Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
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100 CHAPTER 14. HARMONIC FUNCTIONS<br />
Since � 2π<br />
0 P1(e iθ ,z)dθ = 1 for all z ∈ D, we have<br />
� 2π<br />
S<br />
z0<br />
θ0 +2δ0 θ0 θ0 −2δ0<br />
g(z)−g(z0) = (f(e<br />
0<br />
iθ )−f(z0))P1(e iθ ,z)dθ<br />
�<br />
= (f(e<br />
J<br />
iθ )−f(z0))P1(e iθ �<br />
,z)dθ + (f(e<br />
� �� � [0,2π]\J<br />
iθ )−f(z0))P1(e iθ ,z)dθ .<br />
� �� �<br />
Note that<br />
�<br />
|I1|≤<br />
J<br />
I1<br />
|f(e iθ )−f(z0)|<br />
� �� �<br />
< ǫ<br />
2<br />
P1(e iθ ,z)dθ ≤ ǫ<br />
� 2π<br />
P1(e<br />
2 0<br />
iθ ,z)dθ = ǫ<br />
2 .<br />
I2