**Fredholm** is best remembered for his work on integral equations and spectral theory.

- Ludvig Fredholm was a merchant who made his fortune replacing gas lamps with electric lamps.
- They were wealthy, well-educated people who wanted the very best education for their sons Ivar and John Oscar (born 1875), and they were able to afford the very best.
- Ivar, who was born in the Klara area of Stockholm City, proved a brilliant school pupil at the Beskowska School in Stockholm and he was awarded his baccalaureate on 16 May 1885.
- Soon after the award of his baccalaureate, Fredholm entered the Royal Technological Institute in Stockholm (known as KTH), studying there for a year.
- After the year at the Royal Technological Institute, Fredholm enrolled as a student at the University of Uppsala.
- 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.
- Fredholm's first publication came in 1890 when he published On a special class of functions in the Royal Swedish Academy of Sciences.
- 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.
- We should note, however, that there is an error in Fredholm's paper, but fortunately one which 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.
- 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.
- 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.
- 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 Ⓣ(On a new method for solving the Dirichlet problem).
- 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 Ⓣ(On a class of functional equations) which appeared in Acta Mathematica in 1903.
- Hilbert extended Fredholm's work to include a complete eigenvalue theory for the Fredholm integral equation.
- 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.
- 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.
- Fredholm also served on many government committees and he also served on the International Committee for Weights and Measures.
- 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.
- 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.
- 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.
- 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.

Born 7 April 1866, Stockholm, Sweden. Died 17 August 1927, Danderyd, County of Stockholm, Sweden.

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Origin Sweden

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive