# Question:Rayleigh's identity

## Question:Rayleigh's identity

Maple 12

Rayleigh's identity is listed below:

infinity
-----
\
)             2      /      2        1      1  \
/        |f(k)|  = int||f(t)| , t = - - T .. - T|
-----                  \               2      2  /
k = -infinity

sum(abs(f(k))^2, k = -infinity .. infinity) = int(abs(f(t))^2, t = -(1/2)*T .. (1/2)*T);

This identity is an extension from Parseval's theorem for the case where the function of interest is periodic.  The link below provides a worksheet that confirms for a finite series that Rayleigh's identity is valid to within so many significant figures as the frequency parameter, k, increases for CASE 1.  However, for CASE 2 concurrence between the integral and the finite series is not that great.  I suspect I have an error somewhere that is causing the discrepancy.  I thought it might be useful if I get other sets of eyes on this to help isolate the discrepancy.  How I came up with Ck for CASE 2 I can create another worksheet with that derivation if requested.

Rayleighs_identity.mw

Appreciate any useful feedback

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