Linear Methods of Applied Mathematics

Evans M. Harrell II and James V. Herod*

This document contains some brief biographical and historical notes as part of the hypertext, Linear Methods of Applied Mathematics. When a name such as Fourier in the text is highlighted, it usually indicates a link to part of this document.

Much more historical information is available from the History of Mathematics Archive or the Museum of the History of Science.

Sir George Biddell

Actually, these are equivalent to Bessel functions with a fractional index.

Jean le Rond d'

In our text we encounter his original idea for solving the
wave equation, which evolved into the *method of characteristics*.
He also contributed to celestial mechanics, geometry, hydrodynamics,
and complex analysis. His most important scientific writing was his Traité
de Dynamique (1743), but perhaps his greatest contribution to mankind was
his development of the theory of the construction of eyeglasses.

Daniel

Friedrich Wilhelm

V. Ya.

Lennart Carleson, Swedish mathematician working at the Royal Institute of Technology in Stockholm. In a 1966 article, he proved that the Fourier series for a square-integrable function converges almost everywhere.

Baron Augustin-Louis

Unlike most of the mathematicians of his time, who were revolutionaries, Cauchy was ardently conservative. He supported Catholic and royalist causes throughout the revolutionary period in France, and even went into exile with the Bourbons in 1830.

James

More about Clerk Maxwell

Edward Salisbury

Paul Adrien Marie

Pierre Louis

Leonhard

Euler had a large family and although he was early recognized as a genius, he could not find good employment in his native Switzerland. Fortunately, during his early career Catherine the Great founded the Petersburg Academy, the ancestor of the Russian Academy of Sciences, and Euler accepted an invitation to be one of the founding professors of this institution. He became a good friend of Frederick the Great of Prussia - the original "enlightened despot" - and carried on correspondence with him even while Russia and Prussia were at war.

Euler's contributions to our subject include:

- Analyzing vibratory motion
- Introducing the formula exp(i a x) = cos(a x) + i sin(a x)
- Discovering how to solve many ordinary differential equations, including Bessel's equation, before Bessel's birth
- Contributing to the theory of infinite series

Baron Joseph

Fourier was also politically talented, becoming at various times the French Commissioner to the Sultan, the Prefect of Isère (comparable to being a governor), an academician, and the chief engineer of many projects, including the construction of a major route between Grenoble and Turin (Torino), still in use today. He managed to remain influential when Napoleon came to power, after Napoleon's defeat, during his return, and after Waterloo. Another interesting accomplishment of Fourier was his discovery and sponsorship of a young linguist named Champollion, who went on to decipher the Rosetta stone.

Fourier's contributions to our subject include:

- Founding the theory of heat flow and deriving the heat equation
- Realizing that trigonometric series could represent
*arbitrary*functions - Discovering many of the basic formulae applying to trigonometric series
- Expanding the idea of a function beyond that of a formula
- popularizing the modern sign for the definite integral

Among his other contributions to science was his realization, in 1827, that atmospheric gases were responsible for keeping the temperature on earth warm enough for life.

Erik Ivar

Carl Friedrich

Josiah Willard

George

Evans

Oliver

Charles

David

Gustav

L.

Count Joseph Louis

Marquis Pierre Simon de

Adrien Marie

Legendre's contributions to our subject include:

- Discovering the notion of mean-square approximation (also discovered by Gauss)
- Inventing the Legendre polynomials and spherical harmonics
- Understanding the analysis of PDEs with spherical or ellipsoidal symmetry.

Gustav

Franz E.

John

Sir Isaac

Alfred

Denis

Frigyes

Hermann Amandus

Brook

Vito

These historical notes are culled from many sources, but the ones I have used somewhat systematically are:

Solomon Bochner, The Role of Mathematics in the Rise of Science, Princeton: Princeton Univ. Press, 1981.

William Bridgewater and Seymour Kurtz, editors, The Columbia Encyclopedia, Third edition. New York: Columbia University Press, 1963.

W.F. Bynum, E.J. Browne, and Roy Porter, eds., Dictionary of the History of Science, Princeton: Princeton University Press, 1981.

Charles Coulston Gillispie, editor-in-chief, Dictionary of Scientific Biography, New York: Scribner, 1970-.

Felix Klein, Vorlesungen über die Entwicklung der mathematik im 19. Jahrhundert, New York, Chelsea, 1967 (reprint of two volumes published in Berlin, 1926-1927).

Nouveau Petit Larousse, Paris: Librairie Larousse, 1971.

Since writing most of these biographical sketches, I have discovered a very nice site with biographies of many mathematicians and scientists. From it there is a link to a historical discussion of the concept of abstract vector spaces

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Evans M. Harrell II (correct my scholarship!)