Laplace and the era of differential equations

Abstract

Between about 1790 and 1850 French mathematicians dominated not only mathematics, but also all other sciences. The belief that a particular physical phenomenon has to correspond to a single differential equation originates from the enormous influence Laplace and his contemporary compatriots had in all European learned circles. It will be shown that at the beginning of the nineteenth century Newton's “fluxionary calculus” finally gave way to a French-type notation of handling differential equations. A heated dispute in the Philosophical Magazine between Challis, Airy and Stokes, all three of them famous Cambridge professors of mathematics, then serves to illustrate the era of differential equations. A remark about Schrödinger and his equation for the hydrogen atom finally will lead back to present times.

Keywords:

• history of science

Notes

Notes

1. In fact he was the “teacher” of Laplace.

2. Independently of the German philosopher and mathematician (Gottfried Wilhelm) Leibniz (1646–1716).

3. http://en.wikipedia.org/wiki/Adam_Ries

4. http://en.wikipedia.org/wiki/James_Challis

5. http://en.wikipedia.org/wiki/George_Biddell_Airy

6. http://en.wikipedia.org/wiki/Sir_George_Stokes,_1st_Baronet

7. http://en.wikipedia.org/wiki/John_Napier

8. Nap.log (10⁷) = 0; Nap.log y = 10⁷.log_1/e (y/10⁷)

9. Edmond Laguerre (1834–1886), http://en.wikipedia.org/wiki/Edmond_Laguerre

10. http://en.wikipedia.org/wiki/Hermann_Weyl

is a second-order partial differential equation with f being a real-valued function. If

where g is also a real-valued function, then the Laplace equation is called the Poisson equation

x, y and z being real variables

where x is a random variable, b > 0 and α is a so-called “location parameter”, f(x; 0, 1) = exp(−x)/2

where f(t) is a function locally integrable on [0, ∞). In short: a Laplace transformation converts a function f(t) with a real argument t into a function F(s) with a complex argument s