Vivanti–Pringsheim theorem

The Vivanti–Pringsheim theorem is a mathematical statement in complex analysis, that determines a specific singularity for a function described by certain type of power series. The theorem was originally formulated by Giulio Vivanti in 1893 and proved in the following year by Alfred Pringsheim.

More precisely the theorem states the following:

A complex function defined by a power series

${\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}z^{n}}$

with non-negative real coefficients ${\displaystyle a_{n}}$ and a radius of convergence ${\displaystyle R}$ has a singularity at ${\displaystyle z=R}$.

A simple example is the (complex) geometric series

${\displaystyle f(z)=\sum _{n=0}^{\infty }z^{n}={\frac {1}{1-z}}}$

with a singularity at ${\displaystyle z=1}$.

The example of the geometric series gives an isolated singularity. As an example where the singularity at ${\displaystyle z=1}$ is not isolated you may consider ${\displaystyle f(z)=\sum _{n=0}^{\infty }z^{n!}.}$

References

• Reinhold Remmert: The Theory of Complex Functions. Springer Science & Business Media, 1991, ISBN 9780387971957, p. 235
• I-hsiung Lin: Classical Complex Analysis: A Geometric Approach (Volume 2). World Scientific Publishing Company, 2010, ISBN 9789813101074, p. 45