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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