 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-2/4/b/solution

The only denominator zero is

z_{0} = - 1

, which lies strictly inside the given circle. The numerator is an entire function. Apply the derivative form of the Cauchy integral formula with

n = 6

\oint_{C} \frac{s i n z}{( z + 1 ) ^{7}} d z = \frac{2 πi}{6 !} \frac{d ^{6}}{d z ^{6}} sin z_{z = - 1} .

The sixth derivative is

- sin z

, so its value at

- 1

sin 1

. Consequently

\oint_{C} \frac{sin z}{( z + 1 ) ^{7}} d z = \frac{πi sin 1}{360} .

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