Analysis is a major branch of mathematics that has to do with approximating certain mathematical objects, such as numbers or functions, by other objects that are easier to understand or to handle. A simple example of analysis is the calculation of the first few decimal places of π by writing it as the limit of an infinite series. The origins of analysis go back to the 17th century, when people such as Isaac Newton began investigating how to approximate locally – in the neighborhood of a point – the behavior of quantities that vary continuously. This led to an intense study of limits, which form the basis of understanding infinite series, differentiation, and integration.
Modern analysis is subdivided into several areas: real analysis (the study of derivatives and integrals of real-valued functions); functional analysis (the study of spaces of functions); harmonic analysis (the study of Fourier series and their abstractions); complex analysis (the study of functions from the complex plane to the complex plane that are complex differentiable); and non-standard analysis (the study of hyperreal numbers and their functions, which leads to a rigorous treatment of infinitesimals and of infinitely large numbers).
Complex analysis is the study of functions of a complex variable. Often, the most natural proofs for statements in real analysis or even number theory use techniques from complex analysis. Unlike real functions, which are commonly represented as two-dimensional graphs, complex functions have four-dimensional graphs and may usefully be illustrated by color-coding a three-dimensional graph to suggest four dimensions.
Harmonic analysis is the method of expressing periodic functions as sums of sines and cosines. Important aspects of harmonic analysis are Fourier series, and the expansion of discontinuous functions and of odd or even functions.
In a broad sense, nonstandard analysis is the study of the infinitely small; more specifically, it is the study of hyperreal numbers, their functions and properties. Nonstandard analysis, which was pioneered by Abraham Robinson in the 1960s, puts the concept of infinitesimals on a firm mathematical footing and is, for many mathematicians, more intuitive than real analysis.
Real analysis is a branch of analysis dealing with the set of real numbers and functions of real numbers. It is effectively a rigorous form of calculus and studies concepts such as sequences and their limits, continuity, differentiation, integration and sequences of functions.