uvres de Charles Hermite (Cambridge Library Collection - Mathematics)

(Charles Hermite (1822-1901) was a French mathematician wh...)

Charles Hermite (1822-1901) was a French mathematician who made significant contributions to pure mathematics, and especially to number theory and algebra. In 1858 he solved the equation of the fifth degree by elliptic functions, and in 1873 he proved that e (the base of natural logarithms) is transcendental. The legacy of his work can be shown in the large number of mathematical terms which bear the adjective 'Hermitian'. As a teacher at the École Polytechnique, the Faculté des Sciences de Paris and the École Normale Supérieure he was influential and inspiring to a new generation of scientists in many disciplines. The four volumes of his collected papers were published between 1905 and 1908.

Sur la Théorie des Équations Modulaires et la Résolution de l'Équation du Cinquième Degré (French Edition)

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Correspondance D'hermite Et De Stieltjes, Volume 2 (French Edition)

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Cours D'analyse De L'école Polytechnique (French Edition)

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uvres de Charles Hermite (Cambridge Library Collection - Mathematics)

(Charles Hermite (1822-1901) was a French mathematician wh...)

Charles Hermite (1822-1901) was a French mathematician who made significant contributions to pure mathematics, and especially to number theory and algebra. In 1858 he solved the equation of the fifth degree by elliptic functions, and in 1873 he proved that e (the base of natural logarithms) is transcendental. The legacy of his work can be shown in the large number of mathematical terms which bear the adjective 'Hermitian'. As a teacher at the École Polytechnique, the Faculté des Sciences de Paris and the École Normale Supérieure he was influential and inspiring to a new generation of scientists in many disciplines. The four volumes of his collected papers were published between 1905 and 1908.

uvres de Charles Hermite (Cambridge Library Collection - Mathematics)

(Charles Hermite (1822-1901) was a French mathematician wh...)

Charles Hermite (1822-1901) was a French mathematician who made significant contributions to pure mathematics, and especially to number theory and algebra. In 1858 he solved the equation of the fifth degree by elliptic functions, and in 1873 he proved that e (the base of natural logarithms) is transcendental. The legacy of his work can be shown in the large number of mathematical terms which bear the adjective 'Hermitian'. As a teacher at the École Polytechnique, the Faculté des Sciences de Paris and the École Normale Supérieure he was influential and inspiring to a new generation of scientists in many disciplines. The four volumes of his collected papers were published between 1905 and 1908.

Ouvres de Charles Hermite (Cambridge Library Collection - Mathematics)

(Charles Hermite (1822-1901) was a French mathematician wh...)

Charles Hermite (1822-1901) was a French mathematician who made significant contributions to pure mathematics, and especially to number theory and algebra. In 1858 he solved the equation of the fifth degree by elliptic functions, and in 1873 he proved that e (the base of natural logarithms) is transcendental. The legacy of his work can be shown in the large number of mathematical terms which bear the adjective 'Hermitian'. As a teacher at the École Polytechnique, the Faculté des Sciences de Paris and the École Normale Supérieure he was influential and inspiring to a new generation of scientists in many disciplines. The four volumes of his collected papers were published between 1905 and 1908.

Prof Charles Hermite was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.

Background

Hermite was born in Dieuze, The Moselle on 24 December 1822, with a deformity in his right foot which would affect his gait for the rest of his life.

He was the sixth of seven children of Ferdinand Hermite, and his wife Madeleine Lallemand. His father worked in his mother's family drapery business, and also pursued a career as an artist.

The drapery business relocated to Nancy in 1828 and so did the family. Hermite was the sixth of the seven children of Ferdinand Hermite and the former Madeleine Lallemand.

Education

His father, a man of strong artistic inclinations who had studied engineering, worked for a while in a salt mine near Dieuze but left to assume the draper’s trade of his in-laws—a business he subsequently entrusted to his wife in order to give full rein to his artistic bent.

They were not much interested in the education of their children, but all of them attended the Collège of Nancy and ived there.

Instead of seriously preparing for his examination Hermite read Euler, Gauss’s Disquisitiones arithmeticae, and Lagrange’s Traité sur la résolution des équations numériques, thus prompting Richard to call him un petit Lagrange.

Hermite decided to continue his studies at the École Polytechnique; during the preparation year he was taught by E. C. Catalan.

In the 1842 contest of the Paris colleges Hermite failed to win first prix de mathématiques spéciales section but received only first “accessit. ”

He was admitted to the École Polytechnique in the fall of 1842 with the poor rank of sixty-eighth.

Career

Hermite’s first two papers, published in the Nouvelles annales de mathématiques, date from this period.

Still unfamiliar with the work of Ruffini and Abel, he tried to prove in one of these papers the impossibility of solving the fifth-degree equation by radicals.

After a year’s study at the École Polytechnique, he was refused further study, because of a congenital defect of his right foot, which obliged him to use a cane.

Owing to the intervention of influential people the decision was reversed, but under conditions to which Hermite was reluctant to submit.

At this time, Hermite—a cheerful youth who, according to some, resembled a Galois resurrected—was introduced into the circle of Alexandre and Joseph Brtrand.

He took his examinations for the baccalauréat and licence in 1847.

At that time Hermite must have become acquainted with the work of Cauchy and Liouville on general function theory as well as with that of C. G. J. Jacobi on elliptic and hyperelliptic functions.

Hermite was better able than Liouville, who lacked sufficient familiarity with Jacobi’s work, to combine both fields of thought.

In 1832 and 1834 Jacobi had formulated his famous inversion problem for hyperelliptic integrals, but the essential properties of the new ranscendents were still unknown and the work of A. Göpel and J. G. Rosenhain had not yet appeared.

Through his first work in this field, Hermite placed himself, as Darboux says, in the ranks of the first analysts.

He generalized Abel’s theorem on the division of the argument of elliptic functions to the case of hyperelliptic ones.

In January 1843, only twenty years old, he communicated his discovery to Jacobi, who did not conceal his delight.

The correspondence continued for at least six letters; the second letter, written in August 1844, was on the transformation of elliptic functions, and four others of unknown dates (although before 1850) were on number theory.

Extracts from these letters were inserted by Jacobi in Crelle’s Journal and in his own Opuscula.

and are also in the second volume of Dirichlet’s edition of Jacobi’s work.

The next ten years were his most active period.

He occupied that position until 1869, when he took over J. M. C. Duhamel’s chair as professor of analysis at the École Polytechnique and at the Faculté des Sciences, first in algebra and later in analysis as well.

His textbooks in analysis became classics, famous even outside France.

He resigned his chair at the École Polytechnique in 1876 and at the Faculté in 1897.

His scientific work was collected and edited by Picard.

Throughout his lifetime and for years afterward Hermite was an inspiring figure in mathematics.

His name also occurs in the solution of the fifth-degree equation by elliptic functions (1858).

One of the best-known facts about Hermite is that he first proved the transcendence of e (1873).

In a sense this last is paradiematic of all of Hermite’s discoveries.

By a slight adaptation of Hermite’s proof, Felix Lindemann, in 1882, obtained the much more exciting transcendence of π.

If Hermite’s work were scrutinized more closely, one might find more instances of Hermitean preludes to important discoveries by others, since it was his habit to disseminate his knowledge lavishly in correspondence, in his courses, and in short notes.

His correspondence with T. J. Stieltjes, for instance, consisted of at least 432 letters written by both of them between 1882 and 1894.

Jacobi reformulated the problem by simultaneously inverting p integrals—if the irrationality is a square root of a polynomial of the (2p − 1)th or 2 pth degree.

This new approach proved successful in the case of elliptic functions, when Hermite introduced the theta functions of n th order as a means of constructing doubly periodic functions.

In the hyperelliptic case he was less successful, for he did not find the badly needed theta functions of two variables.

In 1855 Hermite took advantage of Göpel’s and Rosenhain’s work when he created his transformation theory (see below).

Meanwhile, Hermite turned to number theory.

For definite quadratic forms with integral coefficients, Gauss had introduced the notion of equivalence by means of unimodular integral linear transformations; by a reduction process he had proved for two and three variables that, given the determinant, the class number is finite.

Hermite generalized the procedure and proved the same for an arbitrary number of variables.

He applied this result to algebraic numbers to prove that given the discriminant of a number field, the number of norm forms is finite.

By the same method he obtained the finiteness of a basis of units, not knowing that Dirichlet had already determined the size of the basis.

Finally, he extended the theorem of the finiteness of the class number to indefinite quadratic forms, and he proved that the subgroup of unimodular integral transformations leaving such a form invariant is finitely generated.

Hermite did not proceed to greater depths in his work on algebraic numbers.

He was an algebraist rather than an arithmetician.

Later he made many contributions to the theory of invariants, in which Arthur Cayley, J. J. Sylvester, and F, Brioschi were active at that time.

One of his nvariant theory subjects was the fifth-degree equaion, to which he later applied elliptic functions.

Armed with the theory of invariants, Hermite returned to Abelian functions.

Meanwhile, the badly needed theta functions of two arguments had been found, and Hermite could apply what he had learned ibout quadratic forms to understanding the trans-ormation of the system of the four periods.

Later, Hermite’s 1855 results became basic for the transformation theory of Abelian functions as well as for Camille Jordan’s theory of “Abelian” groups.

They also led to Herrnite’s own theory of the fifth-degree equation and of the modular equations of elliptic functions.

It was Herrnite’s merit to use ω rather than Jacobi’s q = eπiω as an argument and to prepare the present form of the theory of modular functions.

He again dealt with the number theory applications of his theory, particularly with class number relations or quadratic forms.

His solution of the fifth-degree equation by elliptic functions (analogous to that of third-degree equations by trigonometric functions) was the basic problem of this period.

In the 1870’s Hermite returned to approximation problems, with which he had started his scientific career.

Gauss’s interpolation problem, Legendre functions, series for elliptic and other integrals, con-nued fractions, Bessel functions, Laplace integrals, and special differential equations were dealt with in this period, from which the transcendence proof for e and the Lamé equation emerged as the most remarkable results.

Hermite was seriously ill with smallpox in 1856, and under Cauchy’s influence became a devout Catholic.

Views

Probably he never assimilated the much more profound ideas that developed in the German school in the nineteenth century, and perhaps he did not even realize that the notionof algebraic integer with which he had started was wrong.

Some of his arithmetical ideas were carried on with more success by Hermann Minkowski in the twentieth century.

In the reduction theory of quadratic and binary forms Hermite had encountered invariants.

Quotations:
In a letter to Thomas Joannes Stieltjes in 1893, Hermite remarked: "I turn with terror and horror from this lamentable scourge of continuous functions with no derivatives. "

Connections

Hermite married a sister of Joseph Bertrand Louise Bertrand in 1848; one of his two daughters married Émile Picard and the other G. Forestier.