- #1

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## Homework Statement

Hi All. I am given this integral:

[tex]\int_{-\infty}^{\infty}A\Theta e^{i\omega t}dt[/tex]

I need to show that it's equal to the following:

[tex]=A(\pi \delta(\omega)+\frac{i}{\omega})[/tex]

## Homework Equations

Theta is the Heavyside step function.## The Attempt at a Solution

The step function changes the lower bound of the integral to 0:

[tex]\int_{0}^{\infty}Ae^{i\omega t}dt[/tex]

Elementary integration then gives:

[tex]\lim_{t=\infty}\frac{e^{i\omega t}}{i\omega} +\frac{i}{\omega}[/tex]

I'm not sure how to relate that first limit to any of the delta function definitions I know. A hint was given to use the following definition:

[tex]\pi\delta(x)=\lim_{A=0}\frac{A}{x^2+A^2}[/tex]

However, I'm not sure how to apply this limit to help get to the result.

Thanks for any hints/help!