Vivanti-Pringsheim's theorem

In mathematics , Vivanti-Pringsheim's theorem is a theorem from function theory about singularities of power series on the edge of their convergence circle : It says that the positive, real point on the edge of the convergence circle is always a singularity. A simple example is the geometric series , which has its only singularity in . ${\displaystyle \textstyle \sum _{n=0}^{\infty }z^{n}={\frac {1}{1-z}}}$ ${\displaystyle z=1}$

The theorem was formulated in 1893 by Giulio Vivanti and proved a year later by Alfred Pringsheim . Edmund Landau later proved a generalization to Dirichlet series .

statement

Let be a power series with nonnegative , real coefficients and a positive radius of convergence . Then is a singularity of . ${\displaystyle \textstyle \sum _{n=0}^{\infty }a_{n}z^{n}=f(z)}$ ${\displaystyle a_{n}}$ ${\displaystyle R}$ ${\displaystyle z=R}$ ${\displaystyle f}$

literature

• Remmert-Schumacher: Function theory 1, Springer-Verlag 2002, ISBN 978-3-642-56281-5

Individual evidence

1. ↑ G. Vivanti: Sulle serie di potenze , Rivista di Matematica 3, 111-114
2. ↑ A. Pringsheim: On functions which have finite differential quotients of every finite order in certain points, but no Taylor series expansion , Mathematische Annalen 44, 41-56