Math 113 (Spring 2009) Yum-Tong Siu 1
Derivation of the Poisson Kernel from
Standard Measure of Circle and
Fractional Linear Transformation
.
Consider
(∗)eiϕ =eiθ −a
1−eiθ¯a
which is obtained by restricting the fractional linear transformation
w=z−a
1−¯az
to the circle z=eiθ and w=e−iϕ. Applying logarithm to (∗) to get
log ¡eiθ −a¢−log ¡1−eiθ¯a¢=iϕ
and differentiating, we get
idϕ =eiθ
eiθ −aidθ −−eiθ¯a
1−eiθ¯aidθ
=µeiθ
eiθ −a+¯a
e−iθ −¯a¶idθ
=1−eiθ¯a+eiθ¯a−a¯a
|eiθ −a|2idθ
=1−a¯a
|eiθ −a|2idθ.
Hence
dϕ =1−a¯a
|eiθ −a|2dθ
and the Poisson kernel at the point zis given by
1
2π
1−z¯z
|eiθ −z|2dθ.