1628 CHAPTER 51. FUNDAMENTALS OF COMPLEX ANALYSIS

Suppose f is analytic on |z|< R and

f (z) =∞

∑n=0

anzn (51.4.15)

with the series converging uniformly on |z|= R. Then letting |w|= R,

2u(w) = f (w)+ f (w)

and so

2u(w) =∞

∑k=0

akwk +∞

∑k=0

ak (w)k . (51.4.16)

Now letting γ (t) = Reit , t ∈ [0,2π]∫γ

2u(w)w

dw = (a0 +a0)∫

γ

1w

dw

= 2πi(a0 +a0) .

Thus, multiplying 51.4.16 by w−1,

1πi

∫γ

u(w)w

dw = a0 +a0.

Now multiply 51.4.16 by w−(n+1) and integrate again to obtain

an =1πi

∫γ

u(w)wn+1 dw.

Using these formulas for an in 51.4.15, we can interchange the sum and the integral(Why can we do this?) to write the following for |z|< R.

f (z) =1πi

∫γ

1z

∞

∑k=0

( zw

)k+1u(w)dw−a0

=1πi

∫γ

u(w)w− z

dw−a0,

which is the Schwarz formula. Now Rea0 =1

2πi∫

γ

u(w)w dw and a0 = Rea0− i Ima0.

Therefore, we can also write the Schwarz formula as

f (z) =1

2πi

∫γ

u(w)(w+ z)(w− z)w

dw+ i Ima0. (51.4.17)

7. Take the real parts of the second form of the Schwarz formula to derive the Poissonformula for a disk,

u(reiα)= 1

2π

∫ 2π

0

u(Reiθ

)(R2− r2

)R2 + r2−2Rr cos(θ −α)

dθ . (51.4.18)