# Real part of this complex quantity

• giuliopascal

#### giuliopascal

Hi everyone,

I have a dispersive wave packet of the form:
##\frac{1}{\sqrt{D^2 + 2i \frac{ct}{k_0}}} e^{-y^2/(D^2+2i\frac{ct}{k_0})}##
The textbook says that the enlargement of the package, on the y direction, is:
##L=\frac{1}{D}\sqrt{D^4+4\left(\frac{ct}{k_0}\right)^2} ##
However I have some problems extracting the real part; I write:
##\frac{1}{\sqrt{D^2 + 2i \frac{ct}{k_0}}} = \frac{\sqrt{D^2 - 2i \frac{ct}{k_0}}}{\sqrt{D^4+4\left(\frac{ct}{k_0}\right)^2}}##
And I use the fact that:
##\Re{\sqrt{a+bi}}=\sqrt{\frac{a \pm \sqrt{a^2 + b^2}}{2}} ##
But I can't find the correct expression.

Do you have any suggestion?

Thank you very much

I don't know what you've tried. However the exponential has to be broken down into real and imaginary parts. Then combine the real parts to get one term of the real part of the product and combine the imaginary parts to get the other term.

Try letting u(y)= the wave packet
Form the probability density abs(u^2) = u u*
You should be able to read the dispersion characteristics from there

If I understand your problem is something like : find ##\delta \in \mathbb{Z}## such that ##\delta^2 = z##, ##z\in \mathbb{Z}##

Assume ##\delta = x + iy ## and ## z = X + i Y = \frac{D^2 \frac{ct}{k_0}}{D^4+4\left(\frac{ct}{k_0}\right)^2} + i \frac{ -2 \frac{ct}{k_0}}{D^4+4\left(\frac{ct}{k_0}\right)^2}##.

You must solve the system of equation:

## X = x^2 - y^2 ##
## Y = 2xy ##
## \sqrt{ X^2 + Y^2 } = x^2 + y^2##