Mathematical Investigations
In mathematical analysis, Pringsheim studied real and complex functions, following the power-series-approach of the Weierstrass school. Pringsheim published numerous works on the subject of complex analysis, with a focus on the summability theory of infinite series and the boundary behavior of analytic functions.
One of Pringsheim's theorems, according to Hadamard earlier proved by E. Borel, states that a power series with positive coefficients and radius of convergence equal to 1 has necessarily a singularity at the point 1. This theorem is used in analytic combinatorics and the Perron–Frobenius theory of positive operators on ordered vector spaces
Another theorem called after Pringsheim gives an analyticity criterium for a C∞ function on a bounded interval, based on the behaviour of the radius of convergence of the Taylor expansion around a point of the interval. However, Pringsheim's original proof had a flaw (related to uniform convergence), and a correct proof was provided by Ralph P. Boas.
Pringsheim and Ivan Śleszyński, working separately, proved what is now called the Śleszyński–Pringsheim theorem on convergence of certain continued fractions.
Besides his research in analysis, Pringsheim also wrote articles for the Enzyklopädie der mathematischen Wissenschaften on the fundamentals of arithmetic and on number theory. He published papers in the Mathematische Annalen. As an officer of the Bayerische Akademie der Wissenschaften, he recorded the minutes of its scientific meetings.
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