Convergence of Fourier series $\frac 1 {2i} \sum_{n \neq 0} \frac { \exp (inx)} n$

Question

Let $$ f(x) := \begin{cases} -\frac \pi 2 - \frac x 2 && x \in (-\pi,0) \\ \frac \pi 2 - \frac x 2 && x \in (0, \pi) \\ 0 && x = 0 \end{cases} $$ I have to show that $\frac 1 {2i} \sum_{n \neq 0} \frac { \exp (inx)} n$ converges to $f$ pointwise. I already have shown that $f$ converges pointwise (with dirichlet-test). I also have shown that the sum indeed is the fourier-series of $f$. How can I conclude that the limit must be $f$ ?

Answer 1

A theorem of Dirichlet to which julien alluded implies that the trigonometric Fourier series $\sum_{n=0}^\infty (a_n\cos nx+b_n\sin nx)$ converges to $\frac12 (f(x+)+f(x-))$ for all $x$. You function is piecewise continuous with finitely many jumps and therefore is of bounded variation. (Historically, the original formulation of Dirichlet's theorem was exactly for such functions.)

The fact that $\frac{1}{2i}\sum_{n\ne 0}e^{inx}/n$ diverges for $x=0$ may look like a contradiction to the above. But this is why I emphasized trigonometric above. After conversion from exponential to trigonometric form, the series becomes $\sum_{n=1}^\infty (\sin nx)/n$ which trivially converges to zero at $x=0$.

The conversion to trigonometric form makes it easier for the series to converge, because it effectively changes the method of summation to the Cauchy principal value $\lim_{N\to\infty} \sum_{n=-N}^N$.