From Wikipedia, the free encyclopedia
In mathematics, the Parseval–Gutzmer formula states that, if
f
{\displaystyle f}
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}.}
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 . Also for
x
∈
C
,
{\displaystyle x\in \mathbb {C} ,}
x
¯
x
=
|
x
|
2
.
{\displaystyle {\overline {x}}{x}=|x|^{2}.}
∫
0
2
π
|
f
(
r
e
i
θ
)
|
2
d
θ
=
∫
0
2
π
f
(
r
e
i
θ
)
f
(
r
e
i
θ
)
¯
d
θ
=
∫
0
2
π
f
(
r
e
i
θ
)
(
∑
k
=
0
∞
a
k
(
r
e
i
θ
)
k
¯
)
d
θ
Using Taylor expansion on the conjugate
=
∫
0
2
π
f
(
r
e
i
θ
)
(
∑
k
=
0
∞
a
k
¯
(
r
e
−
i
θ
)
k
)
d
θ
=
∑
k
=
0
∞
∫
0
2
π
f
(
r
e
i
θ
)
a
k
¯
(
r
e
−
i
θ
)
k
d
θ
Uniform convergence of Taylor series
=
∑
k
=
0
∞
(
2
π
a
k
¯
r
2
k
)
(
1
2
π
i
∫
0
2
π
f
(
r
e
i
θ
)
(
r
e
i
θ
)
k
+
1
r
i
e
i
θ
)
d
θ
=
∑
k
=
0
∞
(
2
π
a
k
¯
r
2
k
)
a
k
Applying Cauchy Integral Formula
=
2
π
∑
k
=
0
∞
|
a
k
|
2
r
2
k
{\displaystyle {\begin{aligned}\int _{0}^{2\pi }\left|f\left(re^{i\theta }\right)\right|^{2}\,\mathrm {d} \theta &=\int _{0}^{2\pi }f\left(re^{i\theta }\right){\overline {f\left(re^{i\theta }\right)}}\,\mathrm {d} \theta \\[6pt]&=\int _{0}^{2\pi }f\left(re^{i\theta }\right)\left(\sum _{k=0}^{\infty }{\overline {a_{k}\left(re^{i\theta }\right)^{k}}}\right)\,\mathrm {d} \theta &&{\text{Using Taylor expansion on the conjugate}}\\[6pt]&=\int _{0}^{2\pi }f\left(re^{i\theta }\right)\left(\sum _{k=0}^{\infty }{\overline {a_{k}}}\left(re^{-i\theta }\right)^{k}\right)\,\mathrm {d} \theta \\[6pt]&=\sum _{k=0}^{\infty }\int _{0}^{2\pi }f\left(re^{i\theta }\right){\overline {a_{k}}}\left(re^{-i\theta }\right)^{k}\,\mathrm {d} \theta &&{\text{Uniform convergence of Taylor series}}\\[6pt]&=\sum _{k=0}^{\infty }\left(2\pi {\overline {a_{k}}}r^{2k}\right)\left({\frac {1}{2{\pi }i}}\int _{0}^{2\pi }{\frac {f\left(re^{i\theta }\right)}{(re^{i\theta })^{k+1}}}{rie^{i\theta }}\right)\mathrm {d} \theta \\&=\sum _{k=0}^{\infty }\left(2\pi {\overline {a_{k}}}r^{2k}\right)a_{k}&&{\text{Applying Cauchy Integral Formula}}\\&={2\pi }\sum _{k=0}^{\infty }{|a_{k}|^{2}r^{2k}}\end{aligned}}}
Using this formula, it is possible to show that
∑
k
=
0
∞
|
a
k
|
2
r
2
k
⩽
M
r
2
{\displaystyle \sum _{k=0}^{\infty }|a_{k}|^{2}r^{2k}\leqslant M_{r}^{2}}
where
M
r
=
sup
{
|
f
(
z
)
|
:
|
z
|
=
r
}
.
{\displaystyle M_{r}=\sup\{|f(z)|:|z|=r\}.}
This is done by using the integral
∫
0
2
π
|
f
(
r
e
i
θ
)
|
2
d
θ
⩽
2
π
|
max
θ
∈
0
,
2
π
)
(
f
(
r
e
i
θ
)
)
|
2
=
2
π
|
max
|
z
|
=
r
(
f
(
z
)
)
|
2
=
2
π
M
r
2
{\displaystyle \int _{0}^{2\pi }\left|f\left(re^{i\theta }\right)\right|^{2}\,\mathrm {d} \theta \leqslant 2\pi \left|\max _{\theta \in [0,2\pi )}\left(f\left(re^{i\theta }\right)\right)\right|^{2}=2\pi \left|\max _{|z|=r}(f(z))\right|^{2}=2\pi M_{r}^{2}}
![{\displaystyle {\begin{aligned}\int _{0}^{2\pi }\left|f\left(re^{i\theta }\right)\right|^{2}\,\mathrm {d} \theta &=\int _{0}^{2\pi }f\left(re^{i\theta }\right){\overline {f\left(re^{i\theta }\right)}}\,\mathrm {d} \theta \\[6pt]&=\int _{0}^{2\pi }f\left(re^{i\theta }\right)\left(\sum _{k=0}^{\infty }{\overline {a_{k}\left(re^{i\theta }\right)^{k}}}\right)\,\mathrm {d} \theta &&{\text{Using Taylor expansion on the conjugate}}\\[6pt]&=\int _{0}^{2\pi }f\left(re^{i\theta }\right)\left(\sum _{k=0}^{\infty }{\overline {a_{k}}}\left(re^{-i\theta }\right)^{k}\right)\,\mathrm {d} \theta \\[6pt]&=\sum _{k=0}^{\infty }\int _{0}^{2\pi }f\left(re^{i\theta }\right){\overline {a_{k}}}\left(re^{-i\theta }\right)^{k}\,\mathrm {d} \theta &&{\text{Uniform convergence of Taylor series}}\\[6pt]&=\sum _{k=0}^{\infty }\left(2\pi {\overline {a_{k}}}r^{2k}\right)\left({\frac {1}{2{\pi }i}}\int _{0}^{2\pi }{\frac {f\left(re^{i\theta }\right)}{(re^{i\theta })^{k+1}}}{rie^{i\theta }}\right)\mathrm {d} \theta \\&=\sum _{k=0}^{\infty }\left(2\pi {\overline {a_{k}}}r^{2k}\right)a_{k}&&{\text{Applying Cauchy Integral Formula}}\\&={2\pi }\sum _{k=0}^{\infty }{|a_{k}|^{2}r^{2k}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4226b99415ef1d1ce74760cd96c184e0ddd44b91)
