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Vivanti–Pringsheim theorem

From Wikipedia, the free encyclopedia

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

with non-negative real coefficients and a radius of convergence has a singularity at .

A simple example is the (complex) geometric series

with a singularity at .

References

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  • 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