Preface to analysis of functions of a single variable: a detailed  (Page 3/4)

unlike the elementary numbers $-1,2,$ and $i,$ the definitions of the real numbers $e$ and $\pi$ are quite a different story. In fact, one cannot make sense of either $e$ or $\pi$ until a substantial amount of analysis has been developed, for they both are necessarily defined somehow in terms of a limit process.I have chosen to define $e$ here as the limit of the rather intriguing sequence $\left\{{(1+\frac{1}{n})}^n\right\},$ in some ways the first nontrivial example of a convergent sequence, and this is presented in [link] . Its relation to logarithms and exponentials, whatever they are, has to be postponed to [link] . [link] also contains a section on the elementary topological properties (compactness, limit points, etc.)of the real and complex numbers as well as a thorough development of infinite series.

To define $\pi$ as the ratio of the circumference of a circle to its diameter is attractive, indeed was quite acceptable to Euclid, but is dangerously imprecise unless wehave at the outset a clear definition of what is meant by the length of a curve, e.g., the circumference of a circle.That notion is by no means trivial, and in fact it only can be carefully treated in a development of analysis well after other concepts.Rather, I have chosen to define $\pi$ here as the smallest positive zero of the sine function. Of course, I have to define the sine function first, and this is itself quite deep.I do it using power series functions, choosing to avoid the common definition of the trigonometric functions in terms of “ wrapping” the real line around a circle,for that notion again requires a precise definition of arc length before it would make sense.I get to arc length eventually, but not until [link] .

In [link] I introduce power series functions as generalizations of polynomials, specifically the three power series functions that turn out to bethe exponential, sine, and cosine functions. From these definitions it follows directly that $expiz=cosz+isinz$ for every complex number $z.$ Here is a place where allowing the variable to be complex is critical, and it has costus nothing. However, even after establishing that there is in fact a smallest positive zeroof the sine function (which we decide to call $\pi$ , since we know how we want things to work out),one cannot at this point deduce that $cos\pi =-1,$ so that the equality $e^{i\pi }=-1$ also has to wait for its derivation until [link] . In fact, more serious, we have no knowledge at all at this point of the function $e^z$ for a complex exponent $z.$ What does it mean to raise a real number, or even an integer, to a complex exponent?The very definition of such a function has to wait.

[link] also contains all the standard theorems about continuous functions, culminating with a lengthy section onuniform convergence, and finally Abel's fantastic theorem on the continuity of a power series function on the boundary of its disk of convergence.

The fourth chapter begins with all the usual theorems from calculus, Mean Value Theorem, Chain Rule, First Derivative Test, and so on. Power series functions are shown to be differentiable, from which the law of exponentsemerges for the power series function exp. Immediately then, all of the trigonometric and exponential identities arealso derived. We observe that $e^r=exp\left(r\right)$ for every rational number $r,$ and we at last can define consistently $e^z$ to be the value of the power series function $exp\left(z\right)$ for any complex number $z.$ From that, we establish the equation $e^{i\pi }=-1.$ Careful proofs of Taylor's Remainder Theorem and L'Hopital's Rule are given, as well as an initial approach to the general Binomial Theorem for non-integer exponents.