Integral takes too long without output?

Questions
1

Hello all,

I am trying to solve the integral shown in the image below. The solution gives the non-linear signal produced from the four-terminal geometry used to investigate spin-transfer torque. I am using Wolfram Mathematica (via the Wolfram Cloud) and although I feel as though I have inputted the equation correctly it tries to evaluate for 5 minutes (I have the student plan so this is equal to my allowed run limit) before not returning any answer.

The image below is a screenshot from a thesis (https://www.rug.nl/research/portal/files/3035585/thesis.pdf, page 41) where they managed to solve this equation using Wolfram Mathematica so I know it can be done. This shows the integral I want to solve and the solution they obtained (further algebra may have been used to get it in the form shown here):

integral3.PNG

Integral2.PNG
Integral2.PNG546×500 26.1 KB

This is a screenshot of my input into Mathematica:

wolfram.PNG

This is a link to the notebook:
https://www.wolframcloud.com/obj/226571a7-e61f-4ae3-9942-5f84202ff6af

This is my first experience using Wolfram Mathematica so there is a chance I am inputting it wrong. Or I may just need more run time. I’d be very grateful if someone could give me any advice as to whether this is something Mathematica will be able to solve with more run time?

Thank you in advance!

2

Using:

\cos (w t)=\Re(\exp (i w t))

Re[Integrate[
  1/Sqrt[4 Pi d t] Exp[-L^2/(4 t d)] Exp[-t/x] Exp[I w t], {t, 0, 
   Infinity}]]

(*1/2 Re[E^(-(Sqrt[-I w + 1/x]/Sqrt[(d/L^2)]))/(
  Sqrt[d] Sqrt[-I w + 1/x])]*)

if: Re[L^2/d] > 0 && Im[w] + Re[1/x] > 0

We can speed up computation if:

Re[Integrate[
  1/Sqrt[4 Pi d t] Exp[-L^2/(4 t d)] Exp[-t/x] Exp[I w t], {t, 0, 
   Infinity}, Assumptions -> {d > 0, L > 0, x > 0, w > 0}]]
3

You are amazing! I never would have thought to do that! You have no idea how much it means to me that you’ve helped me with this, I cannot thank you enough!