Maaß was primarily concerned with the theory of modular forms, being influenced in particular by Carl Ludwig Siegel (according to Maaß in his inaugural address for admission to the Heidelberg Academy, he met him in the early 1950s), whose Gesammelte Werke he also co-edited with K. S. Chandrasekharan, in addition to Hecke and Hans Petersson - Hecke's assistant at the time, who suggested the topic of his dissertation. He became known for his introduction of non-analytic automorphic forms in the 1940s (Maaß waveforms). Instead of satisfying Laplace's equation (as analytic functions do), they are eigenfunctions of the invariant Laplace operator; Maaß therefore called them waveforms. Internationally, these forms are known by his name. The motivation for the introduction came in part from Maaß's interest in connections of the theory of modular forms to number theory. Maaß was also concerned with automorphic functions in several variables, Siegel modular functions, and associated zeta functions.
Publications
Maass, Hans (1949), "Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen", Mathematische Annalen, 121: 141–183, doi:10.1007/BF01329622, ISSN 0025-5831, MR 0031519, S2CID 119494842
Maass, H. (1949), "Automorphe Funktionen von mehreren Veranderlichen und Dirichletsche Reihen", Abh. Math. Sem. U. Hamburg 16:72–100.
Maass, Hans (1950), "Modulformen zweiten Grades und Dirichletreihen", Mathematische Annalen, 122: 90–108, doi:10.1007/BF01342953, ISSN 0025-5831, MR 0037870, S2CID 121901325
Maass, Hans (1955), On Siegel's Modular Functions (PDF), Tata Institute of Fundamental Research Lectures on Mathematics, vol. 3
Maass, Hans (1964), Lal, Sunder (ed.), Lectures on modular functions of one complex variable (PDF), Tata Institute of Fundamental Research Lectures on Mathematics, vol. 29, Bombay: Tata Institute of Fundamental Research, ISBN 978-3-540-12874-8, MR 0218305
Maass, Hans (1971), Siegel's modular forms and Dirichlet series, Lecture Notes in Mathematics, vol. 216, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0058625, ISBN 978-3-540-05563-1, MR 0344198
References
Busam, R.; Freitag, E. (1999), "Hans Maass(17. 6. 1911-15. 4. 1992)", Jahresbericht der Deutschen Mathematiker-Vereinigung, 101 (3): 135–150, ISSN 0012-0456, MR 1713154