Real analysis
Real analysis is a branch of mathematics that systematically investigates real-valued functions, focusing specifically on the set of real numbers. It explores the properties and behaviors of these functions through concepts such as limits, continuity, differentiation, and integration. The roots of real analysis can be traced back to the development of calculus, a field credited to mathematicians like Gottfried Wilhelm Leibniz and Isaac Newton, which unveiled the foundational connection between derivatives and integrals through the fundamental theorem of calculus.
Real analysis typically starts with the real number system and progresses to functions of a single variable, introducing key topics such as differential calculus and integral calculus, including methods like Riemann and Lebesgue integration. As learners advance, they explore functions of several variables, addressing continuity and differentiability, and eventually connect with vector-valued functions. The discipline also lays the groundwork for complex analysis, which mirrors real analysis but operates within the complex number system. Overall, real analysis serves as a critical framework in both pure and applied mathematics, enhancing our understanding of functions and their applications.
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Subject Terms
Real analysis
Analysis is an area of mathematics that focuses on the systematic investigation of functions, specifically real- and complex-valued functions. Real analysis deals specifically with the set of real numbers and real-valued functions and is often referred to as the theory of functions of a real variable. Real numbers are quantities that can be represented on a continuous number line. The set of real numbers include rational, irrational, and transcendental numbers. Functions of a real variable include functions whose domain is the real numbers. Real analysis focuses on the analytic properties connecting real numbers and real-valued functions.
Overview
Analysis touches on various areas related to the study, development, and modern application of mathematics. The study of real analysis came as a result of attempts to better understand, advance, and further develop topics in its subfield of calculus, a subject widely credited to Gottfried Wilhelm Leibniz and Isaac Newton. Advances made during this period were based largely on the content of the fundamental theorem of calculus and other advanced logic-based assumptions specific to functions of real values, such as continuity and the emergence of uniform convergence. The fundamental theory of calculus is a major theorem connecting the derivative of a function and the integral; namely, it sets the foundation that the two are inverse processes of one another. This two-part theorem also provides the principle method for evaluating a definite integral, which is the limit of a Riemann sum, named after Bernhard Riemann who made significant contributions to real analysis.
The expansion of real analysis by eighteenth-century mathematicians was grounded by the work of former mathematicians. During this period, there was uncertainty regarding the logic surrounding infinitesimal calculus and limits. Less formal developments, beginning with Greek mathematicians such as Eudoxus of Cnidus and Archimedes of Syracuse, had been more implicit and less controversial. Leibniz, Newton, and others, such as Bernard Bolzano, Augustin-Louis Cauchy, Richard Dedekind, Leonhard Euler, and Karl Weierstrass, helped to formalize mathematical analysis and, resultantly, the other branches of analysis.
The roots of real analysis are widespread in the field of pure and applied mathematics. Generally, an introductory course in real analysis begins with the real number system. Using functions of a single variable, a transition is usually made to differential calculus, which encompasses limits, continuity, and differentiation. The topic of integration is then introduced, often making use of Riemann and Lebesque integration. The next topics usually cover connections to sequences and series in detail before transitioning to functions of several variables. In the discussion of functions of several variables, continuity, differentiability, and other topics are introduced. Then follows vector and real-valued functions of several variables. Topics on vector-valued functions focus on differentiation while those on real-valued functions contend with integration. Finally, connections to complex analysis can be discussed.
Complex analysis is closely related to real analysis. In the nineteenth century, Cauchy used infinitesimals combined with geometric ideas to formulate calculus from an alternate logical perspective. These and other developments ultimately led Cauchy to generate a theory of functions of a complex variable, or complex analysis, another classical branch of mathematical analysis. Complex analysis mirrors the study of real analysis but uses the complex number system to do so.
Bibliography
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