 Solution

ID: past-exam-of-the-metric-spaces-and-complex-analysis-course-of-the-university-of-oxford/2017/4/b/ii/solution

For

∣ z_{0} ∣ < 1

and

k \geq 0

f^{(k)} (z_{0}) = \frac{k !}{2 πi} \int_{γ} \frac{f ( z )}{( z - z _{0} ) ^{k + 1}} d z .

The formula applies because the closed unit disc lies inside the domain of holomorphy. For

k = 0

this is the ordinary value formula, with

0! = 1

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