## Wednesday, April 16, 2008

### Joseph Liouville - Our Mystery Mathematician Was One of a Kind...

Two winners this week in identifying Joseph Liouville (1809-1882), a brilliant mathematician whose work continues to have an impact today in so many branches of math and physics.

Here is an excerpt from an excellent site for biographies of mathematicians:

Liouville's mathematical work was extremely wide ranging, from mathematical physics to astronomy to pure mathematics. One of the first topics he studied, which developed from his early work on electromagnetism, was a new topic, now called the fractional calculus. He defined differential operators of arbitrary order Dt. Usually t is an integer but in this theory developed by Liouville in papers between 1832 and 1837, t could be a rational, an irrational or most generally of all a complex number.

Liouville investigated criteria for integrals of algebraic functions to be algebraic during the period 1832-33. Having established this in four papers, Liouville went on to investigate the general problem of integration of algebraic functions in finite terms. His work at first was independent of that of Abel, but later he learnt of Abel's work and included several ideas into his own work.

Another important area which Liouville is remembered for today is that of transcendental numbers. Liouville's interest in this stemmed from reading a correspondence between Goldbach and Daniel Bernoulli. Liouville certainly aimed to prove that e is transcendental but he did not succeed. However his contributions were great and led him to prove the existence of a transcendental number in 1844 when he constructed an infinite class of such numbers using continued fractions. In 1851 he published results on transcendental numbers removing the dependence on continued fractions. In particular he gave an example of a transcendental number, the number now named the Liouvillian number

0.1100010000000000000000010000...

where there is a 1 in place n! and 0 elsewhere.

His work on boundary value problems on differential equations is remembered because of what is called today Sturm-Liouville theory which is used in solving integral equations. This theory, which has major importance in mathematical physics, was developed between 1829 and 1837. Sturm and Liouville examined general linear second order differential equations and examined properties of their eigenvalues, the behaviour of the eigenfunctions and the series expansion of arbitrary functions in terms of these eigenfunctions.

Liouville contributed to differential geometry studying conformal transformations. He proved a major theorem concerning the measure preserving property of Hamiltonian dynamics. The result is of fundamental importance in statistical mechanics and measure theory.

In 1842 Liouville began to read Galois's unpublished papers. In September of 1843 he announced to the Paris Academy that he had found deep results in Galois's work and promised to publish Galois's papers together with his own commentary. Liouville was therefore a major influence in bringing Galois's work to general notice when he published this work in 1846 in his Journal. However he had waited three years before publishing the papers and, rather strangely, he never published his commentary although he certainly wrote a commentary which filled in the gaps in Galois's proofs. Liouville also lectured on Galois's work and Serret, possibly together with Bertrand and Hermite, attended the course.

And our winners are...

Vlorbik:

okay ... liouville ...
most notable in *my* mind for having founded
"liouville's journal" ... major mathematicians
who were also great *editors* are very few
(& as a former self-publisher, i have a soft spot
for people involved in getting the word out).
best *known*, i suppose, for having proved
that a bounded entire function on {\Bbb C}
is a constant function ...
the first time i looked at the picture, i thought,
"oh, gee ... another one of those photos
i've seen a hundred times in textbooks and so on;
beats me who the devil it actually *is*" ....
but just now, i checked back (looking for comments
on your math-panel post) and at a glance thought,
"hmm ... liouville. maybe i'll just go verify that
at google images." ... funny the tricks one's mind plays ...

Kevin:

Dave: Joseph Liouville. My personal favorite Liouville theorem: Conformal mappings for E^n, and S^n, n > 2 are restrictions of moebius transformations. Probably all anecdotes, well told, are in Lützen, J. Joseph Liouville 1809-1882: Master of Pure and Applied Mathematics. But I observe that his "mathematical" path has crossed that of some other of the "mystery mathematicians": Kaplansky, the previous mystery mathematician, wrote a small and elegant monograph "An Introduction to Differential Algebra" which discusses Liouville extensions and in particular the example: y'' + xy = 0 which is not integrable in finite terms ("the solutions of this equation cannot be obtained from the field of rational functions of x by any sequence of finite algebraic extensions, adjunction of integrals and adjunction of exponentials of integrals"). Liouville wrote a number of items in the first volume of the journal he founded in 1836 . Included in that first volume is an article by Jacobi - another week's mystery mathematician. According to MacTutor History: they also share fellowship in the Royal Society, fellowship in the Royal Society of Edinburgh, and they each have a lunar crater named after them. Lastly, they share the Jacobi-Liouville formula in dynamical systems.
Best regards, Kevin

Lena said...

Well said.