 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/b/solution

For fixed

r < 1

\sum r^{n} cos (n θ)

converges absolutely and uniformly. Thus for integrable

f

, one may integrate the kernel series term by term. The cosine addition formula gives

cos [n (θ - t)] = cos (n θ) cos (n t) + sin (n θ) sin (n t) .

Substitute this into the defining integral. The constant kernel term gives

\frac{1}{2 π} \int f = a_{0} /2

; the remaining terms give

y_{f} (r, θ) = \frac{a _{0}}{2} + n = 1 \sum \infty r^{n} [a_{n} cos (n θ) + b_{n} sin (n θ)],

where

a_{n} = \frac{1}{π} \int_{- π}^{π} f (t) cos (n t) d t, b_{n} = \frac{1}{π} \int_{- π}^{π} f (t) sin (n t) d t .

These are the same Fourier coefficients as in part (a). The kernel

2 K

r^{n}

suppress the higher angular frequencies inside the disk.

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