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# Calculus and Real analysis

Differentiation, integration, series, and so on are familiar to students of elementary calculus. But these topics lead in a number of distinct directions when pursued with greater care and in greater detail. The central location of these fields in the MathMap is indicative of the utility in other branches of mathematics, particularly throughout analysis.

• 26: Real functions are those studied in calculus classes; the focus here is on their derivatives and integrals, and general inequalities. This category includes familiar functions such as rational functions. This seems the most appropriate area to receive questions concerning elementary calculus.
• 28: Measure theory and integration is the study of lengths, surface area, and volumes in general spaces. This is a critical feature of a full development of integration theory; moreover, it provides the basic framework for probability theory. Measure theory is a meeting place between the tame applicability of real functions and the wild possibilities of set theory. This is the setting for fractals.
• 33: Special functions are just that: specialized functions beyond the familiar trigonometric or exponential functions. The ones studied (hypergeometric functions, orthogonal polynomials, and so on) arise very naturally in areas of analysis, number theory, Lie groups, and combinatorics. Very detailed information is often available.
• 39: Finite differences and functional equations both involve deducing functions, as in differential equations, but the premises are different: with difference equations, the defining relation is not a differential equation but a difference of values of the function. Functional equations have as premises (usually) algebraic relationships among the values of a function at several points.
• 40: Sequences and series are really just the most common examples of limiting processes; convergence criteria and rates of convergence are as important as finding "the answer". (In the case of sequences of functions, it's also important to find "the question"!) Particular series of interest (e.g. Taylor series of known functions) are of interest, as well as general methods for computing sums rapidly, or formally. Series can be estimated with integrals, their stability can be investigated with analysis. Manipulations of series (e.g. multiplying or inverting) are also of importance.

You might want to continue the tour with a trip through complex analysis.