complex variables and definite integrals

[sammy]

Hey,
I'm trying to do the following definite integrals using complex variables (Jordan's lemma, etc.)
(1) from 0 to inf of (sin x/x)*exp(-itx) dx for t real
(2) from -inf to inf of sin(x^2) dx
(3) from 0 to inf of log(x^2 + 1)/(x^2 + 1) dx
(4) from -inf to inf of cosh ax/cosh x dx for -1<a<1
(5) from -inf to inf of exp(-kx^2)exp(-itx) dx for k>0, t real

 

[stapel]

(1) from 0 to inf of (sin x/x)*exp(-itx) dx for t real
(2) from -inf to inf of sin(x^2) dx
(3) from 0 to inf of log(x^2 + 1)/(x^2 + 1) dx
(4) from -inf to inf of cosh ax/cosh x dx for -1<a<1
(5) from -inf to inf of exp(-kx^2)exp(-itx) dx for k>0, t real

What are your thoughts? What have you tried? How far have you gotten? Where are you stuck?

Please be complete. Thank you!

Eliz.

 

[sammy]

Hey, sorry I'm new to this board.
Firstly I can show they converge.
For (1) we only need to consider t>0 by symmetry (I think) but I can't come up with the correct function to choose (or the path). sin z isn't bounded so I can't use Jordan's lemma directly. I've considered splitting it up into real and complex parts and working with those but it proved harder than the original question.
For (2) i've tried to use the function sin(z^2) around a semicircle but sin(z^2) is unbounded so i had problems with the curved path.
For (3) i haven't had any bright ideas. Do I need to split it up?
For (4) i got into a real mess trying to work with the exponential forms of cosh and i'm missing the relevance of the -1<a<1.
Finally, for (5) I tried using Jordan's lemma on something like exp(-kz^2). Also i'm not sure how to generalise to t real anyway.

Thanks!