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}

, vanishes. For the sine Fourier coefficients,

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.

 Ancestors (13)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version