Augustin-Louis Cauchy
Rigorized calculus and founded complex analysis
Most quoted
"I am a Christian, that is to say, I believe in the divinity of Jesus Christ, like Bossuet and Pascal, like Corneille and Racine, and like so many other great men who have been illustrious in the sciences and in letters. The more I study nature, the more I am amazed at the works of the Creator. The more I study mathematics, the more I admire the wisdom of God."
"The mean value theorem for derivatives states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point in the open interval where the derivative of the function is equal to the average rate of change of the function over the interval."
— from Cours d'Analyse de l'École Royale Polytechnique, 1821
"A function is continuous if, for every value of the variable between given limits, the numerical value of the difference between two successive values of the function becomes indefinitely small with the numerical value of the difference between the corresponding values of the variable."
— from Cours d'Analyse, 1821
All quotes by Augustin-Louis Cauchy (546)
The derivative of a function at a point is the limit of the ratio of the increment of the function to the increment of the variable, as the latter increment tends to zero.
Every continuous function on a closed interval is uniformly continuous on that interval.
The mean value theorem for derivatives states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point in the open interval where the derivative of the function is equal to the average rate of change of the function over the interval.
The Cauchy integral formula states that if a function is analytic inside and on a simple closed contour, then the value of the function at any point inside the contour can be expressed as a contour integral of the function around the contour.
The Cauchy integral theorem states that if a function is analytic in a simply connected domain, then the integral of the function around any simple closed contour in that domain is zero.
The residue theorem provides a powerful method for evaluating contour integrals of complex functions.
A complex function is analytic at a point if it is differentiable in a neighborhood of that point.
The concept of a group is fundamental to abstract algebra.
The order of a subgroup divides the order of the group.
If a prime number p divides the order of a finite group, then the group has an element of order p.
The theory of permutations is an essential part of group theory.
The concept of a determinant is crucial in linear algebra.
The determinant of a product of matrices is the product of their determinants.
A system of linear equations has a unique solution if and only if the determinant of its coefficient matrix is non-zero.
The characteristic equation of a matrix is used to find its eigenvalues.
The study of differential equations is fundamental to many branches of science and engineering.
The existence and uniqueness of solutions to ordinary differential equations can be established under certain conditions.
The method of variation of parameters can be used to find particular solutions to non-homogeneous linear differential equations.
The Laplace transform is a powerful tool for solving linear differential equations.
The theory of elasticity is concerned with the deformation of solid bodies under stress.
Contemporaries of Augustin-Louis Cauchy
Other Mathematicss born within 50 years of Augustin-Louis Cauchy (1789–1857).