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 $\displaystyle

\lim_{x\to\infty}

$ in the following expression(F).

F = $\displaystyle

e^b \frac{\partial e^a}{\partial x}-e^a \frac{\partial e^b}{\partial x}

$

Where $\displaystyle 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 .

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

$

$\displaystyle 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;

$\displaystyle \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 :

$\displaystyle

\lim_{x\to\infty}

$$\displaystyle 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.