic functions

This question was transferred here following Mathoverflow suggestions. Let us consider two functions $f(z)$ and $g(z)$, both holomorphic on a domain $U$ (a simply connected subset of $\mathbb{C}$). ...
6
votes
3answers
84 views

Calculate $\int_0^{\infty}\frac{1}{(x+1)(x-2)}dx$ using residues

I'm supposed to calculate $$\int_0^{\infty}\frac{1}{(x+1)(x-2)}dx$$ using residues. The typical procedure on a problem like this would be to integrate a contour going around an upper-half ...
0
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3answers
63 views

Contour Integral of $\int\limits_0^{2\pi}\frac{d\theta}{1+a\cos\theta}$ for $a^2<1$ (textbook wrong?)

My book is telling me that the answer is $\frac{2\pi}{\sqrt{1-a^2}}$. I'm getting an extra a on the numerator. Could somebody verify if I'm wrong, or if it's my book (it has been wrong numerous ...
0
votes
1answer
69 views

Schwarz Function of an Ellipse

I want to find the Schwarz function of the ellipse define by $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad a > b > 0. $$ To do so, substitute $$ x = \frac{z+\bar{z}}{2}, \quad y = \frac{z - ...
1
vote
0answers
51 views

Question about integral over $\cos^3(\theta)$ on complex plane

I had an integral of $$\int_{0}^{2\pi}\cos^3(\theta) d\theta$$ The answer came out to be integral over the curve $$\int_{C} \dfrac{(z^2+1)^3}{8iz^4} dz$$ $$=-i* \int_{C}\dfrac{(z^2+1)^3}{8z^4} ...
0
votes
1answer
26 views

Prove the integral is always imaginary

Show that if f is analytic on D and γ is a closed curve in the region then the integral $$\int \overline{f(z)}f'(z)$$ is purely imaginary. I think this problem would use some extension of cauchy ...
2
votes
0answers
33 views

$\frac{\omega}{2\pi}\int_0^{\frac{2\pi}{\omega}}\frac{\sin^2\theta \cos^2\omega t}{(1+\beta\cos\theta \sin\omega t)^5} dt$

I'm going to write out the whole problem as it is given to me (bad grammar and all) even though some of the info may be irrelevant to finding a solution. A charge $e$ moving along a straight line ...
0
votes
1answer
30 views

Cauchy integral formula with singularities

I am stuck on this question. $$\int \frac{e^{sin z^{2}}}{(z^{2}+1)(z-2i)^{3}}dz $$ along the path γ where γ is a circle centered at the origin of radius different from 1 or 2. I initially thought of ...
1
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0answers
17 views

Find volume and surface of a body [closed]

Find volume and surface of a body $$G=\{(x,y,z)\in\mathbb R^3|2x\le x^2+y^2\le1,-\sqrt{ x^2+y^2}\le z\le4- x^2-y^2, y\ge0\}$$ Some directions will be helpfull.
1
vote
1answer
12 views

Big-O Notation?

The problem is to to evaluate the following contour integral along a path $C$ defined/parameterized as $z(t)=εe^{it}$: $ \int_C \frac{e^{iz}}{z} dz$ The solution for the problem proceeds to say ...
0
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0answers
26 views

Evaluation of complex integral?

I'd like to verify the result of this integral, or find if I've made a mistake. In the following, $\mathbf x, \mathbf a, \mathbf b$ are all real vectors in $\mathrm R^3$. I do the following: group ...
0
votes
1answer
9 views

$x^2+y^2\le 1$; $z=\sqrt{x^2+y^2}$; and $x^2+y^2=4-z$

I need to find a value and "surface" of a body which is contained in the following contours: $x^2+y^2\le 1$; $z=\sqrt{x^2+y^2}$; and $x^2+y^2=4-z$. Some hints and directions will be helpful. Sorry for ...
2
votes
1answer
35 views

Modulus of roots of polynomial tend to infinity

Define $f_n:\mathbb{C}\to\mathbb{C}$ and $(\alpha_n)$ such that:$$f_n(z)=\sum_{k=0}^n \frac{z^k}{k!}$$ and $f_n(\alpha_n)=0$. Prove $|\alpha_n|\to\infty$ as $n\to\infty$. I guess this makes sense ...
1
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3answers
47 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 ...
2
votes
2answers
43 views

Contour Integration of $\sin^2(x)/(1+x^2)$

How should I calculate this integral $$\int\limits_{-\infty}^\infty\frac{\sin^2x}{(1+x^2)}\,dx\quad?$$ I have tried forming an indented semicircle in the upper half complex plane using the residue ...
0
votes
2answers
44 views

Basic contour integration

I am trying to evaluate $$\int_{\gamma(0;2)}\frac {e^{i\pi z/2}}{z^2-1}\, \mathrm d z$$ using the Cauchy integral formula The problem is it is not holomorphic at $1$ and $-1$. My textbook suggests ...
0
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1answer
29 views

Contour Integral over a Closed Circle (Complex Analysis)

I'm having trouble understanding the difference, other than notation, between a contour integral over an open curve and a contour integral over a closed curve. So far, it seems to me that the ...
2
votes
1answer
51 views

What do $\int_{-1}^1\frac{dx}{2x+1-2i}$ and $\frac12\log(2x+1-2i)$ mean?

Suppose we want to evaluate $$I=\oint_C\frac{dz}{z+\frac12}$$ where $C$ is the unit square with diagonal corners at $-1-i$ and $1+i$. If we let $z:=re^{it}-\frac12$, then ...
3
votes
3answers
43 views

Improper integral with complex limits

I would like to compute an integral of the form ($a,b \neq 0$) $$\int_{-\infty}^{\infty} e^{-(ax+ib)^2} dx = \frac{1}{a} \int_{-\infty+ib}^{\infty+ib} e^{-z^2} dz$$ where we made the substitution $z ...
2
votes
1answer
54 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
votes
0answers
9 views

To get boundary of xy coordinates around object inside an image using active contour method

Trying to extract the XY coordinates around the boundary of an lesion object inside an image of plain background. I am using the active contour method as suggested to find x and y coordinates along ...
2
votes
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 ...
1
vote
1answer
33 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 ...
0
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0answers
21 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 ...
-4
votes
1answer
56 views

Complex Integrals (No Residue allowed)

Complete the integrals along curve $C$ a) $\displaystyle\int_C\frac1z\ \mathrm dz$ b) $\displaystyle\int_Cf(z)\ \mathrm dz;\quad[f(a)]^2=z\ \&\ f(1)=1$ c) ...
0
votes
0answers
25 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
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0answers
19 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 ...
0
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0answers
19 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 ...
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0answers
26 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
30 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
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1answer
35 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
34 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
46 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 ...
0
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1answer
27 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. $$ z + 3 - i = 0$$ ...
0
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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 ...
6
votes
5answers
462 views

Evaluate $\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$ using complex analysis

How do I compute $$\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$$ What I am doing is take $$f(z)=\frac{(\log z)^2}{1+z^2}$$ and calculating $\text{Res}(f,z=i) = \dfrac{d}{dz} \dfrac{(\log ...
1
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1answer
30 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
63 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}} ...
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1answer
35 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
48 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
votes
0answers
13 views

Looking for examples of interesting contours chosen in contour integrals

As the question states, I am looking for questions that have been answered by picking some rather unique looking contours. Some I have found are: ...
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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 ...
4
votes
3answers
651 views

Improper Integral of $x^2/\cosh(x)$

I need to compute the improper integral $$ \int_{-\infty}^{\infty}{\frac{x^{2}}{\cosh\left(x\right)}\,{\rm d}x} $$ using contour integration and possibly principal values. Trying to approach this as ...
2
votes
0answers
131 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 ...
4
votes
3answers
186 views

Evaluate improper integral $\int_0^\infty \frac{x\sin x}{x^2+1}dx$

How to prove that $$\int_0^\infty \frac{x\sin x}{x^2+1}dx=\frac{\pi}{2e}$$ I've tried several basic approaches like substitution and IBP but can't move forward.