504 Chapter^7(( - z)(( - z) =I( - zl^2
Corollary 7.18 As a by-product of the foregoing proof we have formu-
las (7.28-8) and (7.28-9), namely,{u(r,0)=v(r, 0) =(7.28-10)These are the Poisson's formulas expressing u and v at an interior point of
C* in terms of the values of u on the circle (and the value of v at the center
of the circle in the case of the second formula). They solve a special case
of the so-called Dirichlet problem. We return to these formulas in Section
2.5 of Selected Topics.
For r = 0 the first Poisson formula gives
1 {27r
u(O, 0) = u(O) = 2 71" lo u(R, 'I/;) d'ljJ (7.28-11)which yields the value of u at the center of the disk in terms of the values
of u on the boundary (mean value property for harmonic functions). This
formula can be obtained directly from (7.28-3) by taking real parts, as in
Exercises 7.3, problem 5(b).
An alternative and elegant method of deriving the first Poisson formula
starts from (7.28-11). Noting that the bilinear transformation
r = T(l"I) = R(R(' + z)
.., .., R+z(' ' lzl <R
maps j('I :$ 1 onto j(j :$ R with (^1 = 0 going into ( = z = rei^8 and that
u(T((')) is harmonic in j('j :$ 1, we have, by using (7.28-11) as applied
to u(T((')),u(T(O)) = u(rei^8 ) = 2_ f
2
1f' u(T((')) d'ljJ'
27!" lo
where 'I/;' = Arg (^1 • Since;-, = R(( - z)
.., R^2 -z(we find, with ( = Reit/J, (( = R^2 ,
J'ljJI =-i-d(' = -i (-1-+ _z_) d(
(^1 (-z R^2 -z((7.28-12)(7.28-13)