Mathematical analysis

Mathematical analysis is a branch of mathematics that deals with the theories of differentiation, integration, measure, limits, infinite series, and analytic functions. These theories are usually studied in the context of real and complex numbers. However, it also includes the study of sequences and series of functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Mathematical analysis is used in many areas of mathematics, physics, and engineering.

History

The roots of mathematical analysis can be traced back to the ancient civilizations of Egypt, Babylon, and Greece. However, the modern concept of mathematical analysis began to take shape in the 17th century with the work of mathematicians like Isaac Newton and Gottfried Wilhelm Leibniz, who developed the theories of calculus. The 19th century saw the formalization of these concepts by mathematicians such as Augustin-Louis Cauchy, Bernhard Riemann, and Karl Weierstrass.

Branches of Mathematical Analysis

Mathematical analysis can be divided into several branches, each with its own focus and techniques.

• Real analysis deals with the real numbers and real-valued functions. It covers the concepts of limits, continuity, and differentiability.

• Complex analysis is concerned with the study of complex numbers and functions of a complex variable. It is known for its results being more general and elegant than those in real analysis.

• Functional analysis is the study of spaces of functions and uses the methods of real analysis combined with those of linear algebra and topology.

• Non-standard analysis is a branch of mathematical logic that studies the properties of numbers and functions using a more rigorous notion of the concept of "infinitesimally small".

Applications

Mathematical analysis has wide-ranging applications in many fields. In physics, it is used to solve problems in quantum mechanics and general relativity. In engineering, it is used in the analysis of systems and signals. In computer science, it is used in the design and analysis of algorithms.

See Also

• Calculus
• Differential equation
• Fourier series
• Partial differential equation
• Vector calculus