Soon after the award of his baccalaureate, Fredholm entered the Royal Technological Institute in Stockholm (known as KTH), studying there for a year. Bernkopf writes in :-
During this single year he developed an interest in the technical problems of practical mechanics that was to last all his life and that accounted for his continued interest in applied mathematics.After the year at the Royal Technological Institute, Fredholm enrolled as a student at the University of Uppsala. There was a very definite reason why he chose to go to Uppsala rather than Stockholm. At this early stage of his university education he had plans to eventually undertake doctoral studies and Uppsala was the only university in Sweden at that time which granted doctorates. Uppsala awarded him a Master of Science degree on 28 May 1888 but now he had some problems as how to proceed for it was only through Uppsala that he could be awarded a doctorate yet the person who he really wanted to work under was Mittag-Leffler. It was eight years earlier that Mittag-Leffler had been appointed as the first holder of the chair of mathematics at the newly founded University of Stockholm (called Stockholms Högskola at this time) so to be his student Fredholm had to study at Stockholm. He solved his difficulties by remaining registered for his doctorate at Uppsala but studying at Stockholm under Mittag-Leffler.
Fredholm's first publication came in 1890 when he published On a special class of functions in the Royal Swedish Academy of Sciences. In this paper he constructed a function which is analytic on the unit disk, is infinitely differentiable on the closed disk, but has no analytic continuation outside the disk. As was always the case with all the deep mathematical results which Fredholm produced, this result was inspired by mathematical physics, in this case by the heat equation. The result went further than earlier examples by Mittag-Leffler and Weierstrass. It so impressed Mittag-Leffler that he sent a copy of the paper to Poincaré. We should note, however, that there is an error in Fredholm's paper, but fortunately one which can be corrected. He used a result proved by Kovalevskaya in her Habilitation thesis but, through a misinterpretation, applied the result in a case where it was not valid. This paper is discussed in detail in  where the authors show how Fredholm's argument can be corrected.
On 30 May 1893 Fredholm was awarded his Ph.D. from the University of Uppsala and then on 31 May 1898 he received the degree of Doctor of Science from the same university. His 1898 doctoral dissertation involved a study of partial differential equations, the study of which was motivated by an equilibrium problem in elasticity. He solved his operator equation in the particular cases which arise in the study of the physical problem in his thesis (and in the paper which appeared in 1900 based on that thesis) while the general case was solved by Fredholm somewhat later and not published until 1908.
Fredholm is best remembered for his work on integral equations and spectral theory. The basis for his thinking is explained in  (see also ):-
We may ask what in Fredholm's eyes was the essential basis of his work. The answer is immediate: potential theory ... Already in 1895 after a seminar lecture in 1895 he had talked about Dirichlet's problem as one of elimination. ... Two years later in Stockholm a lecture about the 'principal solutions' of Roux and their connections with Volterra's equation led to a vivid discussion Finally, after a long silence Fredholm spoke and remarked in his usual slow drawl: in potential theory there is also such an equation.In fact much of this work was accomplished during the months of 1899 which Fredholm spent in Paris studying the Dirichlet problem with Poincaré, Émile Picard, and Hadamard. In 1900 a preliminary report on his theory of Fredholm integral equations was published as Sur une nouvelle méthode pour la résolution du problème de Dirichlet Ⓣ. Volterra had earlier studied some aspects of integral equations but before Fredholm little had been done. Of course Riemann, Schwarz, Carl Neumann, and Poincaré had all solved problems which now came under Fredholm's general case of an integral equation; this was an indication of how powerful his theory was.
Fredholm's contributions quickly became well known to the world of mathematics when Holmgren lectured on Fredholm's theory at Göttingen in 1901. Hilbert immediately saw the he importance of Fredholm's theory, and during the first quarter of the 20th century the theory of integral equations became a major research topic. Fredholm published a fuller version of his theory of integral equations in Sur une classe d'équations fonctionelle Ⓣ which appeared in Acta Mathematica in 1903. Hilbert extended Fredholm's work to include a complete eigenvalue theory for the Fredholm integral equation. This work led directly to the theory of Hilbert spaces.
Garding writes in :-
Fredholm's work on integral equations was met with great interest and boosted the morale and self-respect of Swedish mathematicians who so far had been working under the shadow of the continental cultural empires Germany and France. Integral equations had now become a new mathematical tool not confined to symmetrical kernels. It was developed during several decades and was seen as a universal tool with which it was possible to solve the majority of boundary value problems of physics. But the qualitative insight that the theory gave could also be achieved in a simpler way. The significance of Fredholm's work was more the qualitative insight than the explicit formulas.We have moved forward in time in our discussions in order to explain something of the importance of Fredholm's mathematical discoveries. As to his career, after the award of his Doctor of Science degree in 1898 he was appointed as a lecturer in mathematical physics at the University of Stockholm. He spent his whole career at the University of Stockholm being appointed to a chair in mechanics and mathematical physics on 28 September 1906. In 1909-1910 he was Pro-Dean and then Dean in Stockholm University.
However, Fredholm also held various other positions, becoming a civil servant in 1899 and then a Head of Department for the Swedish State Insurance Company in 1902. He was an actuary in the Skandia Insurance Company from 1904 to 1907. Fredholm also served on many government committees and he also served on the International Committee for Weights and Measures.
On 31 May 1911 Fredholm married Agnes Maria Liljeblad, the daughter of Protestant clergy parents, in Sankt Olai. At this time Fredholm was 45 years old and his wife was 33. Their eldest son, Bengt Ivar Fredholm, was born in 1912 and he became a major in the Swedish army.
Fredholm wrote papers with great care and attention so he produced work of high quality which quickly gained him a high reputation throughout Europe. However his papers required so much effort from him that he wrote only a few and in fact his Complete Works in mathematics comprises of only 160 pages. After 1910 he wrote little beyond revisiting his earlier work.
In  (see also ), Zeilon, who was a student of Fredholm's, described his style as a lecturer:-
In his Stockholm lectures Fredholm loved to talk at length about the great problems and methods of classical mathematical physics which had been the main theme of his scientific work. But this did not prevent him from talking about all parts of modern physics. Fredholm was not what is usually called a brilliant speaker He talked slowly in a monotone voice and it could happen that he got embroiled in computational mistakes at the blackboard But this had little importance In fact, his lectures revealed an unusual mastery of his subject and he had the ability of communicating to his students a feeling for the unity and logic of physical theory which is so apparent in his own written work.Fredholm also, as we indicated above, had a career in actuarial science and from 1902 onwards he occupied himself with studying various questions in this area. He made a particularly important contribution by proposing an elegant mathematical formula to determine the surrender value of a life insurance policy.
It is tempting to think that with two mathematical careers running in parallel, namely applications to physical applied mathematics and applications to actuarial science, Fredholm would have had little time for other interests. This, however, was not so for he was also a musician and for him this was more than just a leisure activity. As a young boy he played the flute, but later took up playing the violin. He particularly loved to play Bach. Again he combined his talents, applying his mechanical skills to music as well as to mathematics. Unlikely as it sounds, he built his first violin from half a coconut, while he also used his talents at building machines to make one to solve differential equations. His interest in machines and mechanics led to his membership of the Swedish Society of Engineers and he frequently provided scientific advice to that Society. At the time of his death he was working on the mathematics of the acoustics of the violin, but the unfinished work he left on this topic has proved impossible to understand.
Fredholm received many honours for his mathematical contributions, including the V A Wallmarks Prize for the theory of differential equations in 1903, the Poncelet Prize from the French Academy of Sciences in 1908, and an honorary doctorate from the University of Leipzig in 1909.
Article by: J J O'Connor and E F Robertson