 #1
ryanwilk
 57
 0
Homework Statement
I need to show that the unit step function ([tex]\Theta(s) = 0 [/tex] for [tex] s<0, 1 [/tex] for [tex] s>0[/tex]) can be written as [tex]\Theta(s)=\frac{1}{2\pi i} \int_{\infty}^{\infty} dx \frac{e^{ixs}}{xi0}.[/tex]
Homework Equations

The Attempt at a Solution
Firstly, I'm unsure about what "xi0" actually means. I've looked online and couldn't find anything but if it means "x minus an infinitessimal multiple of i", it kinda works.
There will be a pole in the upper half of the complex plane.
For s>0, the pole will be contained, with residue [tex]e^0 = 1[/tex]. Then calculating the integral and dividing by [tex]2\pi i[/tex] will give [tex]\Theta(s) = 1 [/tex] for [tex] s>0.[/tex]
For s<0, the pole won't be contained so the integral will be zero and [tex]\Theta(s) = 0 [/tex] for [tex] s<0.[/tex]
However, if "xi0" just means "x", the pole is on the axis and it won't make a difference whether s is less or greater than 0...