5
votes
2answers
55 views

How to compute $\int_C {e^{3z}-z\over (z+1)^2z^2}$?

I am asked to compute the integral $$ \int_C {e^{3z}-z\over (z+1)^2z^2} $$ where $C$ is a circle with the center at the origin and radius ${1 \over 2}$. My approach was to separate the integral as a ...
1
vote
1answer
52 views

Contour Integral of $\int_0^{\infty} \frac{1}{x^4+1} dx $ - Missing a factor of 2

I'm supposed to evaluate: $$ \int_0^{\infty} \frac{1}{x^4+1} dx $$ Consider $$ \oint \frac{1}{z^4+1} dz = \oint \frac{1}{(z - \frac{1-i}{\sqrt 2})(z + \frac{1-i}{\sqrt 2})(z - \frac{1+i}{\sqrt ...
0
votes
0answers
41 views

Complex Integration: $\int_0^{\infty} \frac{\sin x}{x(k^2x^2 +1)} dx $

I'm supposed to evaluate: $$ \int_0^{\infty} \frac{\sin x}{x(k^2x^2 +1)} dx $$ Attempt Consider $ \int_0^{\infty} \frac{e^{iz}}{x(k^2x^2 +1)} dz $ Simple poles at $z = \pm \frac{i}{k} $, simple ...
1
vote
1answer
48 views

Complex integration Question - Contour Method [duplicate]

I'm asked to find: $$\int_{-\infty}^\infty \frac{\ln(x^2+1)}{1+x^2} dx $$ Attempt Considering $$ \oint \frac{\ln(z^2+1)}{(z+i)(z-i)} dz $$ So first I find the branch points of the function. This ...
0
votes
1answer
34 views

Show that for $|f(z)| \leq C (|z| + 1)\log(|z| + 1)$, there is an $a$ such that $f(z) = az$

Let $f: \mathbb{C} \to \mathbb{C}$ be analytic and suppose a $C \geq 0$ exists such that \begin{align*} |f(z)| \leq C(|z| + 1) \log(|z| + 1) \end{align*} for all $z \in \mathbb{C}$, where $\log: ...
2
votes
2answers
200 views

Problem with Cauchy integrals

Hello everybody I need to solve some integral with the help of the Cauchy Integral Formula (CIF). I'll post near each integral the job that I've done and the question that I can't answer. let $\kappa ...
0
votes
0answers
33 views

Contour integration!Help

I have to integrate a function following the route from the point $(0,0,0)$ to $(1,1,1)$ which consists of the 2 curves $C=(t,t^2,0)$ and $K=(1,1,t)$ $0\leq t\leq 1$ .Is it right to take the 2 ...
3
votes
2answers
172 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 ...
1
vote
0answers
118 views

evaluate the integral $I =\int_0^{+\infty} e^{ix^2}dx$

"Evaluate the integral $I= \int_{0}^{\infty} e^{ix^{2}}\, dx$. Let R > 0 and consider the closed contour $C_R = C(1)_R + C(2)_R + C(3)_R$ where $C(1)_R$ is the segment of the positive real axis from ...
3
votes
1answer
61 views

Find the analytic continuation of the $ f(z) = \int_{0}^{\infty} \frac{exp(-zt)}{1+t^2} dt$

Find the analytic continuation of the function $f(z)$ defined by $ f(z) = \int_{0}^{\infty} \frac{\exp(-zt)}{1+t^2} dt$ , $ |\arg(z)| < \pi/2$ to the domain $ -\pi/2 < \arg(z) < \pi$ I ...
0
votes
1answer
42 views

Finding the types of singularities of $\oint \frac{\sin(\pi \cdot z)}{(z-1)^2}$

I want to find the types of singularities of $$\oint \frac{\sin(\pi \cdot z)}{(z-1)^2}$$ the point is $z=1$ I know that: $$f(z)=\frac{p(z)}{q(z)},q(a)=0,p(a)\neq 0,p(z)$$ so $p(z)$ analytic in $a$ ...
0
votes
1answer
47 views

Prove that $F_a(z) - F_b(z) = F_a(b)$

We had the following statement. Let $D \subset \mathbb C$ be a domain, $f: D \to \mathbb C$ a continuous function and $\gamma : [\alpha, \beta] \to D$ a contour. Assume that $\int_\gamma f$ ...
1
vote
0answers
24 views

How to show $\int_{\sum_{i = 1}^n \gamma_i} f = \sum_{i = 1}^n \int_{\gamma_i} f$

Let $G \subset \mathbb C$ be an open set, $f: G \to \mathbb C$ a continuous function and $\gamma: [\alpha, \beta] \to G$ a contour. Define the contour integral of $f$ along $\gamma$ to be ...
4
votes
2answers
79 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} = ...
0
votes
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 ...
6
votes
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
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 ...
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 ...
2
votes
2answers
123 views

Calculating ${\int_{-\infty}^{\infty} \frac{\cos(\omega x)}{x^{2} + 25}\,{\rm d}x}$ using contour integration

I want to calculate $\displaystyle{% \int_{-\infty}^{\infty}\frac{\cos\left(\omega x\right)}{x^{2} + 25}\,{\rm d}x\,, \quad}$ for $\omega \in \mathbb{R}$ I thought of integrating along the line ...
3
votes
1answer
61 views

Inverse FT of $Z(\omega) = a [- \frac{1}{i\omega}+\pi \delta(\omega)]$ (Contour integration)

Given is the Fourier transform of some function $z(t)$: $Z(\omega) = a [- \frac{1}{i\omega}+\pi \delta(\omega)]$ I now want to invert the tranform using contour integrals. How can I do that? I ...
6
votes
3answers
142 views

Show that $\int_0^\infty e^{-x^2} \sin{2xb}\, dx =e^{-b^2}\int_0^b e^{x^2} \, dx, \, b>0, $?

Show that $$\int_0^\infty e^{-x^2} \sin{2xb}\, dx =e^{-b^2}\int_0^b e^{x^2} \, dx, \, b>0, $$ I need help. I did the following steps: Apply Cauchy's Theorem, being $\varphi (x) = e^{-z^2}$ analytic ...
2
votes
0answers
131 views

The solution of the contour integral for $\epsilon =+1$

I understand the solution for $\epsilon =-1$. And I am trying the solve this question for $\epsilon =+1$. This is important for me. I want really to learn perfectly because I am continuously seein' ...
0
votes
1answer
241 views

Integral of $z^{n} \log z $ on the unit circle under two assumptions

I'm asked to calculate $\int_{|z| = 1} z^{n} \log z dz$ in two ways: (1) if $\log 1 = 0$; (2) if $\log (-1) = i \pi$. I understand it means that in case (1) I have to work with the principal ...
1
vote
0answers
98 views

Contour integration with branch cut

This is an exercise in a course on complex analysis I am taking: Determine the function $f$ using complex contour integration: $$\lim_{R\to\infty}\frac{1}{2\pi ...
2
votes
1answer
76 views

Contour Integral

I have this question: I'm aware that $e^{iz^2}$ is analytic, and hence $I_R = 0$ by Cauchy's Integral theorem. I'm not really sure what to do from there. Thanks!
3
votes
1answer
312 views

Contour Integration: $\int_0^\infty\frac{1}{x^a(1-x)}\,dx$ for $0<a<1$.

I've been trying to calculate $$\int_0^\infty\frac{1}{x^a(1-x)}\,dx\quad\text{with }0<a<1.$$I haven't had much luck. I tried taking the branch cut with of the positive reals and estimating that ...
2
votes
1answer
168 views

Contour integration - Branch cut

I'm asked to show the following equality given $a\in (-1,1)\subset\Bbb R$ $$\int\limits_0^\infty\frac{x^a\ \log(x)}{(1+x)^2}dx=\frac{\pi\sin(\pi a)-a\pi^2\cos(\pi a)}{\sin^2(a\pi)}$$ So I'm trying ...
2
votes
2answers
268 views

Example of Improper integral in complex analysis

I'm doing this example of Cauchy principle value $$ \int_0^\infty \frac{dx}{x^3+1}=\frac{2\pi}{3\sqrt{3}} $$ After some steps i got, $$ \int_{[0,R]+C_R} \frac{dz}{z^3+1}=2\pi i(B_1)\text{ where ...
4
votes
2answers
292 views

Contour integration using Cauchy's integral formula

I need to show that $$\int_{-\infty}^\infty\frac{\sin^2(x)}{x^2+1}dx=\frac{\pi}{2}\left(1-\frac{1}{e^2}\right)$$ but I don't really know why I'm not getting the result using contour integration ...
1
vote
1answer
65 views

Complex contour integrations

Consider the appropriate contour integral (circle $\oint=e^z$, show that $$\int^{2\pi}_{0}e^{cos\theta}cos(sin\theta +\theta)d\theta = 0$$ A more thorough explanation would be for the better.
3
votes
3answers
93 views

Find $I:=\lim\limits_{R\to \infty}\int\limits_{-R}^R \frac{x \sin(3x)}{x ^2+4}dx$ using residues

Find $I:=\lim\limits_{R\to \infty}\int\limits_{-R}^R \frac{x \sin(3x)}{x ^2+4}dx$ using residues. Let $f(z)= \frac{z \sin(3z)}{z ^2+4}$. First define two contours: $$\Gamma_1: z=t \text{ where } ...
2
votes
1answer
223 views

line integral versus complex integral

Let $a\in \mathbb C, r>0$ and $\gamma_r=\partial D(0,r)$. I want to evaluate the following line integral $$I=\int_{\gamma_r}\frac{1}{|z-a|^2}ds.$$ I'm looking for a complex function $g(z)$ such ...
1
vote
0answers
191 views

Prove that if $A$ is null and $f: \mathbb{R} \longrightarrow \mathbb{R}$ has a continuous derivative, then $f(A)$ is null

Prove that if $A$ is null and $f: \mathbb{R} \longrightarrow \mathbb{R}$ has a continuous derivative, then $f(A)$ is null. I think it has something to do with the fact that $f'$ is bounded in any ...
3
votes
0answers
134 views

Satisfying a Differential Equation and complex Laguerre

I have the following problem Show that $$L_n(x)=\frac{e^x}{2 \pi i}\oint \frac{t^n e^{-t}}{(t-x)^{n+1}}dt$$ satisfies $$x\, L_n^{\prime\prime}+(1-x)L_n^\prime+n\, L_n=0$$ where the contour is ...
3
votes
2answers
498 views

Another residue theory integral

this is the last from me I need to evaluate the following real convergent improper integral using residue theory (vital that i use residue theory so other methods are not needed here) I also need to ...
5
votes
1answer
518 views

Evaluating $\int\limits_0^\infty \frac{\log x} {(1+x^2)^2} dx$ with residue theory

I need a little help with this question, please! I have to evaluate the real convergent improper integrals using RESIDUE THEORY (vital that I use this), using the following contour: ...