- #1
ryanwilk
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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}}{x-i0}.[/tex]
Homework Equations
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The Attempt at a Solution
Firstly, I'm unsure about what "x-i0" 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 "x-i0" just means "x", the pole is on the axis and it won't make a difference whether s is less or greater than 0...