Mathematical analysis develops from calculus, which is the oldest and most basic branch of mathematical analysis. Mathematical analysis generally refers to a relatively complete mathematical subject with the general theory of calculus and infinite series as the main content, including their theoretical basis (basic theory of real number, function and limit). Although mathematical analysis is only a branch of mathematics, it has a wide range of applications and is the basis of almost all advanced mathematics. Mathematical analysis was independently founded by Newton and Leibniz in the17th century, and perfected by Cauchy and Weisstras in the19th century. Calculus and its related contents have been called analysis since Newton. Since then, the field of calculus has been expanding, but many mathematicians still use this name. Today, although many contents have been separated from calculus and become independent disciplines, people still call it analysis. Mathematical analysis is also called analysis for short.

Branch field

Mathematical analysis is currently divided into the following sub-fields:

Real analysis: it is a rigorous study of the differential and integral forms of real functions. This includes the study of limit, power series and measure.

Functional analysis: learning function space, introducing concepts such as Banach space and Hilbert space.

Harmonic analysis: dealing with Fourier series and its abstraction.

Complex analysis: it is the study of complex differentiable functions from complex plane to complex plane.