I'm trying to find examples of complex power series $\sum a_n z^n$ with radius of convergence $1$ and where:

(1) The series converges everywhere on the circle $|z| = 1$ except one point; (2) The series diverges everywhere on the circle $|z| = 1$ except one point.

For (1) I found $\displaystyle \sum_{n=1}^\infty \frac{z^n}{n}$ which is a harmonic series when $|z| = 1$. It converges for $|z|=1, z \neq 1$ by Dirichlet test: $\sum_{n=1}^N e^{in\theta}$ bounded and $1/n$ decreasing

Can anyone give an example for part (2)?

  • $\begingroup$ In part (2), should it read $|z| = 1$, since you wrote earlier that the series is converging for $|z|<1$? $\endgroup$ – Isaac Browne Jul 2 '18 at 20:42
  • $\begingroup$ According to An Introduction to Classical Complex Analysis, Vol 1 by R.RB. Burkel, page 81, an example of 2) due to Sierpinksi (who else?) can be found in a 1929 paper by Landau. Unfortunately, I found this on Google books, and it doesn't display the appropriate page of the bibliography, so I can't narrow it down for you any more, but perhaps this will get you started. $\endgroup$ – saulspatz Jul 2 '18 at 20:45
  • $\begingroup$ mathforum.org/kb/message.jspa?messageID=460987 The French translation referred to can be found at plouffe.fr/simon/math/Sierpinski%20Oeuvres%20Choisies%20I.pdf $\endgroup$ – saulspatz Jul 2 '18 at 20:54
  • $\begingroup$ math.stackexchange.com/questions/288765/… $\endgroup$ – saulspatz Jul 2 '18 at 21:11
  • $\begingroup$ @saulspatz Cute! You found a message that I posted at sci.math 15 years ago. $\endgroup$ – José Carlos Santos Jul 2 '18 at 23:12

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