# Improper integral

1. Jul 19, 2011

### Canerg

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

#### Attached Files:

• ###### integral_.pdf
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140
2. Jul 19, 2011

### 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.

Last edited: Jul 19, 2011
3. Jul 19, 2011

### 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

4. Jul 19, 2011

### hunt_mat

Calculus of residues perhaps?

5. Jul 19, 2011

### Canerg

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

6. Jul 19, 2011

### hunt_mat

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

What do you mean by true path?

7. Jul 19, 2011

### Canerg

can you help me to solve using square

8. Jul 19, 2011

### hunt_mat

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

9. Jul 19, 2011

### Canerg

:) ok
i will try

10. Jul 19, 2011

### BackEMF

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

11. Jul 20, 2011

### Canerg

%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

12. Jul 20, 2011

### 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?

13. Jul 20, 2011

### Canerg

MathType5

14. Jul 20, 2011