Further Analysis | Part IB, 2001

Let f(z)f(z) be analytic in the discz<R\operatorname{disc}|z|<R. Assume the formula

f(z0)=12πiz=rf(z)dz(zz0)2,0z0<r<R.f^{\prime}\left(z_{0}\right)=\frac{1}{2 \pi i} \int_{|z|=r} \frac{f(z) d z}{\left(z-z_{0}\right)^{2}}, \quad 0 \leqslant\left|z_{0}\right|<r<R .

By combining this formula with a complex conjugate version of Cauchy's Theorem, namely

0=z=rf(z)dzˉ,0=\int_{|z|=r} \overline{f(z)} d \bar{z},

prove that

f(0)=1πr02πu(θ)eiθdθf^{\prime}(0)=\frac{1}{\pi r} \int_{0}^{2 \pi} u(\theta) e^{-i \theta} d \theta

where u(θ)u(\theta) is the real part of f(reiθ)f\left(r e^{i \theta}\right).

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