Wirtinger studied mathematics at the University of Vienna, beginning his course in 1884. His main lecturers were Emil Weyr and Gustav von Escherich (1849-1935). Gustav von Escherich had studied at the University of Vienna and worked at Graz before taking up a position in Vienna in 1884, the year Wirtinger began his course. He taught Wirtinger about the new approach to analysis which Weierstrass had developed. However, it was Emil Weyr who inspired Wirtinger with his courses on synthetic geometry and this led to Wirtinger's first two publications and his doctoral thesis. He received his doctorate from the University of Vienna in 1887 for his thesis Über eine kubische Involution in der Ebene Ⓣ. His thesis was not published but, as we just noted, he had published two papers before he submitted his thesis, namely Über die Brennpunktkurve der räumlichen Parabel Ⓣ (1886) and Über rationale Raumkurven 4. 0rdnung Ⓣ (1886).
For a fairly complete list of all Wirtinger's mathematical publications see THIS LINK.
After the award of his doctorate, he continued to study at the University of Vienna. However, the largest influence on the direction of his research came from Felix Klein when he studied at the University of Berlin and at the University of Göttingen. He was able to visit Berlin and Göttingen thanks to a travel grant that he was awarded after obtaining his doctorate which funded his studies in Germany for a year. In Berlin he attended lectures by Karl Weierstrass, Leopold Kronecker and Lazarus Fuchs but these did not make nearly such a lasting impression on him as the time he spent at Göttingen. There he attended Felix Klein's lectures on Abelian functions and partial differential equations of physics. Along with William Osgood he attended Klein's seminar and Wirtinger and Klein became life-long friends. Wirtinger habilitated at the University of Vienna in 1890 and, in addition to teaching there, he was appointed as an assistant to Emanuel Czuber at the Technical University in Vienna. Czuber had just been appointed as an ordinary professor. This enabled Wirtinger to get married and, in 1890, he married Amalia Feyertag; they had three sons and two daughters. Sadly two of the sons died just before, or during, World War I. Hans Hornich writes in  that Amalia was a loving wife for Wirtinger but she did not play any role in his professional career. Amalia died a few years before her husband.
The work he undertook after his appointments in Vienna appeared in a number of papers such as Über eine Verallgemeinerung der Kummer'schen Fläche und ihre Beziehungen zu den Thetafunktionen zweier Variablen Ⓣ (1890), Zur Theorie der Abel'schen Funktionen vom Geschlecht 3 Ⓣ (1891) and Untersuchungen über Abel'sche Funktionen vom Geschlecht 3 Ⓣ (1891). His most important work during these years, however, was the book Untersuchungen über Thetafunktionen Ⓣ (1895) which led to Wirtinger receiving the Beneke Prize from the Faculty of Philosophy of Göttingen. This book was a work of major importance on the general theta function. In it Wirtinger combined ideas from Riemann's function theory with ideas from Klein to prove results of great significance. It was this book, which he had developed from work begun at Göttingen, that brought Wirtinger's name to the fore as a leading mathematician.
Constantin Caratheodory writes in :-
However, Wirtinger was not a specialist who only worked on one problem and did not have a sense for the essentials of science. In his lectures he always stressed the historical context and had a remarkable interest in the philosophical basis of mathematics. He was economical with his publications, but every single paper - even if only a few pages long - does not only contain surprising thought of exceptional beauty but also proof that he could combine his perfect geometrical insight with his rare skill of mastering the mathematical symbolism.Wirtinger's range of mathematics was quite exceptional. Not only did he write beautiful papers on function theory, he also wrote on geometry, algebra, number theory, plane geometry and the theory of invariants. He also wrote several important papers on Lie's translation manifolds and their application to abelian integrals. This list would certainly make one believe that Wirtinger's range within pure mathematics was very wide but his interests went well beyond pure mathematics. He published results on Einstein's theory of relativity and other areas of mathematical physics. He also worked and lectured on statistics and wrote a work on rainbows. In fact Erwin Schrödinger attended his lectures on mathematical statistics.
As a consequence of the outstanding work he had published, in 1895 Wirtinger was promoted Extraordinary Professor at Vienna where he lectured on a wide variety of topics and continued to produce outstanding research. Then, later the same year, he accepted a chair at the University of Innsbruck. During his years in Innsbruck he wrote the article Algebraische Funktionen und ihre Integrale Ⓣ for the Encyklopädie der mathematischen Wissenschaften. He also worked on a paper with Adolf Krazer (1858-1926) but Wirtinger became occupied with other work and the paper appeared with Krazer as the only author. Many mathematicians at this stage in their careers wrote textbooks based on lecture courses they were giving but Wirtinger was so full of new ideas and solving problems that he was totally engrossed in research.
Wirtinger returned to a chair at the University of Vienna in 1903. This chair had been held by Leopold Gegenbauer but he had stopped teaching due to ill health in 1901 and the chair became vacant following his death in June of 1903. Wirtinger held this chair for 32 years. When he was appointed his colleagues included Gustav von Escherich, who had taught him as a student, and Franz Mertens who was appointed as an ordinary professor in Vienna in 1894. Mertens retired in 1911 and Philipp Furtwängler was appointed in 1912. Another of Wirtinger's colleagues was Hans Hahn who was a privatdozent in Vienna from 1905 and, after several years away, returned to a chair in Vienna in 1921. Wirtinger also took on administrative duties in the University, in particular he was Dean of the Faculty of Arts in the academic year 1915-16.
Let us say a little about Wirtinger's teaching at the University of Vienna. The lecturing was based on a three-year cycle and each professor had to give the introductory lectures in the first semester once every three years. In the other two years Wirtinger gave advanced lectures on function theory and on algebraic or elliptic functions. In addition to these courses, Wirtinger ran a two-hour seminar but he here found it somewhat harder to interact with students due to his deafness and, as a consequence, usually lectured rather than encouraging student contributions. As a teacher of advanced students, he was highly successful, but his lectures required the students to have a good background knowledge before they could benefit. This meant that when he lectured to large classes with varied backgrounds, he did not get nearly as many attending as others teaching similar topics at Vienna. For example in  Hans Hornich notes that it was a period when the universities were overcrowded and introductory classes by Philipp Furtwängler or Hans Hahn would have around 400 students crowded into the lecture theatre while when Wirtinger taught a similar course there would be barely 50 students. However, his seminar would attract 3 or 4 students who were undertaking research and they would find the sessions both stimulating and extremely profitable. In fact he trained a number of outstanding young mathematicians in these sessions. We should also mention, when describing his teaching, that Wirtinger was interested in the history of mathematics and also in philosophy. Particularly when he was in the later stages of his career he would hold seminars in the history of mathematics, particularly on the ancient Greek mathematicians. His knowledge of ancient Greek allowed him to tackle these topics reading works in the original Greek.
In 1906 Wirtinger published Über die Entwicklung einiger mathematischer Begriffe in neuerer Ⓣ. A Press report from 1906 includes the following:-
The mathematician Wilhelm Wirtinger, who established his scientific reputation over 10 years ago with the solution of a difficult problem (the general theta function), reported today in the Academy of Sciences on "the development of some mathematical concepts in modern times." The definition of these terms is not always easy, as Wirtinger shows with the example of an infinite set: "Even Galileo had mentioned that the infinite set of integers seems to be far greater than the set of square numbers, since the square numbers are becoming increasingly rare the further one progresses along the series of numbers, while, on the other hand, the size of both sets would have to be the same because each number corresponds to a perfect square."For Wirtinger there seems to have been relatively little outside mathematics. He sometimes travelled through Munich to the Achensee for a summer vacation. Often he would be accompanied by Hans Hahn who did some teaching at the university of Innsbruck. Others taking their summer vacations on the Achensee would have relaxed and taken the chance to have a break from academic work. Not so Wirtinger, who sat and watched the waves on the lake and thought about the mathematical theory which lay behind them. After his holiday he wrote up his results to provide an improved theory of capillary waves.
When Kurt Reidemeister was appointed as associate professor of geometry at the University of Vienna in 1923 he became a colleague of Wirtinger. At that time Wirtinger was interested in knot theory and he showed Reidemeister how to compute the fundamental group of a knot from its projection. Wirtinger's method was first published in work of Emil Artin in 1925. Wirtinger certainly did not lessen his mathematical activity as he grew older. At the age of 71 he wrote the first of a series ground-breaking papers on higher dimensional spaces.
Among the mathematicians who Wirtinger taught while he held the chair at Vienna are Otto Schreier, Kurt Gödel, Johann Radon and Olga Taussky-Todd. We note that Hans Hornich, the author of , was also one of Wirtinger's students.
It was not only with his research and teaching that Wirtinger contributed to the mathematical sciences but also through his interest in mathematical education. We quote (with minor additions) from  about his role in the International Commission on Mathematical Instruction:-
When the Austrian sub-committee of the Internationale mathematische Unterrichts-Kommission (IMUK) was established in 1909, Wirtinger was nominated as one of the three delegates and thus became a member of IMUK. He continued in this function until the dissolution in 1920. In 1936, after the reestablishment of IMUK, Wirtinger was elected an honorary member of the committee. For the IMUK meeting in Milan in 1911, he was a member of the committee preparing the reports on the mathematical training of students of the natural sciences and gave the report for Austria. In 1933, he published, together with Hans Hahn and Erwin Kruppa (1885-1967), the report on the training of mathematics teachers in Austria.The report mentioned in this quote was Die Ausbildung der Mathematiklehrer an den Mittelschulen Oesterreichs Ⓣ published in L'Enseignement Mathématique.
Wirtinger received many honours for his achievements. In 1907 the Royal Society of London awarded him their Sylvester Medal and Wirtinger travelled to England to receive the honour. He was the third recipient of the medal which had been previously awarded to Henri Poincaré and Georg Cantor, so indeed this ranked him among the very greatest mathematicians of his day. Among other honours was his election to the Austrian Academy of Sciences (1905), the Berlin Academy of Science, the Göttingen Academy of Sciences, the Accademia dei Lincei (1927) and the Bavarian Academy of Sciences and Humanities (in 1931). He received honorary doctorates from the University of Hamburg (in 1926) and the University of Oslo. He was made an honorary member of the International Commission on Mathematical Instruction on 15 July 1936. The title was conferred on Wirtinger (and others including Castelnuovo, Dickstein, Enriques and Loria) at the International Congress of Mathematicians held in Oslo.
Article by: J J O'Connor and E F Robertson