# Fredholm, Erik Ivar

# Erik Ivar Fredholm

Swedish mathematician and educator Erik Ivar Fredholm (1866–1927) proved in early childhood that he was a brilliant student of numerical theory. By the time Fredholm had completed his doctoral studies in 1898, he had also shown himself to be a brilliant theorist by developing the integral equation on which would be built the quantum theory and consequently a future of remarkable discoveries that have altered the way people live.

Fredholm was born on April 7, 1866, in Stockholm, Sweden, the first son of Ludvig Oscar and Catharina Paulina (Stenberg) Fredholm. Ludvig Fredholm, a merchant, amassed a fortune when his business was able to replace gas lamps with electric lamps. His wife was also the daughter of a wealthy merchant, and the couple had married on May 2, 1861, in Arboga, Sweden. In 1875, nine years after Erik's birth, another son, John Oscar was born to the couple. Well-educated themselves and able to afford the best education for their sons that money could buy, the Fredholm's sent their oldest son to the Beskowska School in Stockholm, where he received his diploma on May 16, 1885. As a child he played the flute and maintained a love of music throughout his life, as is characteristic of many mathematicians. The following year he studied at the Polytechnic Institute in Stockholm. According to M. Bernkopf in the *Dictionary of Scientific Biography,* "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 continuing interest in applied mathematics."

## Pursued Career in Education

Fredholm pursued his education and in 1886 enrolled at the University of Uppsala. At that time, Uppsala was the only institution in Sweden that awarded doctorates. Fredholm received a bachelor of science degree in 1888 and a Ph.D. on May 30, 1893. Ten years later he would receive a Doctor of Science degree from Uppsala, as well. He went to study at the University of Stockholm the same year he received his bachelor's degree, having heard that teaching in Stockholm was superior. His professor there was Mittag-Leffler, a man well known for his unique brand of instruction. Fredholm remained enrolled at Uppsala in order to obtain his doctorate, but he stayed at Stockholm for the rest of his career, becoming first a lecturer in mathematical physics in 1898 and a professor of rational mechanics and mathematical physics on September 28, 1906. He also served the university as pro-dean beginning in 1909 and then as dean the following year.

Fredholm's responsibilities as an educator at the University of Stockholm did not preclude him from pursuing other careers. In addition to his university affiliation, he worked as a civil servant beginning in 1899 and served as a department head at the Swedish State Insurance Company in 1902. From 1904 until 1907 Fredholm worked as an actuary for the Skandia Insurance Company, and it was while he was there that he developed a formula to determine the surrender value of a life insurance policy. According to J. J. O'Connor and E. F. Robertson, in an essay posted on the *University of St. Andrews School of Mathematics and Statistics Web site,* noted that 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." In actuality, Fredholm actively pursued music even into his later years, His focus eventually shifted from the flute to the violin, with the compositions of J. S. Bach among his favorites. With an intellect equally at home with music and with mathematics, Fredholm was also actively engaged in the physical world and took time to build machines that could solve differential equations. Fredholm's occupation with mechanics and machines was encompassing enough for him to become a member of the Swedish Society of Engineers, often adding his own expertise in scientific matters to the society.

## Began to Publish Research

Fredholm first published in 1890; his paper "A Special Class of Functions" was released under the auspices of the Royal Swedish Academy of Sciences. According to O'Connor and Robertson, "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." Mittag-Leffler was so impressed that he sent a copy of Fredholm's paper to the well-known French mathematician Jules Henri Poincaré.

Fredholm's doctoral thesis reveals his first major work in partial differential equations. Two years after he received his degree, the paper was published and revealed to the world the solution to the integral equation for which he became famous. The very first equation he completed had already been investigated for nearly a century by U.S. astronomer George Hill, although no satisfactory results had ever been reached by Hill. According to Bernkopf, Niels Abel had solved a different form of it in 1823, but that also had a different function. In 1884 Carl Neuman reached a partial solution to the equation by use of a particular method called an "iteration scheme," but had to add a condition to guarantee the convergence of his solution, noted Bernkopf. An integral equation is a mathematical equation that includes an unknown function, "f." The integral equation solved by Fredholm—which he went on to further study in two algebraic forms—now bears his name and is widely used in quantum mechanics.

Regarding the true significance of Fredholm's work, N. Zeilon noted in his *Obituary of Erik Ivar Fredholm:* "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.'" Fredholm spent several months in Paris in 1899 studying with French mathematicians Poincaré, Emile Picard, and Hadamard, where he was able to reach many of his conclusions that contributed to the success of his work.

## Far-reaching Implications

When fellow mathematician Erik Holmgren presented 35-year-old Fredholm's equation to the mathematical world in Göttingen, Germany, in 1901, many were immediately aware of its importance, and for the next 25 years integral equations became a major area of mathematical research. Mathematician David Hilbert eventually extended Fredholm's work and his theory of "Hilbert spaces" became an important step in the development of the quantum theory that describes the behavior of particulate matter—atoms, electrons, and the like—during short intervals.

In *Mathematics and Mathematicians: Mathematics in Sweden before 1950* author Lars Garding explained that "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.… 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 and 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 specific formulas."

## Gained Family Later in Life

Dedicating much of his early adulthood to education and research, Fredholm finally married in middle age. His wedding occurred in Sankt Olai, Sweden, on May 31, 1911. Fredholm was now 45 years of age; his wife, Agnes Maria Liljeblad, was 33 at the time of the marriage, and was the daughter of members of the Protestant clergy. The couple had several children.

Fredholm's mathematical contributions brought him many honors. His awards include the V. A. Wallmarks prize, awarded to the mathematician in 1903; the Poncelet prize presented him by the French Academy of Sciences in 1908; and an honorary doctorate from the University of Leipzig presented to Fredholm in 1909.

When he died at the age of 61 in 1927, Fredholm was reportedly at work on calculating the mathematics involved in the acoustics of the violin. The papers he left on this project have proven impossible for other to understand, leaving conclusions undetermined. In addition to his work on integral equations, Fredholm also contributed mathematical insights into spectral theory.

## Books

*Dictionary of Scientific Biography,* Volume 5, Charles Scribner's Sons, 1980.

Garding, Lars, *Mathematics and Mathematicians: Mathematics in Sweden before 1950,* American Mathematical Society, 1998.

Zeilon, N. *Obituary of Erik Ivar Fredholm, Oeuvres complete de Ivar Fredholm,* [Malmo, Sweden], 1955.

## Periodicals

*Physics World,* December 1999.

## Online

"Erik Ivar Fredholm," *University of St. Andrews School of Mathematics and Statistics Web site,*http://www.-history.mcs.standrews.ac.uk/history/Mathematicians/Fredholm.html (December 2002).

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