# Real part of this complex quantity

• giuliopascal
In summary, the conversation is discussing a dispersive wave packet and its enlargement in the y direction. The problem is extracting the real part of the packet and finding the correct expression. Suggestions include breaking down the exponential into real and imaginary parts and using the wave packet's probability density to read the dispersion characteristics. Another approach involves solving a system of equations to find the correct expression.
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##

## What is the real part of a complex quantity?

The real part of a complex quantity is the portion of the number that is not imaginary. It is represented by the horizontal axis on the complex plane.

## How is the real part of a complex quantity calculated?

The real part of a complex quantity is calculated by taking the cosine of the angle between the complex number and the real axis on the complex plane.

## Can the real part of a complex quantity be negative?

Yes, the real part of a complex quantity can be negative if the complex number lies in the left half of the complex plane. This indicates that the number has a negative real value.

## What is the significance of the real part in a complex quantity?

The real part of a complex quantity represents the real-world quantity or value in a mathematical expression. It is used to calculate physical quantities such as voltage, current, and resistance in electrical circuits.

## How does the real part affect the graph of a complex quantity?

The real part of a complex quantity affects the horizontal position of the point on the complex plane. If the real part changes, the point will move horizontally, while the imaginary part determines the vertical position.

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