Karl Theodor Wilhelm Weierstrass was a German mathematician who is often called as the "father of modern analysis".
Weierstrass was interested in the soundness of calculus, and also made significant advancements in the field of calculus of variations. Among the several significant axioms, Weierstrass established a necessary condition for the existence of strong extrema of variational problems.
As we know, Cauchy gave a form of the (ε, δ)-definition of limit, in the context of formally defining the derivative, in the 1820s, but did not correctly distinguish between continuity at a point versus uniform continuity on an interval, due to insufficient rigor. Notably, in his 1821 Cours d'analyse, Cauchy gave a famously incorrect proof that the (pointwise) limit of (pointwise) continuous functions was itself (pointwise) continuous. The correct statement is rather that the uniform limit of uniformly continuous functions is uniformly continuous. This required the concept of uniform convergence, which was first observed by Weierstrass's advisor, Christoph Gudermann, in an 1838 paper, where Gudermann noted the phenomenon but did not define it or elaborate on it. Weierstrass saw the importance of the concept, and both formalized it and applied it widely throughout the foundations of calculus.
The formal definition of continuity of a function, as formulated by Weierstrass, is as follows:
is continuous at if such that for every in the domain of ,
Using this definition and the concept of uniform convergence, Weierstrass was able to write proofs of several then-unproven theorems such as the intermediate value theorem (for which Bolzano had already given a rigorous proof), the Bolzano–Weierstrass theorem, and Heine–Borel theorem.
The next mathematician is Joseph-Louis Lagrange
Joseph-Louis Lagrange was an Italian mathematician and astronomer born in Turin, Piedmont, who lived part of his life in Prussia and part in France. He made significant contributions to all fields of analysis, number theory, and classical and celestial mechanics.
The greater number of his papers were contributed to the Prussian Academy of Sciences. Several of them deal with questions in algebra:
- His discussion of representations of integers by quadratic forms (1769) and by more general algebraic forms (1770).
- His tract on the Theory of Elimination, 1770.
- Lagrange's theorem that the order of a subgroup H of a group G must divide the order of G.
- His papers of 1770 and 1771 on the general process for solving an algebraic equation of any degree via the Lagrange resolvents. This method fails to give a general formula for solutions of an equation of degree five and higher, because the auxiliary equation involved has higher degree than the original one. The significance of this method is that it exhibits the already known formulas for solving equations of second, third, and fourth degrees as manifestations of a single principle, and was foundational in Galois theory. The complete solution of a binomial equation of any degree is also treated in these papers.
- In 1773, Lagrange considered a functional determinant of order 3, a special case of a Jacobian. He also proved the expression for the volume of a tetrahedron with one of the vertices at the origin as the one sixth of the absolute value of the determinant formed by the coordinates of the other three vertices.
His most well-know result in calculus is mean value theorem (Lagrange theorem):
The theorem is used to prove global statements about a function on an interval starting from local hypotheses about derivatives at points of the interval.
More precisely, if a function f is continuous on the closed interval [a, b], where a < b, and differentiable on the open interval (a, b), then there exists a point c in (a, b) such that
The third mathematician we discuss in our summary is Johann Peter Gustav Lejeune Dirichlet
Johann Peter Gustav Lejeune Dirichlet was a German mathematician with deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and other topics in mathematical analysis; he is credited with being one of the first mathematicians to give the modern formal definition of a function.
In a couple of papers in 1838 and 1839 he proved the first class number formula, for quadratic forms (later refined by his student Kronecker). The formula, which Jacobi called a result "touching the utmost of human acumen", opened the way for similar results regarding more general number fields. Based on his research of the structure of the unit group of quadratic fields, he proved the Dirichlet unit theorem, a fundamental result in algebraic number theory.
He first used the pigeonhole principle, a basic counting argument, in the proof of a theorem in diophantine approximation, later named after him Dirichlet's approximation theorem. He published important contributions to Fermat's last theorem, for which he proved the cases n = 5 and n = 14, and to the biquadratic reciprocity law. The Dirichlet divisor problem, for which he found the first results, is still an unsolved problem in number theory despite later contributions by other researchers.
