4
votes
1answer
64 views

Complex Integration: theorem of Residues

I am trying to prove a theorem that is doing my head in a bit. I have tried to simplify the problem as much as possible and leave out the details, even though it might look a bit too big. The ...
5
votes
2answers
111 views

Find the Fourier Transform of $\dfrac{x}{x^4+4}$

I have a problem here that becomes quite difficult to manage. I have to find the fourier transform of: $$f(x)=\frac{x}{x^4+4}$$ I'm sure there will be many ways to do this and I'll post my method ...
4
votes
1answer
120 views

Residue theorem with exponential and trig functions

The following integral should be doable using the residue theorum: $$\frac1{2\pi}\int_{0}^{2\pi}e^{\cos\theta}\cos(n\theta) \,d\theta$$
0
votes
0answers
33 views

Integration of a complex function having square rooted denominator

How to evaluate this integral ? $$\int_{-\infty}^{\infty}\frac{1}{(w-ia)\sqrt{w^2-ibw+b^2c}}dw$$ where $a$, $b$ and $c$ are real and greater than zero. Does the residue theorem work on it? Please ...
1
vote
2answers
135 views

$\int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx$ with residue calculus

I'm trying to compute $\displaystyle \int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx$, $(0<a<1)$ Let $f$ denote the integrand. I'm using the rectangular contour given by the following curves: ...
3
votes
1answer
132 views

Would like help with a contour integral.

Disclaimer: the knowledge I have about contour integration is solely from the book "Mathematical Methods in the Physical Sciences" by Mary L. Boas. I am trying to understand how the following ...
7
votes
1answer
197 views

To calculate residue of the function $f(z) = \frac{z^2 + \sin z}{\cos z - 1}$.

I was trying to find the residue of the function $$f(z) = \frac{z^2 + \sin z}{\cos z - 1}.$$ Here is the my attempt: The given function has a pole of order two at $z = 2n\pi$. So, we use the ...
1
vote
1answer
136 views

$\int_{0}^{+\infty}\frac{\sin x}{x^{k}(1+x^{2})}dx \ $ via residue calculus

I want to evaluate with calculus of residues $$\int_{0}^{+\infty}\frac{\sin x}{x^{k}(1+x^{2})}dx \ $$ $ k \in \mathbb{N}, k \geq 1$ If $k = 1$ we have $$\int_{0}^{+\infty}\frac{\sin ...
1
vote
3answers
131 views

Find the residue of $\frac{e^{iz}}{(z^2+1)^5}$ at $z = i$ and evaluate $\int_0^{\infty} \cos x/(x^2+1)^5 dx$

I know the evaluation of $\int_0^{\infty} \cos x/(x^2+1)^5 dx$ requires that I solve the first part, but for some reason I'm stumped. I get that I should use $\lim_{z \to ...
0
votes
1answer
47 views

Integrating Real Function in the Complex Plane

Question: Evaluate the integral $$\int_{-\infty}^{\infty}\frac{\sin(x)}{x(x^2+a^2)}=Im\left ( \frac{e^{ix}}{x(x^2+a^2)} \right)$$ ...
4
votes
2answers
289 views

Why do we use only upper half plane to do Residue Integration?

In the class of mathematics my professor showed me that "how to use the residue integration method?". And i doubt at almost the last step that professor did. he made a contour over the upper half ...
1
vote
1answer
119 views

Complex integral using residue theorem

I have $$\int_{|z|=1} z^m \sin\left(\frac{1}{z}\right)~dz,$$ for $m = 0,1,2,\dots$ I know that there is a singularity at $z=0$, and this singularity is within the curve, thus the residue theorem ...
0
votes
1answer
120 views

Evaluate this integral without using the residue theorem

I want to evaluate this integral $$ I = \int_c \tan z + \frac{\csc z}{z} dz $$ $$ c :|z| = 1 $$ apparently tanz is analytic in this region so its integral equals to zero now $$ I = ...
1
vote
0answers
229 views

Laplace transform of a product of functions

While trying to compute the Laplace transform of a certain product, part of the calculation leaves me with a Bromwich integral which has the form: ...
0
votes
1answer
115 views

Calculating Residues

I want to calculate this integral $$I:=\int dk^{0}\frac{e^{-ik^{0}(x^{0}-x'^{0})}}{\left(\left(k^{0}\right)^{2}-|\vec{k}|^{2}\right)} $$ for that I recall the Residue Theorem: $$I=2\pi i \left\{ ...
1
vote
1answer
301 views

finding $\int_0^\infty \dfrac{dx}{1+x^4}$ through complex analysis

I am trying to find $\int_0^\infty \dfrac{dx}{1+x^4}$ by setting it equal to $\dfrac{1}{2}\oint_C \dfrac{dz}{1+z^4}$ and solving that. By a computer program I've calculated it to be $\approx 1.11072$; ...
2
votes
1answer
244 views

How to evaluate this complex integral !?

We have the following complex integral : $$\frac{1}{2\pi i}\int_{-i\infty}^{i\infty}e^{-\frac{\pi}{2}\cot\left(\frac{\pi}{s}\right)}\frac{x^{s}}{s}ds$$ Where $x\in\mathbb{R}:x>1$. i tried closing ...
4
votes
2answers
388 views

Evaluating real integral using residue calculus: why different results?

I have to evaluate the real integral $$ I = \int_0^{\infty} \frac{\log^2 x}{x^2+1}. $$ using residue calculus. Its value is $\frac{\pi^3}{8}$, as you can verify (for example) introducing the function ...
3
votes
1answer
232 views

Complex analysis integration with residues.

I have to show that $$\int_{0}^{2\pi}\frac{d\theta}{(a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta)^{2}} =\frac{ \pi(a^{2}+b^{2})}{a^{3}b^{3}}$$ where $a,b>0$. I have tried using double angle formulas ...
1
vote
0answers
205 views

Residue of $\mathrm{exp} (z+1/z)/((z-z_1)(z-z_2))$ at z=0

The ultimate aim is to solve the following integral: \begin{equation} \label{eq:Icos1} \begin{aligned} I = \int\limits_{0}^{2\pi} \frac{\mathrm{exp}\left(c \cos(\varphi)\right)\mathrm{d}\varphi}{a - ...
-1
votes
2answers
91 views

$\int_{-1}^{1}\exp(-at^2)/t^2 dt$ using residue theorem

How to calculate $$\int_{-1}^{1}\frac{e^{-at^2}}{t^2}dt$$ where $a>0$, using residue theorem?