Quote:
Originally Posted by wildrabbitt
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.

Obvious substitution. You said you knew how to make substitutions in integrals.
It is perhaps unfortunate that the variables in the integrals on both sides have the same name.