Baggett — Analysis of Functions of a Single Variable: A Detailed Development

Lawrence W. Baggett, University of Colorado, dated 29 October 2006. 233 PDF pages (229 numbered), set in TeX, served from the author's faculty page. T1 — a research mathematician's own text on his department's server, and the same content was published through OpenStax CNX (collection col11249) under CC BY 3.0; the PDF held here carries no licence notice of its own, so the open licence comes from the CNX edition rather than from this file.

Written for "serious and talented mathematics scholars, seniors or first-year graduate students" — the highest audience level in this corpus, above trench-real-analysis and well above illowsky-introductory-statistics.

The organising idea: a list of truths, not a list of topics

Most texts here are built from a syllabus. This one is built from seven results the author thinks a mathematician should be able to explain, listed in the preface as his own list, with the note that another professor's list would differ:

√2 is real but not rational
the area of a circle is A = πr²
its circumference is C = 2πr
e^(iπ) = −1
the Fundamental Theorem of Calculus
the Fundamental Theorem of Algebra
an arbitrary angle cannot be trisected with compass and straight edge

Six of the seven are then derived over the course of the book. The seventh is dropped in the preface, and the reason is a curriculum judgement worth recording: it looks like geometry but "turns out to be an algebraic result that is not appropriate for this course in analysis" — Baggett says it belongs in an undergraduate algebra course, which in this hub is abstract-algebra, where judson-abstract-algebra carries the Galois theory that settles it.

The consequence is that the book's definitions are chosen to make those six provable rather than to match convention, and Baggett says so each time:

e is defined as the limit of (1 + 1/n)ⁿ (Chapter II), which he calls "in some ways the first nontrivial example of a convergent sequence."
sin and cos are defined as power series (Chapter III), explicitly to avoid the usual wrapping of the real line around a circle — that definition would need arc length first, and arc length only arrives in Chapter VI.
π is then defined as the smallest positive zero of the sine function. Defining it as the ratio of circumference to diameter, "quite acceptable to Euclid," is called "dangerously imprecise" without a prior definition of the length of a curve.
area is defined only for geometric sets — regions bounded above and below by graphs of continuous functions — to stay out of measure theory while still covering the disk, which is all truth 2 needs.

Ordering the definitions this way costs him the identity e^(iπ) = −1 in Chapter III, where the power series already give exp(iz) = cos z + i sin z: cos π = −1 cannot be deduced yet, and the milestone waits for Chapter IV. He points that out rather than hiding it.

Real and complex, refused as separate subjects

The book's other structural decision: it does not separate real analysis from complex analysis. Baggett's stated reason is historical — "separating these two subjects does not at all jive with their development historically, and to me it seems unnecessary and potentially confusing to do so." What he does limit is dimension: functions of several variables are "a very different kettle of fish," so the book stays with one variable, real or complex, throughout.

That is a live disagreement with the shelf this spoke has built. trench-real-analysis and carroll-geometric-function-theory are the two halves Baggett declines to cut apart, and this corpus had them as two subject pages (real-analysis, complex-analysis) before this book arrived.

The payoff appears in Chapter VII. Green's Theorem, proved in Chapter VI over geometric sets, is what Cauchy's Theorem is then deduced from — so the real-variable machinery is not a prerequisite recalled from another course but the previous chapter.

What is in it

Seven chapters and an appendix; at least 201 numbered exercises and 135 numbered theorems, the exercises set inline between results rather than gathered at chapter ends, and spread evenly (21–35 per chapter).

Ch.	Subject	Notable
I	The real and complex numbers	least upper bound; existence of ℝ deferred to the appendix; geometric progression, Binomial Theorem, triangle inequality
II	Limits of sequences	ε–N from the start; e; compactness and limit points; infinite series
III	Functions and continuity	power series as generalised polynomials, giving exp/sin/cos; uniform convergence; Abel's theorem on the boundary of the disk of convergence
IV	Differentiation	Mean Value Theorem, Chain Rule, Taylor; laws of exponents from the power series; two-variable differentiability and partials, in preparation for holomorphy
V	Integration	integrable = uniform limit of step functions; FTC; area of plane regions; `A = πr²`
VI	Integration over smooth curves	arc length; `C = 2πr`; contour integrals; vector fields and differential forms; Green's Theorem
VII	"The Fundamental Theorem of Analysis"	Cauchy's Theorem and Integral Formula; Fundamental Theorem of Algebra; Maximum Modulus; Open Mapping and Inverse Function; Laurent expansion, isolated singularities, Residue Theorem
App.	Complete ordered fields	ℝ constructed from Dedekind cuts, plus uniqueness up to isomorphism

The integration choice is the one to keep. Baggett defines the integrable functions as the uniform limits of step functions, and states plainly that this is a smaller class than the Riemann-integrable functions — "they suffice for my purposes, and this approach serves to emphasize the importance of uniform convergence." A deliberate loss of generality, taken to make one earlier idea carry more of the book.

Where it sits against the rest of the corpus

It is the first-course complex analysis the spoke was missing — the hole carroll-geometric-function-theory opened and complex-analysis was written around. Against Carroll's account of the settled first-course syllabus, Baggett has Cauchy's Theorem, the Cauchy Integral Formula, Taylor and Laurent expansions, Liouville, the Fundamental Theorem of Algebra, the Maximum Modulus Theorem, isolated singularities and the Residue Theorem. He does not have the Argument Principle or Rouché's Theorem, and there is no conformal mapping anywhere in the book — which is exactly the ground Carroll's second course begins from. So the hole is mostly closed and the two texts do not quite meet.

It is the third text here to build on ε–δ without mentioning the alternative — and the most significant silence of the three. The word infinitesimal does not appear once in 229 pages, in a book that goes further into foundations than anything else in this corpus: an appendix that constructs the reals by Dedekind cuts and proves a complete ordered field unique up to isomorphism. If a text was going to weigh the standard construction against Robinson's hyperreals, this is the one that had the room. See real-analysis and synthesis → the foundational fork.

On the rigour spectrum it sits at the top, and it gets there by a different route than carroll-geometric-function-theory: not by assuming more, but by refusing to assume anything — "we of necessity must spend quite a lot of energy determining what the relevant words themselves really mean."

Caveats

The whole text was read as extracted PDF text, not as typeset pages. Displayed formulas survive extraction poorly, so counts of exercises and theorems are lower bounds and no formula here is quoted from a display.
The licence claim rests on the CNX edition, not on this file. Anyone reusing it should check the CNX/Internet Archive copy rather than the faculty-page PDF.
Twenty years old and never revised as far as the file shows — dated October 2006, with a pdfTeX-0.13d producer string. Mathematics does not date, but the pedagogy claims around it can.

complex-analysis · real-analysis · vector-calculus · mathematics · lawrence-baggett · university-of-colorado-boulder · carroll-geometric-function-theory · trench-real-analysis · keisler-elementary-calculus · openstax