20200317, 12:45  #1 
Jul 2014
19×23 Posts 
something about a sum
Hi, can anyone explain this?
\[S=\sum_{v=\infty}^{\infty}\int_0^Ne^{2\pi ivx+2\pi i\frac{x^2}{N}} \mathrm{d}x\] \[=N\sum_{v=\infty}^{\infty}\int_0^1e^{2\pi iN(x^2+vx)} \mathrm{d}x\] What I'm hoping for is some intermediate steps which get from the first, step by step to the second that make sense. Please help. Last fiddled with by wildrabbitt on 20200317 at 12:53 
20200317, 13:19  #2  
Aug 2004
201_{8} Posts 
Quote:
If so, try setting x = Ny and rewrite the integral in terms of y instead of x. Chris 

20200317, 13:19  #3  
Feb 2017
Nowhere
2×1,523 Posts 
Quote:
It is perhaps unfortunate that the variables in the integrals on both sides have the same name. 

20200317, 14:36  #4 
Jul 2014
437_{10} Posts 
Thanks to both of you. I do understand integration by substitution but I didn't know what the substitution required was.
I should be able to do it now. 