Alfred Pringsheim's father was Rudolf Pringsheim and his mother was Paula Deutschmann. It was a Jewish family. Rudolf was rich, having made his money in the construction industry. Alfred grew up in Breslau where he became passionate about music, art and mathematics. He became an excellent pianist and his musical tastes were much influenced by Richard Wagner who, although much older than Pringsheim, became friendly with the talented young man.
Having studied mathematics in Berlin in 1868, Pringsheim became a follower of Weierstrass's mathematics. Following the usual practice of German students he moved between different universities and after studying in Berlin he went to Heidelberg. After completing his undergraduate studies, Pringsheim undertook research at Heidelberg with Leo Königsberger as his thesis advisor. Königsberger had been appointed to a chair of mathematics at Heidelberg in 1869 and, having studied under Weierstrass himself, was at this stage in his career concentrating his research on the theory of elliptic functions. This was a perfect match with Pringsheim's mathematical interests, and he was awarded his doctorate in 1872.
In 1877 Pringsheim submitted his habilitation thesis to the University of Munich and began teaching there as a privatdozent. Two years later, in 1879, he married Hedwig Dohm. Hedwig's father was Ernst Dohm, a well known Berlin journalist, and her mother was Hedwig Schleh. Hedwig Dohm was an actress in Berlin before her marriage. Alfred and Hedwig had four sons, Erik, Peter, Heinz, and Klaus, and a daughter Katja. Klaus and Katja were twins; Klaus became a conductor, composer, music writer, while Katja studied physics and mathematics but agreed to give up her studies after marrying the famous writer Thomas Mann on 11 February 1905 in Munich. Peter became a physics professor after studying under Nernst in Berlin.
Pringsheim worked at the Ludwig-Maximilians University of Munich for his whole career. He was promoted to extraordinary professor in 1886, was elected to the Bavarian Academy of Sciences in 1898, and became a full professor in 1901 (the year in which his father died at the age of 80). Pringsheim retired in 1922. Freudental writes in  that Alfred and Hedwig Pringsheim made their:-
... home into a centre of Munich's social and cultural life. The novelist Thomas Mann, who was his son-in-law, wrote a novel based on the Pringsheim family. Pringsheim's refined wit was famous. His sprightly Bierrede was the acme of the yearly meeting of the Deutsche Mathematiker-Vereinigung and was mentioned by mathematicians throughout the year. His puns were famous ... Pringsheim was a brilliant lecturer and conversationalist ...
Pringsheim was extremely well off; as well as having considerable financial resources, he owned an extremely valuable collection of Italian Majolica earthenware. However, he was a patriotic German and in order to support his country during World War I, he had purchased a considerable quantity of War loans. He lost a lot of money with the War loans as he did with the hyperinflation of the first few years of the 1920s. Despite this he still had a mansion in the Arcisstrasse and his Majolica collection.
From 1933 to 1939 his life was made impossible as a non-Aryan. His house was taken away and he was forced to sell it to the Nazi party. It was not because the Nazis wanted to use the grand mansion for they demolished it and built a party headquarters on the site. In 1938 Pringsheim was forced out of the Bavarian Academy of Sciences by the president Alexander Mueller. In March 1939 the German Ministry of Trade authorized export of Pringsheim's Majolica collection to London for auction at Sotheby's, provided that 80% of the proceeds up to 20,000 pounds and 70% of the remainder be paid to the German Gold Discount Bank in foreign currency. Pringsheim was to receive the remaining proceeds. In exchange, Pringsheim and his wife were allowed to emigrate to Switzerland. The sale took place at Sotheby's, London, on 7 June 1939.
Before he left Germany to go to Zurich, Pringsheim gave to his friend Carathéodory a present of a very rare text from Jacob Bernoulli to his brother Johann Bernoulli containing the solution to the isoperimetric problem. Alfred and Hedwig Pringsheim moved to Zurich in 1939. He died there two years later, his wife dying in the following year.
Pringsheim worked on real and complex functions. His work :-
... is characterised by meticulous rigour rather than by great ideas.
He gave a very simple proof of Cauchy's integral theorem. He also has important results on the singularities of power series with positive coefficients. In 1893 he proved that a function is analytic if it is infinitely differentiable on an open interval and the radius of convergence r(x) of the Taylor series centred at x is bounded away from 0. There was a problem with the proof which was discovered and repaired much later by R P Boas.
Pringsheim criticised attempts by du Bois-Reymond to establish ideal boundaries between convergence and divergence. He also suggested that the paradoxes of the infinitary calculus arose from transferring properties of real numbers to infinite-dimensional domains where they fail, and agreed with Cantor that any use of infinitesimals in analysis would necessarily lead to inconsistencies.
An interesting comment concerning Pringsheim appears in . In 1761 Lambert made a first step in solving the old problem of squaring the circle by proving the irrationality of π. But the authors of many monographs and textbooks claimed that there was a gap in Lambert's proof concerning the convergence of the continued fraction expansion of the tangent function. Pringsheim was the first to note in 1898 that Lambert's proof was absolutely correct and exceptional for its time, since the expansion of the tangent function was not only written down formally, but also proved to be a convergent continued fraction. Pringsheim did a lot of work on continued fractions: he introduced the term 'unconditional convergence' of a continued fraction and also gave what is now known as the Pringsheim criterion which insures the convergence of a continued fraction in 1898.
A major work by Pringsheim was Vorlesungen über Zahlen- und Funktionenlehre which appeared in five parts. Part 1 was Reelle Zahlen und Zahlenfolgen and appeared in 1916 as did Part 2 Unendliche Reihen mit reellen Gliedern. Part 3 was entitled Komplexe Zahlen, Reihen mit komplexen Gliedern. Unendliche Produkte und Kettenbrüche and was published in 1921. Part 4 Grundlagen der Theorie der analytischen Funktionen einer komplexen Veränderlichen appeared in 1925 while Part 5 Eindeutige analytische Funktionen appeared in 1932.
Article by: J J O'Connor and E F Robertson