His daughter Katia was the first female in Munich to earn the qualifications for university admission and was one of the first active women students at Munich University. She later became the wife of the author and Nobel Prize winner Thomas Mann. In 1889 Pringsheim and his family moved into a Neo-Renaissance villa at Arcisstrasse 12 designed by the Berlin architects Kayser & von Großheim with interior furnishings provided by Joh. Wachter and the court furniture manufacturer O. Fritsche of Munich. On major social nights the Munich elite was hosted here in what was known as the Pringsheim Palace. Besides mathematics, ever since his youth Pringsheim was also intensively occupied with music, and adapted various compositions of Richard Wagner for the piano. Later he became interested in the theory and history of art, building up important collections of majolica earthenware and paintings. In his novel "Königliche Hoheit", Thomas Mann portrayed his father-in-law as the character Samuel Spoelman. 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 criterion 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".