## Integrating along the imaginary axis...

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?
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 Quote by Cairo 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?
 Would it be kiR for -oo

## Integrating along the imaginary axis...

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.
 Hmmmm... I'm not sure how to proceed here then. Would it not be valid to have kiR where oo
 I'm talking nonsense! Forgive me! Would it be ik where R