Complex Methods | Part IB, 2001

State the Cauchy integral formula.

Assuming that the function f(z)f(z) is analytic in the disc z<1|z|<1, prove that, for every 0<r<10<r<1, it is true that

dnf(0)dzn=n!2πiξ=rf(ξ)ξn+1dξ,n=0,1,\frac{d^{n} f(0)}{d z^{n}}=\frac{n !}{2 \pi i} \int_{|\xi|=r} \frac{f(\xi)}{\xi^{n+1}} d \xi, \quad n=0,1, \ldots

[Taylor's theorem may be used if clearly stated.]

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