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Old 2020-03-17, 12:45   #1
wildrabbitt
 
Jul 2014

2·13·17 Posts
Default 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.

Last fiddled with by wildrabbitt on 2020-03-17 at 12:53
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Old 2020-03-17, 13:19   #2
Chris Card
 
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Aug 2004

3×43 Posts
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Quote:
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.
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
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Old 2020-03-17, 13:19   #3
Dr Sardonicus
 
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Feb 2017
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Quote:
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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Old 2020-03-17, 14:36   #4
wildrabbitt
 
Jul 2014

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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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