Burkhardt was appointed to the post of assistant at the University of Göttingen in 1887 and he habilitated there in 1889. During his time in Göttingen he also spent a few months in Paris (winter 1893/94) in order to attend lectures by Poincaré, Picard and Félix Tisserand.
In 1897 Burkhardt was appointed to an ordinary professorship at the University of Zurich, succeeding Arnold Meyer (1844-1896). His inaugural lecture was entitled Mathematisches und naturwissenschafltiches Denken . In the same year of 1897 he married Mathilde Büdinger, daughter of Max Büdinger (1828-1902) who had been professor of history at Zürich and Vienna. Perhaps Burkhardt is best known as one of Albert Einstein's examiners. In 1905 Einstein submitted his thesis On a new determination of molecular dimensions to the University of Zürich and Burkhardt was appointed to examine the thesis. In his report he stated:-
The mode of treatment demonstrates fundamental mastery of the relevant mathematical methods. ... What I checked, I found to be correct without exception.From Zürich, Burkhardt moved to a chair at the Technical University of Munich in 1908. In Munich he was made an extraordinary member of the Bavarian Academy of Sciences in 1909 and then an ordinary member in 1912.
His main work was in analysis, particularly the theory of trigonometric series, and on the history of mathematics. Other topics on which Burkhardt published papers included groups, differential equations, differential geometry and mathematical physics. One of his papers, published in 1888, was Hyperelliptic sigma functions . Other papers include: Theorie der Cremonatransformationen (1892), Über Vectoranalysis (1896), Mathematisches und naturwissenschaftliches Denken (1902), Über Reihenentwicklungen nach oszillierenden Funktionen (1903), Zu den Funktionen des elliptischen Zylinders (1906), and Mathematische Miszellen aus der Vorlesungspraxis (1913).
He published a two-volume work Funktionentheoretische Vorlesungen in 1897 and 1899. The first volume is subtitled Einführung in die Theorie der analytischen Functionen einer complexen Verânderliehen and was reviewed by Maxime Bôcher  who writes:-
The object of the author in writing the little volume before us has been to furnish an introduction to the theory of functions which is not confined to the presentation of the methods of any one school (Cauchy, Weierstrass, Riemann) but blends these methods as far as possible into an organic whole. The author has been very successful in making his book an introduction not merely to those parts of the theory which have long been classical (algebraic, elliptic, and Abelian functions) but also to the many other important developments of the last thirty years. The mathematical public may well congratulate itself that a mathematician so thoroughly familiar with all sides of the subject as is Professor Burkhardt has undertaken the task of writing an elementary work along these lines.The second volume, which was published in 1899, was subtitled Elliptische Functional . This volume was reviewed by James Pierpont who writes :-
The theory of elliptic functions has developed so rapidly and in so many different directions in recent years that an elementary treatise of moderate compass which would afford a rapid survey of its many and heterogeneous parts has been a long felt want. ... The present volume meets the wants ... most successfully .... [of] those who regard the theory of elliptic functions as merely one division of a greater theory and who thus study the elliptic functions not only on account of the interesting properties they offer per se, but also as a means of becoming more familiar with the principles and methods of the theory of functions, or as a stepping stone to the more abstruse theories of the abelian transcendents and automorphic functions. We are so impressed with its many merits that we do not hesitate to predict for it a rapid and widespread popularity.This work was very popular and further editions where published by Burkhardt in 1903, 1907, and 1912. Then in 1921, G Faber, who succeeded Burkhardt at the Technische Hochschule in Munich, supervised the publication of a further edition of Burkhardt's text. The fourth edition of 1912 was translated into English by S E Rasor, Professor of Mathematics at Ohio State University, and published as Theory of functions of a complex variable in 1913.
The characteristic feature of the book is the predominance it gives to the ideas of Riemann. It is indeed remarkable, as Professor Burkhardt observes, that up to the present time no work on the elliptic functions has treated the theory from Riemann's standpoint. In several works on this subject we find reference to some of Riemann's ideas; but with the exception of Thomae's 'Abriss' they are cursory and inadequate. We feel sure that this novel and valuable feature will be widely appreciated. Another feature of the work is its comprehensiveness, accompanied by very moderate proportions. There is something so encouraging to the student in a textbook of moderate size. The main divisions of the theory have received attention in accordance with their relative importance. By seeking everywhere the simplest form of treatment, Professor Burkhardt has succeeded in compressing a great deal into a very small compass. The student who reads this book with care will gain a very good idea of the modern theory of elliptic functions, in spite of the gigantic size this theory has assumed.
Included among Burkhardt's historical works are Die Anfange der Gruppentheorie und Paolo Ruffini (1892) and Entwicklungen nach oscillierenden Funktionen und Integration der Differentialgleichungen der mathematischen Physik (1904-8). This latter work is rather chaotic, with unconnected textual quotations following each other, but nevertheless it is very useful as a historical source. It includes details of Cauchy's work in a number of different areas. In 1907 Burkhardt published Vorlesungen über die Elemente der Differential-und Integralrechnungen und ihre Anwendungen zur Beschreibung von Naturerscheinungen which was written as an introductory text for students of chemistry, mineralogy, and statistics.
The founders of the Encyclopädie der Mathematischen Wissenschaften were Wilhelm Meyer, Heinrich Weber, Walther von Dyck and Felix Klein. Early on, before even the title of the encyclopaedia had been decided, Klein proposed that Burkhardt be invited to join the committee. This committee had quite a difficult time since Burkhardt, Klein, von Dyck and particularly Meyer, never seemed to agree readily on any of the important decisions. However, the encyclopaedia proved highly important and influential. Burkhardt collaborated with Wilhelm Meyer as joint authors of the article on potential theory which they wrote for the Encyclopädie in 1899. Burkhardt wrote several other important articles for the Encyclopädie including Trigonometrische Reihen und Integrale .
The first International Congress of Mathematicians was held in Zürich in 1897.  At the committee meeting for the first International Congress of Mathematicians on 21 January 1897, Geiser could announce that Burkhardt had agreed to sign the invitation to the congress. After that meeting he was also asked to join the reception committee. He attended most of the committee meetings in 1897. Burkhardt was the only professor from the University of Zürich on the organising committee. Burkhardt also attended the 1904 ICM in Heidelberg and the 1912 ICM in Cambridge. As a member of the international committee of the latter he represented the Royal Bavarian Academy of Sciences in Munich.
Zygmunt Janiszewski, Isabel Maddison, Ludwig Berwald, and Edward Van Vleck are among the many students that Burkhardt taught, and these four are included in this archive.
Article by: J J O'Connor and E F Robertson