Integrating along the imaginary axis

[Cairo]

I'm really confused with how to prove this result...could anybody help please?

Let I_{1} be the line segment that runs from iR (R>0) towards a small semi-circular indentation (to the right) at zero of radius epsilon (where epsilon >0) and I_{2} a line segment that runs from the indentation to -iR.

Define

f(z)=\frac{e^{2\pi iz^{2}/m}}{1-e^{2\pi iz}}

Prove that

I_{1}+I_{2}=-i\intop_{\varepsilon}^{R}e^{-2\pi iy^{2}/m}dy

How can I do this?

 

[snipez90]

Define

$$f(z)=\frac{e^{2\pi iz^{2}/m}}{1-e^{2\pi iz}}$$

Prove that

$$I_{1}+I_{2}=-i\intop_{\varepsilon}^{R}e^{-2\pi iy^{2}/m}dy$$

How can I do this?

First, draw out the contour. It's clear you're going to have to work from the definition of the complex line integral. So, how would you parametrize I_1 and I_2?

 

[Cairo]

Would it be kiR for -oo<k<00 ?

 

[snipez90]

I_1 is a segment running from iR to $i\varepsilon$ right? So it wouldn't make sense for your parameter to range over the reals.

 

[Cairo]

Hmmmm...

I'm not sure how to proceed here then. Would it not be valid to have kiR where
oo<k<epsilon?

This would give the line segment, right?

 

[Cairo]

I'm talking nonsense! Forgive me!

Would it be ik where R<k<epsilon?