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

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Contour integral with trigonometric functions

Does the following integral, where $n$ is an integer, have an analytic solution? $I_1 = \int_{0}^{2 \pi} \frac{\sin(n x) \sin(x) \cos(x)}{\cos(x)^2 + a} \mathrm{d}x $ I tried writing $\sin(n x)$ as ...
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1answer
88 views

Evaluate the integral $\int_0^\infty \frac {x^{1/2} dx}{x^2 + 1}$ using method of residues

I am trying to evaluate the integral $\int_0^\infty \frac {x^{1/2} dx}{x^2 + 1}$ using method of residues. I can solve this very easily without the $x^{1/2}$ on top, but I do not know what to do when ...
3
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1answer
64 views

Show that $f(0) + f'(0)=\frac{1}{2\pi i} \int_C\frac{f(w)e^w}{w^2}dw$

Let f be holomorphic on a region containing the closed unit disk $D(0, 1)$ and C be the unit circle traversed anticlockwise. Let n be a positive integer. Show that $$f(0) + f'(0)=\frac{1}{2\pi i} ...
2
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1answer
81 views

Calculating convolution integral analytically

How can i compute convolution integral analytically, without using graphs. I hate using graphs, shiftings which are error prone. If this is possible can you explain what way i must follow? For ...
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2answers
80 views

Application of Green's Theorem leading to different solutions of area integral

I have an area integral problem over an irregular convex polygon. I use Green's Theorem to convert the area integral to a contour integral, and solve using standard methods. Green's Theorem says I ...
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3answers
108 views

Show that $\int_{0}^{+\infty} \frac{\sin(x)}{x(x^2+1)} dx = \frac{\pi}{2}\left(1-\frac{1}{e}\right) $

I'm trying to show that $$\int_{0}^{+\infty} \frac{\sin(x)}{x(x^2+1)} dx = \frac{\pi}{2}\left(1-\frac{1}{e}\right) $$ using Jordan's lemma and contour integration. MY ATTEMPT: The function in ...
2
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1answer
96 views

Show $\int_{0}^{\infty} \frac{x^{-z}}{(1 + x)^{2}} ~ \mathrm{d}{x} = \frac{\pi z}{sin(\pi z)}$

I need to solve the following integral: $$ I = \int_{0}^{\infty} \frac{x^{-z}}{(1 + x)^{2}} ~ \mathrm{d}{x}. $$ Wolfram Alpha gives the answer as $ \frac{\pi z}{sin(\pi z)}$, or equivalently, $\pi ...
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1answer
28 views

Limit of contour integrals over semi-circles

For each $r>0$ let $I(r)=\int_{\gamma} \frac{e^{iz}}{z}dz$ where $\gamma(t)=re^{it}, t \in [0,\pi]$. Show that $\lim_{r \to \infty} I(r)=0$ I used $\int_{\gamma} ...
2
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3answers
64 views

Computing the integral using cauchy's theorem

I need to integrate $$\oint _{|z|=1} \frac {\sin z}{z}\, dz$$ I write $\sin z=z-\frac{z^3}{3!}+\frac{z^5}{5!}-...$, then I get by dividing by $x$, the series $1-\frac{z^2}{3!}+\frac{z^4}{5!}-...$. I ...
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3answers
188 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.
3
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1answer
54 views

Determine the complex contour integral $\oint \limits_{C} \frac{2}{z^3+z}dz$ without using Residue Theorems

Without residue theory, determine $$\oint \limits_{C} \frac{2}{z^3+z}dz$$ if $C: \big|~z~-~\frac{i}{2}~\big|=1$ is positively oriented. We first find that our integrand has three distinct ...
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1answer
42 views

Show $\frac {1}{2\pi i}\int_\gamma\frac {f'(z)}{f(z)}$ is sum of poles and zeroes times their order

Let $f: B(a,r) \to {\Bbb C} \cup \{\infty\}$ be a meromorphic function. For $p$ a zero or pole of $f$, let $\mu(f,p)$ denote its order. Let $\gamma$ be a closed curve in $B(a,r)$. Then $$\frac {1}{2 ...
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1answer
70 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 - ...
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3answers
73 views

Integrate: $\int_0^{2\pi} \frac{d\theta}{1+\cos^2\theta}$

I'm trying to integrate this here fella: $\int_0^{2\pi} \frac{d\theta}{1+\cos^2\theta}$ from examples in Ablowitz I know that for $|A|^2>|B|^2$ and $A>0$, $\int_0^{2\pi} ...
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1answer
44 views

Complex Laurent Series and Contour Integral

Let $f(z) = \sin{(\frac{1}{z})}$, where $z \neq 0$. Find a Laurent Series expansion of $f$ around the annulus $D: 1< |z|<3$. Use the result to find $$\oint \limits_C ...
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1answer
65 views

Complex Integral Over an Ellipse

Suppose we had a function defined as $$ S(\zeta) = \zeta - \sqrt{\zeta^2 - c^2} $$ and we wish to evaluate $$ \oint_\gamma \frac{S(\zeta)}{z-\zeta}\, d\zeta, $$ where $\gamma$ is the (positively ...
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1answer
62 views

2 ways for $\int_{|z|=r} x^2dz$

As the title says, I have to compute $$\int_{|z|=r} x^2dz$$ for the circle traversed anti-clockwise in 2 different ways. If I use a parametrisation I quickly get to 0 - which might be wrong though ...
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1answer
30 views

An inequality from complex contour integral on page 81 of Stein and Shakarchi's Complex Analysis

On page 81 of Stein and Shakarchi's Complex Analysis, there is an inequality, \begin{equation} \left\lvert \int_{A_{R}} f \right\rvert \leq \int_{0}^{2\pi}\left\lvert\frac{e^{a(R+it)}}{1+e^{R+it}} ...
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1answer
56 views

How to Solve the Contour Integral $\int_{|z|=r}x^{2}dz$

I have been asked to compute the integral $$\int_{|z|=r}x^2dz$$ for the circle traversed anticlockwise, without parametrisation, by observing that $$x = ...
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1answer
34 views

Almost Gamma function with imaginary exponential: substitution/contour trick

In a physics course, this integral showed up: $\int_0^\infty dp\, p^2\, e^{ibp}\; , \quad b \in \mathbb{R}$ My prof proceeded with the seemingly insane substitution $x=-ib$ to yield $\Gamma(3)x^{-3} ...
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1answer
53 views

Evaluate $\int_{-1}^{1}\mathop{dx}\frac{1}{(x^2+a^2)^k}$?

My original question was what is the condition for the following integral to converge: $$\int_{-1}^{1}\mathop{dx}\left[\frac{1}{x^{2k}}-\frac{1}{(x^2+a^2)^k}\right].$$ I know $k>0$. By motivation ...
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0answers
91 views

On $\sum a^n \tan(n\theta)$

It is well known that $$\sum_{n=0}^{\infty} a^n \cos(n\theta) = \frac{1-a\cos(\theta)}{1-2a\cos(\theta)+a^2}$$ $$\sum_{n=0}^{\infty} a^n \sin(n\theta) = \frac{a\sin(\theta)}{1-2a\cos(\theta)+a^2}$$ ...
2
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2answers
73 views

Why doesn't this contour work for complex integration?

I am asked to integrate: $$\int_0^\infty \frac{1}{(1+x)x^{1/3}} dx$$ Complexification of this integral leads to: $$f(z)=\frac{1}{(1+z)z^{1/3}}$$ singularities: $z=0$ and $z=-1$. So I thought, let's ...
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0answers
28 views

Jordan's lemma for different exponential functions/integration directions

Okay, lets say we have this complex integral $\int_{-\infty}^{\infty}$ $\frac{e^{-iaz}}{(z+i)(z-i)}$ where a>0. If I were to specify a semicircular contour in the lower half of the real line going ...
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3answers
77 views

Conversion into contour integral and poles

Say I have this integral $\int_{-\infty}^\infty$ $\frac{x^2}{x^6+1}$ dx . Now I know that it has six poles according to this denominator which are the six roots for -1. The question is after I split ...
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1answer
40 views

How to show the existence of an entire function

I have been working on this problem for quite sometime. For part (i), I obtained the Taylor series for $4\sin(z) - \sin(4z)$. At $z = -\pi$, the Taylor series is: $4\sum_{n=0}^{n} \frac{(z + ...
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1answer
17 views

Evaluating a contour-integral.

Consider the ellipse $C$ given by $x^2 + y^2/4 = 1$. How to evaluate $$\int_C x^2 \, \nu(d(x,y))$$ where $\nu$ is the Lebesgue length measure on $C$? I am not sure if this can be computed like a ...
2
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2answers
66 views

What is the use of Dirichlet Integral? [closed]

How can I find the value of $$\large\int_0^\infty\left(\dfrac{\sin x}x\right)^5dx$$ using Contour Integrals? I attempted it using Integration by Parts and got the an got the answer. I have studied ...
3
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0answers
85 views

Contour Integral of Square root Function. Branch Cuts

I am doing a physics problem and have come across a contour integral that I just don't know how to solve. I do not have the complex analysis background and I am wondering if anyone can explain how to ...
3
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0answers
44 views

Can I split this integral to a sum over three contours?

I have the following integral $$ Z = \frac{1}{2\pi i} \int dx \, \frac{1}{(x-a_1)(x-a_2)(x-a_3)}\times \frac{1}{(x+\epsilon - a_1)(x + \epsilon - a_2)(x+ \epsilon - a_3)} $$ and this integral has ...
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0answers
60 views

Find the different values of this integral when all paths of integration are possible

This is the question.. I only know how to do the question from infinity to infinity.. enter image description here Find the different values of this integral when all paths of integration are ...
2
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1answer
76 views

How do I evaluate the following Integral

The integral is $$\int_{0}^{\infty}dx\frac{1}{\sqrt{1+a(1+x^2)^m+b(1+x^2)^{m-2}x^2}},$$ where $m, a$ and $b$ are real numbers such that the integral is definitely convergent. Any ideas on how to solve ...
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2answers
85 views

Why should I take this Contour for $I = \int_0^{\infty} \frac{dx}{1+x^3} $? (Analytic Continuation)

When discussing analytic continuation, my lecturer used the following example, $$ I = \int_0^{\infty} \frac{dx}{1+x^3} $$ I have in my notes that the contour was taken as below. I must admit I was ...
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0answers
35 views

Is this integral automatically zero?

If I integrate $\int e^{iz}\,dz$ for z complex, along the positive real line, then is the imaginary part of the integral $i\int \sin(x)\,dx$ automatically equal to zero (integration only along the ...
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0answers
22 views

Inverse Laplace transform of the form $F(s) = \frac{s^{m}}{(1+a \cdot s)^{n}(1+b \cdot s)^{h}}$

I am trying to solve the inverse Laplace transform of the form \begin{equation} F(s) = \frac{s^{m}}{(1+a \cdot s)^{n}(1+b \cdot s)^{h}} \end{equation} where, $a$ and $b$ are known constants, $m$, $n$, ...
3
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2answers
107 views

How do I evaluate the following integral $\int_{-\infty}^\infty\mathop{dx} \frac{x^n}{(x^2+a^2)^m}$?

I am interested in the following integral $$\int_{-\infty}^\infty\mathop{dx} \frac{x^n}{(x^2+a^2)^m},$$ given that $m>n/2$ (this is just what I wrote so that the integral converges. If this is not ...
2
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1answer
73 views

Use contour integration to compute the Fourier transform,

The problem statement is: Use contour integration to determine the Fourier transform, $\large \hat f(ξ)=∫_{-\infty}^{\infty}f(x)e^{−iξx}dx$, of $\large f(x)=\frac{1}{2−2x−x^2}$. Some issues that I ...
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33 views

In P.V. contour integration in complex analysis, using a wedge vs. a keyhole contour

When is it clearly better -- perhaps even necessary -- to use a keyhole contour, instead of a wedge contour? The wedge contour minimizes computation of residues, as we can choose it so that it ...
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0answers
31 views

show that $\int_{C_R} \frac{z e^{iz}}{1+z^2}dz$ tends to zero

Show that $\int_{C_R} \frac{z e^{iz}}{1+z^2}dz$ where $C_R$ is the half circle in the upper half plane with radius $R$ tends to $0$ as $R$ goes to infinity. My professor showed me something with ...
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2answers
73 views

Compute $\int _0^\infty\frac{x \sin x}{1+x^2}dx$ with the residue theorem

Compute $\int _0^\infty\frac{x \sin x}{1+x^2}dx$ with the residue theorem Ok so I have done a couple of these but I'm stuck on this one. I want to use $$ \int_0^\infty \frac{ze^{iz}}{1+z^2}dz $$ ...
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25 views

On the right half-plane, what is an upper bound for $\frac{1}{\log(z+2)}$?

I am trying to estimate some factors in my integrand in complex integration, and I think the upper bound for $\frac {1}{log(z+2)}$ on the semicircle in the right half plane is just $\frac ...
2
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2answers
65 views

Contour integral of $\log(z)/(z+a)^2$ around z=0

My question is primarily conceptual: Consider a function $f(z)$ which has a branch cut from $z=0$ to $z=\infty$ along the positive Re(z) axis. If I wish to integrate it along a small, clockwise circle ...
2
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2answers
134 views

Complex integration using singularities

I'm working on Ablowitz and Fokas' Complex Variables. On section 3.5 on singularities, problem 2 reads: Evaluate the integral of f(z) over the unit circle centered at the origin: a) $f(z)=z/(z^2 + ...
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26 views

The use of Cauchy Goursat Theorem for a contour

I have been attempting this question for some time. My reason is as follow: The contour C (−i, −2 − i, −2 −4i, −4i) is a rectangle on the left of the Imaginary axis (vertical axis). For any z ...
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1answer
47 views

Is a binomial expansion of the denominator valid here?

I am trying to prove this integral tends towards zero (a variation of the Hankel contour problem). I am not sure if my approximation is valid here, My integral round a circular pole at the origin is, ...
2
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2answers
92 views

Evaluating the improper integral $\int_{0}^{\infty} \frac{x^3}{e^{x}-1} dx$ [duplicate]

I read somewhere that $$ \int_{0}^{\infty} \frac{x^3}{e^{x} - 1} = \frac{\pi^4}{15}$$ Does anyone see a way to prove this? My first idea was doing a contour integration and use the residue theorem, ...
1
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2answers
85 views

Cauchy Integral Formula (When f(z) is not analytic everywhere inside C)

enter image description here For the question in the photo, I have thought about it for a long time. My idea is as follow: C (|z| = 2) is a positively oriented simple connected contour. The point ...
3
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1answer
55 views

Inverse Laplace transform of $\frac{\exp(\frac{\lambda s}{1 - 2s})}{(1 - 2s)^{k/2}}$ (MGF of noncentral chi-squared distribution)

I am trying to use the countour integral to calculate the inverse Laplace transform of the function $$F(s) = \frac{\exp(\frac{\lambda s}{1 - 2s})}{(1 - 2s)^{k/2}} \hspace{1cm}\mathrm{for} \hspace{1cm} ...
4
votes
1answer
104 views

Inverse Laplace transform of one complicated function

I want to ask the inverse Laplace transform of the following function: $$F(s) = \frac{1}{s \cdot (1 + a \cdot s)^{m} \cdot (1 + b \cdot s)^{m-k}} \cdot \Bigl[\exp{(\frac{- c \cdot s}{ 1 + b \cdot s } ...
0
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1answer
47 views

How to Integrate$\int_C\frac{Log~z}{(z-i)^2}$ where $C:|z-i|=1$?

Let $C:|z-i|=\alpha$,where $0<\alpha<1$ I need to evaluate this expression $~\int_C\frac{Log~z}{(z-i)^2}$ what I tried to do was representing $Log~z$ as a Laurent series on ...