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

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1
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1answer
65 views

Computation of a certain integral involving cyclotomics

How would one compute $\frac{1}{2\pi i }\oint_{|z| = \frac{1}{2}} \frac{\Phi_{n}(z)}{z^{k + 1}} dz$ in terms of k and n. If this is not possible, how would someone find a good approximation for this. ...
1
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5answers
116 views

Indented Path Integration

The goal is to show that $$\int_0^\infty \frac{x^{1/3}\log(x)}{x^2 + 1}dx = \frac{\pi^2}{6}$$ and that $$\int_0^\infty \frac{x^{1/3}}{x^2 + 1}dx = \frac{\pi}{\sqrt{3}}.$$ So, we start with the ...
1
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1answer
60 views

Contour integration in the complex plane gone wrong

Considering a function of complex variable $z$: $$f(z)=\frac{e^z}{z}$$ and a contour integral: $$\oint_C dz f(z)$$ such that the contour $C$ encircles the origin counterclockwise, it is clear from the ...
1
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1answer
257 views

Integrals involving Hermite Polynomials

Could you please tell me, How to evaluate this integral which involve hermite polynomials, $\int_{-\infty}^\infty e^{-ax^2}x^{2q}H_m(x)H_n(x)\,dx=?$ where $H_n$ is the $n$-th Hermite polynomial ...
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0answers
80 views

Computing the logarithmic derivative of the numerator and denominator of a rational function.

Consider the rational function $R(z)=N(z)/D(z)$ where $N(z)$ and $D(z)$ are polynomials of $z$ with real coefficients. Furthermore, $N(0) \neq 0$, $D(0) \neq 0$, and $N(z)$ and $D(z)$ are relatively ...
4
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2answers
81 views

Establish $\int_0^{\infty} \frac{x^a}{x^2 + b^2}dx = \frac{\pi b^{a-1}}{2 \cos(\pi a /2)}$ when $-1 < a < 1$

My attempt at a solution: (this is homework, btw) Let $f(z) = \frac{z^a}{z^2 + b^2}dz$ then the singularities of $f$ occur at $\pm ib$. $$ Res(f; ib) = \frac{z^a}{z + ib} \biggr |_{ib} = ...
3
votes
2answers
129 views

How to find this integral using Cauchy integral formula

How to obtain that $$\int\limits_{|z|=r} (\bar{z})^{-m} z^{-n-1}\, dz = \begin{cases} 2\pi ir^{-2m} &\text{if}\,\,n=m, \\ 0 &\text{if}\,\,n \neq m, \end{cases}$$ for $r>0$. I suppose I ...
3
votes
1answer
209 views

Analytic continuation of the Riemann zeta function using contour integration

To find the analytic continuation of the Riemann zeta function using contour integration one can integrate $\displaystyle f(z) = \frac{z^{s-1}}{e^{-z}-1}$ around a contour that consists of rays just ...
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0answers
30 views

Argue away the contribution from the contour integral

$$J(\lambda)=\int\frac{exp(i\lambda z^2)}{z-2-i}dz$$ Consider the contour integral above, consisting of a straight line C1 at $y=0$ between $1<x<r$, C2 given by $x=r$ between $0<y<r$ and ...
1
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2answers
177 views

Integral through Fourier Transform and Parseval's Identity

$$ \int_{-\infty}^{\infty}{\rm sinc}^{4}\left(\pi t\right)\,{\rm d}t\,. $$ Can you help me evaluate this integral with the help of Fourier Transform and Parseval Identity. I could not see how it is ...
0
votes
5answers
102 views

Complex Integration Problem. Please help.

Please help me with this one. Calculate the integral: $$\int_0^{2\pi} \frac{\mathrm{d}t}{a\cos t+b\sin t+c}$$ as $\sqrt{a^2+b^2}=1<c$. I'm working on it for quite a while and somehow I can't ...
2
votes
2answers
197 views

Complex Line Integral along the unit circle

Show that: $$\oint_\gamma\frac{1}{z}\left(z+\frac{1}{z}\right)^{2n}\mathrm{d}z=2\pi i\cdot\binom{2n}{n},\quad\text{while > }\gamma=\{z\in\mathbb{C}\,|\,|z|=1\}\,\,(\text{unit circle})$$ So ...
1
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1answer
68 views

Please help me with this Integral

Calculate the integral (complex): $$\oint_{D(0,1)}\overline ze^z \mathrm dz$$ While $D(0,1)$ is the unit circle.
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1answer
61 views

Work done by gravitational force

In my calculus class we learned about line integrals, and for homework we have exercise to find work done by gravitational force on material dot with mass $m$ which follows path of the elipse ...
2
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2answers
219 views

What are the reasons for using a semi-circle in upper half plane of $\mathbb{C}$ for contour integration?

Why is it that when one in considering contour integration of a real function, such as $$ \int_{-\infty}^{\infty} \frac{dx}{1+x^2}$$ the contour in the complex plane used is the following: ...
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2answers
50 views

Inverse Laplace of $\frac{s^3}{2+s^3}$

How I can find the Inverse Laplace of $\displaystyle \frac{s^3}{2+s^3}$ Thanks
4
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1answer
63 views

Show $\int_{\gamma}e^{iz}e^{-z^2}dz$ same value on every line parallel to $\mathbb{R}$

From an old qualifier: Show that $$\large\int_{\gamma}e^{iz}e^{-z^2}\mathrm dz$$ has the same value on every straight line path $\gamma$ parallel to the real axis. Justify the estimates involved. My ...
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0answers
52 views

Integration of rational function on Banach algebra

I do not follow the proof of this Theorem Theorem Suppose$R(\lambda) = P(\lambda) + \sum_{m,k}c_{m,k}(\lambda - \alpha_m)^{-k}$ is a rational function with poles at the points $\alpha_m$. ($P$ ...
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4answers
461 views

For which $L^p$ is $\pi=3.2$?

This is a silly question that came to mind after watching numberphile video on How Pi was nearly changed to 3.2. For which $p\in(1,+\infty)$ is the ratio of the perimeter of the $L^p$ disc in $\Bbb ...
1
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1answer
136 views

Evaluating series by contour integration, the residue theorem, and cotangent

I'm trying to understand this section in Tristan Needham's book Visual Complex Analysis about what he says is a standard method for evaluating series via a contour integral. My specific question is ...
5
votes
3answers
169 views

Applications of the Residue Theorem to the Evaluation of Integrals and Sums

Evaluate the integral $$\int_{-\infty}^{\infty} \frac{1}{(1 + x^2)^{n+1}} dx. $$ I know that it equals $2\pi i$(the sum of the residues; at $z_k$) where $z_k$ are the poles of the function. I ...
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0answers
76 views

Contour Integrals Complex Analysis

Evaluate $$\int_C\!\frac{\cos(z)}{z(z+2)}dz$$ where $C$ is the square with side $4$ centered at $z=0$ oriented clockwise.
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1answer
125 views

Compute $\int_{|z|=1}\frac{\log z}{z}dz$.

Here is a question about contour integration in complex analysis: Compute $$\int_{|z|=1}\frac{\log z}{z}dz$$ I am not sure if I understand the question since the logarithm must be defined in a ...
0
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0answers
136 views

Difficult Fourier transform

While looking at non-local modifications to wave propagation in 2d I have run into the following integral $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}d\omega dk \ln(k^2-\omega^2)e^{-i\omega ...
3
votes
1answer
41 views

Integration of exponential with square

It is known that $\int_\mathbb{R}e^{-tx^2}dx=\sqrt{\pi/t}$. What about $\int_\mathbb{R}e^{-t(x+ai)^2}dx$ for $a\in\mathbb{R}$? Is it still also $\sqrt{\pi/t}$? I can't simply change the variable ...
0
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1answer
52 views

Complex Analysis: Principle Part and Evaluating Integrals

I have two quick questions. Identify the pole in the following function and find the res of said function at it's pole. $(1)$\ $G(z)=\frac{\cos(z)}{\sin(z)}$ Here, ...
0
votes
1answer
169 views

Evaluate the integral $\int_{\gamma}\frac{z^2+2z}{z^2+4}dz$

Evaluate $$\int_{\gamma}\frac{z^2+2z}{z^2+4}dz$$ where the contour $\gamma$ is 1.) the circle of radius $2$ centered at $2i$, traversed once anti-clockwise. 2.) the unit circle centered at the ...
6
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2answers
114 views

Evaluate $\int\limits_0^\pi \frac{\sin^2x}{2-\cos x}\ \mathrm dx$ by complex methods

find integral $$\int\limits_0^\pi \frac{\sin^2x}{2-\cos x}dx$$ what I had in mind is to use Euler formula, to turn it into a complex integral and change the limits of integration from $ -\pi$ to ...
0
votes
1answer
102 views

Question regarding integral of the form $\oint\limits _{s_{R}}e^{i\cdot k\cdot z}f\left(z\right)dz$.

I came across the following (unproven) claim in the lecture notes of a course I'm reading. For the purpose of this claim given $a,R,R_{0}\in\mathbb{R}$ such that $R,R_{0}>0$ we mark $$E=\left\{ ...
135
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5answers
34k views

Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \ \mathrm dx$

I need help with this integral: $$I=\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right)\ \mathrm dx.$$ The integrand graph looks like this: $\hspace{1in}$ The ...
0
votes
1answer
55 views

Using Contour Integrals to Calculate Other Integrals

How would I show that $$\int_{0}^{\pi /2}\sin^{2n}(t)\, dt = \frac{\pi}{2} \frac{1(3)(5)...(2n-1)}{2(4)(6)(8)...(2n)}$$ using contour integration? My guess is that I would integrate along the unit ...
2
votes
2answers
70 views

Residue/Contour integration problem

Supposedly, $\displaystyle\int_{-\infty}^\infty \frac{\cos ax}{x^4+1}dx=\frac{\pi}{\sqrt{2}}e^{-a/\sqrt{2}}\left(\cos\frac{a}{\sqrt{2}}+\sin\frac{a}{\sqrt{2}}\right)$, $a>0$. Using ...
1
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1answer
43 views

Starting point of Schwarz-Christoffel integral

In the Schwarz-Christoffel mapping, we write the integral as $$F(z) = \int_{z_0}^z (s-x_1)^{-k_1}\dotsb(s-x_n)^{-k_n} ds$$ How do we choose $z_0$ in this case? On the wikipedia article, it writes ...
2
votes
1answer
73 views

How to approach/solve this integral?

Could somebody suggest how to approach or solve this integral: $$ \int_{0}^\infty e^{-a t}{2+t-2\sqrt{1+t}\over t^2}{\rm d\,}t, $$ where $a>0$ ? It is not a homework. I tried to use residuum ...
1
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1answer
149 views

Evaluate an Integral involving Gaussian divided by square root of a quartic polynomial

Could you please tell me, How to evaluate the integral, $\int_{-\infty}^\infty\int_{-\infty}^\infty\dfrac{e^{-a(x^2+y^2)}}{\sqrt{k^2+\beta^2(x^2-y^2)^2}}dx~dy$ I already have obtained a series ...
0
votes
2answers
65 views

How to compute a certain contour integral.

How can I compute the contour integral $ \displaystyle \oint_{C} \frac{1}{z^{3} + 9 z} \, d{z} $, where $ C := \{ z \in \mathbb{C} \,|\, |z| = 4 \} $ is the counterclockwise-oriented circle with ...
3
votes
2answers
50 views

find $\int _\gamma \frac{1}{z+\frac {1}{2}}dz$

I'm asked to find $$\int _\gamma \frac{1}{z+\frac {1}{2}}$$ where $\gamma (t)=e^{it}, 0\leq t\leq 2\pi$. To do this I deal with two different logarithms, one without the negative imaginary axis ...
0
votes
1answer
54 views

Complex integral square

Let $\alpha$ be the closed curve along the square with vertices at $1, i, -1, -i$. Give an explicit parametrization for $\alpha$ and calculate $$\frac{1}{2\pi i}\int_\alpha\frac{dz}{z}$$ I ...
6
votes
1answer
233 views

Extended Proof of the Theorem that a bounded analytic function is constant.

I am having trouble feeling convinced by my proof and more importantly - feeling confident in my working out. The question reads (a) Let $f$ be an entire function such that there exist real ...
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0answers
36 views

Function with poles/singularities; Polynomial approximant has no poles

I don't know if i should ask this question or if it makes too much sense. My knowledge of this topic is quite incomplete, so please bear through with me. Any insights are appreciated. A function ...
2
votes
2answers
154 views

Calculate the complex integral $\oint_{|z|=1}\sin{\frac{1}{z}} dz$

How do I calculate this complex integral? $$\displaystyle\oint_{|z|=1}\sin\left ({\displaystyle\frac{1}{z}}\right ) dz$$ I made the Taylor series for this: $$\displaystyle\sum_{n=0}^\infty ...
2
votes
1answer
215 views

Solving integral using contour integration

I am trying to show that $I = \int_0^{+\infty} \frac{x^a}{x^2 + 1} = \frac{\pi / 2} {\cos \frac{\pi a}{2}}$ provided $-1 < a < 1$. So I consider $I = \int_{C_R}\frac{z^a}{z^2 + 1}$ where $C_R$ ...
0
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1answer
92 views

Fourier transform of $\exp(-t^2)$ using contour integration.

I am calculating the Fourier transform of $\exp(-t^2)$ using contour integration. I am left with the integral $\int_{-\infty}^\infty \exp(i\omega t)\exp(-t^2)$. Usually I would now use the residue ...
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0answers
38 views

The part I dont understand while calculating contour integral.

Question is the following; $$\int_{0}^{\infty}[x^{m-1}/(1+x^n)]dx$$ for $m,n=1,2,\dots$ and $n>m>0$ Solution: its poles were found as this $$a_k=e^{i(2k+1)\pi/n}$$ for $k=0,1,...(n-1)$ And ...
2
votes
2answers
52 views

Residues of Complex Functions

I need to find the residues of $f$ at the isolated singular points, namely $z=1,z=0$. Where $f(z)=\dfrac{2z+1}{z(z+1)}$. I already have that the residue at $z=0$ is $1$, and I know I need to do ...
5
votes
1answer
131 views

Help with identity in complex contour

I am dealing with an integral in the complex plane, in particular I want to transform $$ \int_{0}^{2\pi} \frac{d\phi}{\sqrt{1+b^2 -2b \cos \phi}} $$ into another integral near the branch cuts for the ...
0
votes
0answers
33 views

integral of $(x+ia)^{-0.5+ib}(x+ic)^{-0.5-ib}$?

How can I find $$\int_0^{\infty}(x+ia)^{-0.5+ib}(x+ic)^{-0.5-ib}dx,$$ where $a$, $b$ and $c$ are real constants? This looks like it should be doable using the residue theorem on e.g. a keyhole ...
1
vote
0answers
71 views

Prove two equalities about the Cauchy projectors

Let $P_\pm$ be the Cauchy projectors defined by their action on $C^1(S^1)$ function $f$, where $S^1 = \{ \zeta \in \mathbb C \mid |z| = 1 \}$, by the rule $$ P_\pm f(z) = \lim\limits_{\varepsilon ...
14
votes
1answer
307 views

Further our knowledge of a certain class of integral involving logarithms.

$\newcommand{\limitp}{\alpha}\newcommand{\innerp}{\beta}$I am fascinated by definite integrals. Exploring math.stackexchange, I have found many interesting integrals of the form $$ ...
1
vote
1answer
82 views

Difficulties choosing the right contours; e.g. $\int_{0}^{\infty}\frac{x^a}{1+x^2} dx$, with $a\in \mathbb{R}$ and $0<|a|<1$

I'm often stuck at choosing the right contour to integrate along. Are there some general rules of thumb to choose a suitable one? For example, I got stuck at this one: $ \displaystyle ...