watchmath 5 years ago Compute integral sin x /x from 0 to infinity.

1. myininaya
2. anonymous

3. myininaya

no but it could help

4. anonymous

You have to memorize the solutions to these. Don't try to actually solve it. The answer is just \pi/2\

5. watchmath

I know the result. I want to know how :D.

6. anonymous

you'll have hard time getting help here for anything above pre-calculus

7. anonymous

and elementary statistics :)

8. anonymous

If you are currently in cal 1, you probably won't do the series stuff until the end of cal 2

9. anonymous

I'm going to upload an attachment soon, just hold on. Integration by parts SUCKS

10. myininaya

i think this involves advance calculus

11. anonymous

This does involve advanced calculus... this is going to take longer than I thought

12. watchmath

can you tell me the idea?

13. myininaya

you can use laplace transform

14. myininaya
15. anonymous

This is going to use Taylor Series convergence

16. anonymous

Honestly, I don't feel like proving that the Taylor Series converges. That would take about a week by hand for me (because I don't actually know how to do it yet). However, I'm assuming that it converges to pi/2

17. watchmath

That's ok. I found it :D.

18. anonymous

we can do this using complex analysis:$\frac{1}{2}\int\limits_{-\infty}^{\infty}\frac{\sin(x)}{x}dx=\frac{1}{2}I \left( \int\limits_{-\infty}^{\infty}\frac{e^{ix}}{x}dx\right)$ this is the same as: $I(\frac{1}{2} \pi i ( res_{0}\frac{e^{ix}}{x}))=I(\frac{\pi i}{2})=\frac{\pi}{2}$

19. anonymous

we need the residue at zero since we have a simple pole there