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- - **something about a sum**
(*https://www.mersenneforum.org/showthread.php?t=25374*)

something about a sumHi, 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=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 |

[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. |

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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