Analysis

The term analysis is used in two ways in mathematics. It describes both the discipline of which calculus is a part and one form of abstract logic theory.

Analysis is the systematic study of real and complex-valued continuous functions. Important subfields of analysis include calculus, differential equations, and functional analysis. The term is generally reserved for advanced topics which are not encountered in an introductory calculus sequence, although many ideas from those courses, such as derivatives, integrals, and series are studied in more detail. Real analysis and complex analysis are two broad subdivisions of analysis which deal with real-values and complex-valued functions, respectively.

Derbyshire (2004, p. 16) describes analysis as "the study of limits."

Logicians often call second-order arithmetic "analysis." Unfortunately, this term conflicts with the more usual definition of analysis as the study of functions. This terminology problem is discussed briefly by Enderton (1972, p. 287).

REFERENCES

Bottazzini, U. The "Higher Calculus": A History of Real and Complex Analysis from Euler to Weierstrass. New York: Springer-Verlag, 1986.

Bressoud, D. M. A Radical Approach to Real Analysis. Washington, DC: Math. Assoc. Amer., 1994.

Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, 2004.

Ehrlich, P. Real Numbers, Generalization of the Reals, & Theories of Continua. Norwell, MA: Kluwer, 1994.

Enderton, H. B. A Mathematical Introduction to Logic. New York: Academic Press, 1972.Hairer, E. and Wanner, G. Analysis by Its History. New York: Springer-Verlag, 1996.

Royden, H. L. Real Analysis, 3rd ed. New York: Macmillan, 1988.

Weisstein, E. W. "Books about Analysis." http://www.ericweisstein.com/encyclopedias/books/Analysis.html.

Wheeden, R. L. and Zygmund, A. Measure and Integral: An Introduction to Real Analysis. New York: Dekker, 1977.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.