[SOLVED] Need to solve

kurapati · #1 $Old$ July 15th, 2009, 06:46 AM

Hi All, Could any one please help me solve this.

I have big expression with sin(x) and cos(x), which resulted after integrating another function. I need to apply the $\lim_{x\to\infty}$ in the following expression(F).

F = $e^b \frac{\partial e^a}{\partial x}-e^a \frac{\partial e^b}{\partial x}$

Where $e^a,e^b$ are given as follows:

Infact they are the electric fields in two different waveguides a, b. However, The expressions are similar .

$e^a = s^a [ cos(\sigma^a x) + \frac {\kappa^a T^a} {\sigma^a} sin(\sigma^a x) ]$

$e^b = s^b [ cos(\sigma^b x) + \frac {\kappa^b T^b} {\sigma^b} sin(\sigma^b x) ]$

I need to use the following representation of the Dirac delta function;

$\delta( \sigma ) = \lim_{x\to\infty} \frac {\sin(\sigma x)} {\pi \sigma}$

Now, I need to show that the function F, after the limit applied is equal to :

$\lim_{x\to\infty}$ $F = (\sigma^{b^2}-\sigma^{a^2}) \frac {S^a S^b \pi}{2} (1 + \frac{\kappa^a T^a \kappa^b T^b}{\sigma^a \sigma^b}) \delta (\sigma^a - \sigma^b)$

Please I need help to get this done. I would be grateful for the help.

kurapati · #2 $Old$ July 17th, 2009, 06:11 AM

Hi friends,

I got the solution.