Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am trying to compute an integral in an example in my complex analysis textbook:

$$\int_{-\infty}^\infty {xsinx\over x^4+1}dx$$

The book gives some startup hints, but I don't quite follow, I set $f(z)={e^{iz}z\over z^4+1}$. Then the function has poles of order 1 at $\sqrt i$ and $-\sqrt i$, or $e^{i\pi/4}$ and $-e^{i\pi/4}$.

Next, I need to find the residue of $f$ at $e^{i\pi/4}, -e^{i\pi/4}$. After that I think I know what to do next. But the step of finding the residue is getting me.

share|improve this question
    
Might want to be careful about your poles; $z^4+1=0$ will have 4 roots. I'm thinking you'll want to integrate over a semicircular contour in the upper half plane, where there are two poles, but you've only got one of them. You'll have $e^{i\frac{\pi}{4}}$ and $e^{i\frac{3\pi}{4}}$ in the upper half plane. –  FireGarden Apr 13 at 23:30

2 Answers 2

up vote 1 down vote accepted

You can calculate the residue by using $f/f'$.

$\displaystyle \lim_{z\to e^{\frac{\pi i}{4}}}\frac{ze^{iz}}{4z^{3}}$

Then do the case for $e^{3\pi i/4}$ the same way and add them.

share|improve this answer

if the function has pole of order 1 at p

take lim, z goes to p, (z-p)f(z)

Think of f(z) at p in laurent series at p: b/(z-p)+some thing analytic at p

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.