Let 
and 
be the Fourier transforms of 
and 
, respectively. Then
![int_(-infty)^inftyf(t)g^_(t)dt
=int_(-infty)^infty[int_(-infty)^inftyF(nu)e^(-2piinut)dnu][int_(-infty)^inftyG^_(nu^')e^(2piinu^'t)dnu^']dt
=int_(-infty)^inftyF(nu)int_(-infty)^inftyG^_(nu^')[int_(-infty)^inftye^(2piit(nu^'-nu))dt]dnu^'dnu
=int_(-infty)^inftyF(nu)[int_(-infty)^inftyG^_(nu^')delta(nu^'-nu)dnu^']dnu
=int_(-infty)^inftyF(nu)G^_(nu)dnu,](/images/equations/ParsevalsRelation/NumberedEquation1.svg) |
where 
denotes the complex conjugate.
Parseval's Relation
See also
Fourier Transform, Parseval's TheoremExplore with Wolfram|Alpha
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 425, 1985.Referenced on Wolfram|Alpha
Parseval's RelationCite this as:
Weisstein, Eric W. "Parseval's Relation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ParsevalsRelation.html