Parseval–Gutzmer formula
In mathematics, the Parseval–Gutzmer formula states that, if f {\displaystyle f} is an analytic function on a closed disk of radius r with Taylor series f ( z ) = ∑ k = 0 ∞ a k z k , {\displaystyle f(z)=\sum _{k=0}^{\infty }a_{k}z^{k},} then for z = reiθ on the boundary of the disk, ∫ 0 2 π | f ( r e i θ ) | 2 d θ = 2 π ∑ k = 0 ∞ | a k | 2 r 2 k , {\displaystyle \int _{0}^{2\pi }|f(re^{i\theta })|^{2}\,\mathrm {d} \theta =2\pi \sum _{k=0}^{\infty }|a_{k}|^{2}r^{2k},} which may also be written as 1 2 π ∫ 0 2 π | f ( r e i θ ) | 2 d θ = ∑ k = 0 ∞ | a k r k | 2 . {\displaystyle {\frac {1}{2\pi }}\int _{0}^{2\pi }|f(re^{i\theta })|^{2}\,\mathrm {d} \theta =\sum _{k=0}^{\infty }|a_{k}r^{k}|^{2}.} == Proof == The Cauchy Integral Formula for coefficients states that for the above conditions: a n = 1 2 π i ∫ γ f ( z ) z n + 1 d z {\displaystyle a_{n}={\frac {1}{2\pi i}}\int _{\gamma }^{}{\frac {f(z)}{z^{n+1}}}\,\mathrm {d} z} where γ is defined to be the circular path around origin of radius r.