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wildrabbitt 2020-03-17 12:45

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.

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.


Please help.[/QUOTE]

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.


Please help.[/QUOTE]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.

wildrabbitt 2020-03-17 14:36

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.


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