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 wildrabbitt 2020-03-17 12:45

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.

 Chris Card 2020-03-17 13:19

[QUOTE=wildrabbitt;539926]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.

Do you know how to do integration by substitution?
If so, try setting x = Ny and rewrite the integral in terms of y instead of x.

Chris

 Dr Sardonicus 2020-03-17 13:19

[QUOTE=wildrabbitt;539926]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.