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Old 2020-03-17, 13:19   #3
Dr Sardonicus
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Feb 2017

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Originally Posted by wildrabbitt View Post
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.
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