The term ‘mathematical analysis’ refers to the major branch of mathematics which is concerned with the theory of functions and includes the differential and integral calculus. Analysis and the calculus began as the study of curves, calculus being concerned with tangents to and areas under curves. The focus was shifted to functions following the insight, due to Leibniz and Isaac Newton in the second half of the seventeenth century, that a curve is the graph of a function. Algebraic foundations were proposed by Lagrange in the late eighteenth century; assuming that any function always took an expansion in a power series, he defined the derivatives from the coefficients of the terms. In the 1820s his assumption was refuted by Cauchy, who had already launched a fourth approach, like Newton’s based on limits, but formulated much more carefully. It was refined further by Weierstrass, by means which helped to create set theory. Analysis also encompasses the theory of limits and of the convergence and divergence of infinite series; modern versions also use point set topology. It has taken various forms over the centuries, of which the older ones are still represented in some notations and terms. Philosophical issues include the status of infinitesimals, the place of logic in the articulation of proofs, types of definition, and the (non-) relationship to analytic proof methods.
An analytic proof of the Borwein Conjecture