Solve Complicated Integral: Get Professional Help

[Canerg]

Hi this is very complicated integral i couldn't solve
can you help me ? how does it solve

 

[disregardthat]

Hello, is j negative, and do you know if a is non-zero or positive? I think this integral only exists if j is non-positive and a is non-negative.

 

[Canerg]

j=sqrt(-1) and a is positive
I solved this integral numerically and i found the exact result in both Matlab and Mathematica program but I need analytical solution. I tried residue theorem but result didn't match numeric solutions.I asked some mathematicians but they couldn't find true path for the residue and i look ryzik integral book i coulnd't find.

sqrt;squareroot
Thank you for your connection

 

[hunt_mat]

Calculus of residues perhaps?

 

[Canerg]

yes Calculus of residues but true path is important
may be fresnel integral can solve this problem but diffucult to understand. :(

 

[hunt_mat]

You could complete the square and see if that might help you.

What do you mean by true path?

 

[Canerg]

can you help me to solve using square

 

[hunt_mat]

$x^{2}-2rx=(x-r)^{2}-r^{2}$

 

[Canerg]

:) ok
i will try

 

[BackEMF]

What program did you write those equations in if you don't mind Canerg?

 

[Canerg]

Hi BackEMF this is my Matlab code you can use quad instead of quade

%clc; clear all
lamda=1.55e-6;
k=2*pi/lamda;
a=10000;
r=1e-2;L=1000;
f=@(x)(1./(1+a*x.^2).*exp(i*k/(2*L)*(x.^2-2*x*r)));%% integral by numerical solution
numerical=quade(f,-inf,inf)

 

[BackEMF]

Hi Canerg, sorry I wasn't clear enough. I meant the equations you submitted in PDF, do you mind telling me what typsetting program did you use?

 

[Canerg]

MathType5

 

[BackEMF]

Thanks, sorry for imposing on your thread!