Augustin-Louis Cauchy : biography

Wave theory, mechanics, elasticity

In the theory of light he worked on Fresnel’s wave theory and on the dispersion and polarization of light. He also contributed significant research in mechanics, substituting the notion of the continuity of geometrical displacements for the principle of the continuity of matter. He wrote on the equilibrium of rods and elastic membranes and on waves in elastic media. He introducedCauchy, De la pression ou tension dans un corps solide, [On the pressure or tension in a solid body], Exercices de Mathématiques, vol. 2, p. 42 (1827) a 3 × 3 symmetric matrix of numbers that is now known as the Cauchy stress tensor. In elasticity, he originated the theory of stress, and his results are nearly as valuable as those of Simeon Poisson. Other significant contributions include being the first to prove the Fermat polygonal number theorem.

Complex functions

Cauchy is most famous for his single-handed development of complex function theory. The first pivotal theorem proved by Cauchy, now known as Cauchy’s integral theorem, was the following:

oint_C f(z)dz = 0, 

where f(z) is a complex-valued function holomorphic on and within the non-self-intersecting closed curve C (contour) lying in the complex plane. The contour integral is taken along the contour C. The rudiments of this theorem can already be found in a paper that the 24-year-old Cauchy presented to the Académie des Sciences (then still called "First Class of the Institute") on August 11, 1814. In full formCauchy, Mémoire sur les intégrales définies prises entre des limites imaginaires [Memorandum on definite integrals taken between imaginary limits], submitted to the Académie des Sciences on February 28, 1825 the theorem was given in 1825. The 1825 paper is seen by many as Cauchy’s most important contribution to mathematics.

In 1826Cauchy, Sur un nouveau genre de calcul analogue au calcul infinitésimal [On a new type of calculus analogous to the infinitesimal calculus], Exercices de Mathématique, vol. 1, p. 11 (1826) Cauchy gave a formal definition of a residue of a function. This concept regards functions that have poles—isolated singularities, i.e., points where a function goes to positive or negative infinity. If the complex-valued function f(z) can be expanded in the neighborhood of a singularity a as

f(z) = phi(z) + frac + frac{(z-a)^2} + cdots + frac{(z-a)^n},quad B_i, z,a in mathbb,

where φ(z) is analytic (i.e., well-behaved without singularities), then f is said to have a pole of order n in the point a. If n = 1, the pole is called simple. The coefficient B1 is called by Cauchy the residue of function f at a. If f is non-singular at a then the residue of f is zero at a. Clearly the residue is in the case of a simple pole equal to,

underset{mathrm} f(z) = lim_ (z-a) f(z),

where we replaced B1 by the modern notation of the residue.

In 1831, while in Turin, Cauchy submitted two papers to the Academy of Sciences of Turin. In the firstCauchy, Sur la mécanique céleste et sur un nouveau calcul qui s’applique à un grande nombre de questions diverses [On the celestial mechanics and on a new calculus that can be applied to a great number of diverse questions], presented to the Academy of Sciences of Turin, October 11, 1831. he proposed the formula now known as Cauchy’s integral formula,

f(a) = frac oint_C frac dz,

where f(z) is analytic on C and within the region bounded by the contour C and the complex number a is somewhere in this region. The contour integral is taken counter-clockwise. Clearly, the integrand has a simple pole at z = a. In the second paperCauchy, Mémoire sur les rapports qui existent entre le calcul des Résidus et le calcul des Limites, et sur les avantages qu’offrent ces deux calculs dans la résolution des équations algébriques ou transcendantes Memorandum on the connections that exist between the residue calculus and the limit calculus, and on the advantages that these two calculi offer in solving algebraic and transcendental equations], presented to the Academy of Sciences of Turin, November 27, 1831. he presented the residue theorem,