Preface to analysis of functions of a single variable: a detailed  (Page 4/4)

It is in [link] that the first glimpse of a difference between functions of a real variable and functions of a complex variable emerges.For example, one of the results in this chapter is that every differentiable, real-valued function of a complex variable must be a constant function,something that is certainly not true for functions of a real variable. At the end of this chapter, I briefly slip into the realmof real-valued functions of two real variables. I introduce the definition of differentiabilityof such a function of two real variables, and then derive the initial relationships among the partial derivatives of such a function and the derivative of that function thought of as a function of a complex variable. This is obviously donein preparation for Chapter VII where holomorphic functions are central.

Perhaps most well-understood by math majors is that computing the area under a curverequires Newton's calculus, i.e., integration theory. What is often overlooked by students is that the very definition of theconcept of area is intimately tied up with this integration theory. My treatment here of integration differs from most others in thatthe class of functions defined as integrable are those that are uniform limits of step functions.This is a smaller collection of functions than those that are Riemann-integrable, but they suffice for my purposes,and this approach serves to emphasize the importance of uniform convergence. In particular, I include careful proofs of the Fundamental Theorem of Calculus,the integration by substitution theorem, the integral form of Taylor's Remainder Theorem, and the complete proof of the general Binomial Theorem.

Not wishing to delve into the set-theoretic complications of measure theory, I have chosen only to define the area for certain “geometric” subsets of the plane.These are those subsets bounded above and below by graphs of continuous functions. Of course these suffice for most purposes, and in particularcircles are examples of such geometric sets, so that the formula $A=\pi r^2$ can be established for the area of a circle of radius $r.$ [link] concludes with a development of integration over geometric subsets of the plane. Once again, anticipating later needs, we have again strayed intosome investigations of functions of two real variables.

Having developed the notions of arc length in the early part of [link] , including the derivation of the formula for the circumference of a circle, I introduce theidea of a contour integral, i.e., integrating a function around a curve in the complex plane. The Fundamental Theorem of Calculus has generalizations to higher dimensions,and it becomes Green's Theorem in 2 dimensions. I give a careful proof in [link] , just over geometric sets, of this rather complicated theorem.

Perhaps the main application of Green's Theorem is the Cauchy Integral Theorem, a result about complex-valuedfunctions of a complex variable, that is often called the Fundamental Theorem of Analysis.I prove this theorem in [link] . From this Cauchy theorem one candeduce the usual marvelous theorems of a first course in complex variables, e.g.,the Identity Theorem, Liouville's Theorem,the Maximum Modulus Principle, the Open Mapping Theorem,the Residue Theorem, and last but not least our mathematical truth number 6, the FundamentalTheorem of Algebra. That so much mathematical analysis is used to prove thefundamental theorem of algebra does make me smile. I will leave it to my algebraist colleagues to point out how someof the fundamental results in analysis require substantial algebraic arguments.

The overriding philosophical point of this book is that many analytic assertions in mathematics are intellectually very deep;they require years of study for most people to understand; they demonstrate how intricate mathematical thought is and how far it has comeover the years. Graduates in mathematics should be proud of the degree they have earned, and they should beproud of the depth of their understanding and the extremes to which they have pushed their own intellect.I love teaching these students, that is to say, I love sharing this marvelous material with them.