Integral operation

1. Mar 17, 2015

moriheru

My question concerns a integral and why it vanishes:
-nηIJ 1/2π ∫0 dσ ei(m+n)σ=-nηIJ deltam+n=0

Just to justify why this should be on the beyond the standard model forum, this is part of a calulation concerning the comutators of the alpha modes.

2. Mar 17, 2015

RUber

I assume m and n are integers?
If $m+n \neq 0$ then, $m+n = a \in \mathbb{Z}$ and
$\int_0^{2\pi} e^{i (m+n) \sigma} d\sigma = \frac{1}{i(m+n)} e^{i (m+n) \sigma} |_0^{2\pi} = \frac{1}{i(m+n)} - \frac{1}{i(m+n)} = 0.$
If $m+n=0$ then $\int_0^{2\pi} e^{i (m+n) \sigma} d\sigma = \int_0^{2\pi} 1 d\sigma= 2\pi.$

3. Mar 17, 2015

moriheru

Thankyou.

4. Mar 17, 2015

RUber

No problem--this idea is fundamental to Fourier transforms and a host of other applications requiring orthogonality of sines and cosines.

5. Mar 17, 2015

moriheru

Oh... thats an FT I missed that :) would've made that easyer,thanks.

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