Key Contributions:
* Hermite's Identity: A fundamental identity in number theory that expresses the product of two quadratic forms as a sum of four squares.
* Proof of the Transcendence of e: In 1873, Hermite provided the first rigorous proof that the mathematical constant *e* (Euler's number) is transcendental. This was a groundbreaking achievement in the field of number theory.
* Hermitian Forms and Hermitian Matrices: Hermite made significant contributions to the theory of quadratic forms, introducing Hermitian forms and matrices which are now widely used in linear algebra and other areas of mathematics.
* Elliptic Functions: Hermite explored the properties of elliptic functions, which are complex functions with double periodicity. He made substantial contributions to the theory of these functions, including the development of the Hermite polynomials.
Other Works:
* Hermite Polynomials: A sequence of orthogonal polynomials named after Hermite. These polynomials have applications in quantum mechanics, probability, and other fields.
* Hermite Interpolation: A method of polynomial interpolation that uses a set of points and their derivatives to approximate a function.
* Publications in Mathematical Journals: Hermite published numerous research papers in prestigious mathematical journals like the "Journal de Mathématiques Pures et Appliquées" and the "Comptes Rendus de l'Académie des Sciences."
* Correspondence with Other Mathematicians: Hermite maintained extensive correspondence with other leading mathematicians of his time, including Charles Hermite and Carl Gustav Jacobi. This correspondence provides valuable insights into the development of mathematical ideas.
Hermite's work had a profound impact on the development of mathematics, and his legacy continues to influence research in various areas today.