 Solution

ID: past-exam-of-the-algebra-analysis-and-differential-equations-1-course-of-the-university-of-oxford/2010/6/i/solution

For the positively oriented circle

C = {z : ∣ z - z_{0} ∣ = r}

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

The hypothesis that the closed disk lies in the open holomorphy domain ensures the formula applies on and inside this circle.

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