 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-2/18/c/solution

The boundary data are odd except at isolated endpoint choices, which do not change integrals. Thus every cosine Fourier coefficient, including

a_{0}

b_{n} = \frac{2}{π} \int_{0}^{π} sin (n t) d t = \frac{2 [ 1 - ( - 1 ) ^{n} ]}{πn} = {4/ (πn), 0, n odd, n even .

Consequently the extension is

y_{f_{0}} (r, θ) = \frac{4}{π} n \geq 1 n odd \sum \frac{r ^{n} sin ( n θ )}{n} .

It converges absolutely for every

r < 1

; its limiting boundary values at jumps are the midpoint of the two one-sided limits.

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