Mathematical analysis

Mathematical analysis is the branch of mathematics that studies functions, spaces, and operators through methods of approximation and convergence. It grew out of calculus, especially the use of derivatives and integrals to study variable quantities, and in the 19th century its foundations were reformulated with greater rigor. Basic objects of study in mathematical analysis include the real numbers, functions, sequences, series, and limits. Analysis is closely connected with applications in the sciences, providing methods for formulating and analyzing mathematical models, constructing approximations, and estimating their errors. Typical analytical questions concern how functions, sequences, and operators behave under limiting processes and perturbations: convergence and continuity, regularity and stability, quantitative bounds, and the accuracy of approximations. Modern analysis studies these questions in many settings, including Euclidean spaces, metric spaces, topological spaces, measure spaces, and function spaces. Its major areas include complex analysis, functional analysis, measure theory, harmonic analysis, and the theory of ordinary and partial differential equations.