15 July 1865|
Ybbs an der Donau, Austrian Empire
15 January 1945 (aged 79)|
Ybbs an der Donau, Greater German Reich
|Alma mater||University of Vienna|
Complex analysis of one and several variables|
|Awards||Sylvester Medal (1907)|
University of Innsbruck|
University of Vienna
Gustav Ritter von Escherich
|Doctoral students||see the "Teaching activity" section|
He was born at Ybbs on the Danube and studied at the University of Vienna, where he received his doctorate in 1887, and his habilitation in 1890. Wirtinger was greatly influenced by Felix Klein with whom he studied at the University of Berlin and the University of Göttingen.
He worked in many areas of mathematics, publishing 71 works. His first significant work, published in 1896, was on theta functions. He proposed as a generalization of eigenvalues, the concept of the spectrum of an operator, in an 1897 paper; the concept was further extended by David Hilbert and now it forms the main object of investigation in the field of spectral theory. Wirtinger also contributed papers on complex analysis, geometry, algebra, number theory, and Lie groups. He collaborated with Kurt Reidemeister on knot theory, showing in 1905 how to compute the knot group. Also, he was one of the editors of the Analysis section of Klein's encyclopedia.
A partial list of his students includes the following scientists:
- Wirtinger, Wilhelm (1926), "Zur formalen Theorie der Funktionen von mehr komplexen Veränderlichen" [On the formal theory of functions of several complex variables], Mathematische Annalen (in German), 97 (1): 357–375, doi:10.1007/BF01447872, JFM 52.0342.03, available at DigiZeitschirften. In this important paper, Wirtinger introduces several important concepts in the theory of functions of several complex variables, namely Wirtinger derivatives and the tangential Cauchy–Riemann condition. The paper is deliberately written from a formal point of view, i.e. without giving a rigorous derivation of the properties deduced.
- Wirtinger, Wilhelm (1936), "Eine Determinantenidentität und ihre Anwendung auf analytische Gebilde in euklidischer und Hermitescher Maßbestimmung" [A determinant identity and its application to analytic forms in Euclidean and Hermitian distances], Monatshefte für Mathematik (in German), 44 (1): 343–365, doi:10.1007/BF01699328, JFM 62.0815.01, MR 1550581, Zbl 0015.07602.
- Wirtinger, Wilhelm (1936), "Ein Integralsatz über analytische Gebilde im Gebiete von mehreren komplexen Veränderlichen" [An integral theorem on analytic forms on a domain of several complex variables], Monatshefte für Mathematik (in German), 45 (1): 418–431, doi:10.1007/BF01708005, JFM 63.0308.03, MR 1550660, Zbl 0016.40802.
- Wirtinger derivatives
- Wirtinger inequality (2-forms)
- Wirtinger's inequality for functions
- Wirtinger presentation
- Wirtinger's representation and projection theorem
- Hornich, Hans (1948), "Wilhelm Wirtinger", Monatshefte für Mathematik (in German), 52 (1): 1–12, doi:10.1007/BF01320497, MR 0024394, Zbl 0030.10102, available at DigiZeitschirften. An ample commemorative paper containing a list of Wirtinger's publications.