Stefan was brought up in Czestochowa where he attended primary school and then the gymnazie (the local Gymnasium). He completed his studies at the Gymnasium in 1913 and began his university studies in the School of Engineering at the University of Breslau. Following the usual practice of students at this time of studying at more than one institution, he moved to the School of Engineering at the University of Vienna in 1915 and remained there to complete his Diplomingenieur (Engineering degree) in 1920. He decided to continue his studies and, having been very enthusiastic about both pure and applied mathematics while taking his engineering degree, entered the Institute for Applied Mathematics at the Berlin University in 1921. This Institute had only been founded in 1920 with Richard von Mises as its director. Von Mises strongly influenced Bergman and this influence had a continuing impact on his scientific work for the rest of his career. Another of his professors at Berlin also had a major impact on his scientific development. This was Erhard Schmidt who had been awarded his doctorate by the University of Göttingen for a thesis on integral equations written under Hilbert's supervision.
Bergman worked for his doctorate under von Mises' supervision, and was awarded the degree in 1922 for his thesis Über die Entwicklung der harmonischen Funktionen der Ebene und des Raumes nach Orthogonalfunktionen Ⓣ. The authors of  write:-
His thesis dealt with the development of all analytic functions in a given domain in terms of a fixed set of orthogonal functions. Its results were applied, on the one hand, to fluid dynamics, conformal mapping and potential theory and led, on the other hand, to the "Bergman kernel function" which is one of his major achievements in pure mathematics.See Kracht and Kreyszig  for further details on the context of Bergman's work on kernel functions and their later generalisations by Aronszajn and others.
Perhaps it was Bergman's deep understanding of both pure and applied mathematics which let him develop powerful method in potential theory which he applied to electrical engineering, elasticity and fluid flow. Bergman used the theory of integral equations as developed by Erhard Schmidt and David Hilbert :-
To obtain a large number of harmonic functions in space, he applied and generalised the Whittaker method to create such functions by means of integrals over analytic functions. Using algebraic-logarithmic analytic functions as generators in the integral, he created harmonic functions that are multivalued in space and have closed branch lines. This led him further to a general theory of integral operators that map arbitrary analytic functions into solutions of various partial differential equations. he devoted many years of work to this topic, producing a monograph in 1969.While Bergman was working in Berlin on his habilitation his father, after many years of bad eyesight, went completely blind in 1923 and died on 29 December 1929. In 1930 Bergman was appointed as a privatdozent in both the Institute for Mathematics and the Institute for Applied Mathematics at the University of Berlin with an habilitation thesis on the behaviour of kernel functions on the boundary of their domains. It was highly unusual for anyone to obtain a post in both Institutes, but entirely appropriate in Bergman's case. However he was forced from his positions in Berlin in 1933 after Hitler came to power. On 7th April 1933, Hitler introduced a law for the "Restoration of the civil service". This meant that all non-Aryans and Jewish civil servants were dismissed from their positions with the exception of those who either had fought in the Great War or had been in office since August 1914. To escape from the anti-Semitic Nazi regime, Bergman went to Russia in 1934 working at Tomsk in Siberia until 1936, then at Tbilisi in Georgia during 1936-37 :-
The success of his stay in the Soviet Union is best shown by the fact that some of his students became leading mathematicians in their own right, such as Vekua, Fuchs, Kufarev, etc.For the second time in four years, Bergman was forced to move because of political decisions. This time it was the purges by Stalin of foreign scientists which forced him to leave Russia in 1937. He went to the Institut Henri Poincaré in Paris where he wrote an important two-volume monograph on complex analysis.
In 1939 World War II broke out and the German armies invaded France. Bergman was forced to flee for the third time, going to the United States with von Mises as his sponsor. He lectured first at the Massachusetts Institute of Technology in Cambridge, Massachusetts, where he gave a series of lectures on Theory of pseudo-conformal transformations and its connection with differential geometry during 1939-40. Then he taught at Yeshiva College in New York City, moving then to Brown University in Providence, Rhode Island. While at Brown University he participated in the Summer School in 1941 Advanced instruction and research in mechanics which resulted in the publications Partial Differential Equations and Fluid dynamics. In the following summer there was another summer school on Advanced instruction and research in mechanics which resulted in the text The Hodograph Method in the Theory of Compressible Fluid (1942). As part of the war effort he also worked for the National Advisory Committee for Aeronautics producing several reports such as: Graphical and Analytical Methods for the Determination of a Flow of a Compressible Fluid Around an Obstacle (July 1945); On Two-dimensional Flows of Compressible Fluids (August 1945); and On Supersonic and Partially Supersonic Flows (December 1946). In 1945 he joined von Mises in Harvard. In the previous year von Mises wrote the following concerning Bergman's work :-
It has been known for a century that the problem of finding the two-dimensional potential flow of an incompressible fluid can be solved by means of complex variables: To each analytic function of a complex variable corresponds a particular solution of the potential problem and vice versa. Several years ago Stefan Bergman discovered that essentially the same is true for a vast class of partial differential equations which includes the potential equation as the simplest case. Bergman gave explicit formulae which allow a solution of a given differential equation to derive from an arbitrarily chosen analytic function (in some instances from a pair of real functions) and proved that all solutions can be derived in this way. Now, two of Bergman's pupils, Bers and Gelbart, found that in a special case the analogy can be carried much farther. They consider a special type of differential equation, yet more general than the potential equation, and build up a system of solutions in close analogy to the procedure followed in the theory of analytic functions. Fortunately, this restricted type includes the problem of a two-dimensional flow of a compressible fluid which is today in the centre of interest in aviation. Though all solutions obtained by Bers and Gelbart can be derived by Bergman's methods also, it must be expected that the new approach will prove very useful.In 1950 Bergman married Adele Adlersberg and in the same year began a collaboration with Menahem M Schiffer. His stay in Harvard lasted until 1952 when he moved to the Mathematics Department at Stanford University where he spent the rest of his career. He was most interested in research and seldom taught, in fact this made it difficult for him to get a post since he made it known that he required a non-teaching post.
Bergman is best known for his research in several complex variables, as well as the Bergman projection and, as was mentioned above, the Bergman kernel function which he invented in 1922 while at Berlin University. He is also known for applications of the kernel function to conformal mappings which he explains in his classic text The Kernel Function and Conformal Mapping (1950). Donald Spencer writes:-
This book is a survey of methods and results centring around orthogonal analytic functions and associated kernel functions defined for multiply-connected domains of the complex plane. ... The book is written in an elementary style which avoids abstract concepts almost entirely, and much of it can be read by anyone having an elementary knowledge of complex variables.A revised edition of this book which was considerably enlarged and included many new results was published in 1970. In 1953 Bergman and Schiffer published Kernel functions and elliptic differential equations in mathematical physics. P R Garabedian writes:-
In this book the authors collect their researches of the last few years on elliptic partial differential equations. The first part of the book is devoted largely to background material on heat conduction, fluid dynamics, electrostatics, and elasticity, together with the more formal applications of variational formulas and the kernel function. The second part lays more stress on rigor, and treats fundamental solutions, reduction of boundary value problems to integral equations, orthonormal systems and kernel functions, eigenvalue problems associated with the kernels, variational theory of domain functions, comparison domains, basic existence theorems, and dependence of solutions on the boundary data or on the coefficients of the differential equation. The presentation is in an easy flowing style, and the material should prove to be a most useful guide to those interested in the more advanced theory of linear elliptic partial differential equations.Bergman published Integral operators in the theory of linear partial differential equations in 1961. This book was reviewed by Copson (see ) and White (see ). J Mitchell also reviews the book, writing:-
This treatise gives a summary of the author's numerous contributions from 1926 to 1961 to the theory of solutions of linear partial differential equations in two and three real variables by means of integral operators which usually involve analytic functions of one, or sometimes two, complex variables. Results in the theory of one complex variable on such topics as analytic continuation, the residue theorem, Hadamard's theorems on the connection between the coefficients of the power series development of an analytic function and the character and location of the singularities and on Abelian integrals are used to give information concerning domains of regularity, series expansion, singularities and integral relations for the solutions.In 1974 Charles Fefferman found a deep application of Bergman's ideas to biholomorphic mappings and a conference on several complex variables, held in 1975, had Bergman's work as its main theme. Bergman attended the conference, clearly enjoying the central role of his work.
Krantz, writing about Bergman in , explained:-
Bergman was an extraordinarily kind and gentle man. He went out of his way to help many young people begin their careers, and he made great efforts on behalf of Polish Jews during the Nazi terror. He is remembered fondly by all who knew him.Bergman died in 1977, and after his wife died the terms of her will ensured that funds should be used for a special prize to honour her husband's name:-
The American Mathematical Society was asked by the Wells Fargo Bank of California, the managers of the Bergman Trust, to assemble a committee to select recipients of the prize. In addition the Society assisted Wells Fargo in interpreting the terms of the will to assure sufficient breadth in the mathematical areas in which the prize may be given. Awards are made every year or two in: 1) the theory of the kernel function and its applications in real and complex analysis; or 2) function-theoretic methods in the theory of partial differential equations of elliptic type with attention to Bergman's operator method.
Article by: J J O'Connor and E F Robertson