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A Course of Modern Analysis, by G. N. Watson
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A Course of Modern Analysis
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Product details
Paperback: 616 pages
Publisher: Cambridge University Press; 4 edition (January 2, 1962)
Language: English
ISBN-10: 0521091896
ISBN-13: 978-0521091893
Product Dimensions:
6 x 1.2 x 8.9 inches
Shipping Weight: 1.8 pounds
Average Customer Review:
4.5 out of 5 stars
21 customer reviews
Amazon Best Sellers Rank:
#1,818,370 in Books (See Top 100 in Books)
This was a supplementary text for a math physics course I had taken years ago. It was entirely on complex analysis. The main text was Copson which like Whittaker and Watson emphasized power series methods as it was also one of the early British texts. In truth Copson seemed more of an outline as complete proofs many times were relegated to references. Whittaker and Watson began with basic real analysis-Dedekind L and R sections, theory of convergence, uniform convergence, sequence and series convergence, Riemann integration, then complex numbers, Argand diagrams and after this development you're eased into complex analysis proper. The development is rigorous as necessary proofs are given along the way. It's not easy as the notation is antiquated. Metric topological ideas and notation made later texts easier to understand and more systematic. For example I did a review on Anthony Zee's text on quantum field theory here on Amazon and outlined performance of a sample contour integral. I needed to justify an inequality to show an arc integral goes to zero at infinity. It follows since sine is a convex function. This notion isn't in this text nor in Copson. Whittaker and Watson does use this inequality in their proof of Jordan's lemma on p. 115 in the residue chapter. They justify with a mean value theorem argument, sin(x)/x is decreasing since its derivative is negative on zero to pi/2 so it's between 2/pi and 1. Copson just suggests drawing the graphs. Anyway the convexity argument though involving similar ideas is structured in a more comprehensible way. The notation is old but still understandable. Here and there phrasing is a little unclear, e.g., on p.53 in the modified Heine-Borel theorem section, CD is referred to as a straight line when they mean a closed interval. This is forgiveable as there's a massive amount of information in this text. Even in the 2nd edition of Jackson, this text is referred to as maintaining full mathematical rigor and is suggested as a mathematical physics reference. It does do the 2nd order linear partial differential equations of mathematical physics like Laplace, heat and wave equations with solutions. It doesn't go as far as classification into parabolic or hyperbolic, pretty much just ones that will have transcendental functions as solutions. To these functions is devoted a great deal of the text.There is material in the text that is difficult to ferret out elsewhere for instance on pp.141-142 of chapter 7, the Borel integral is proven as a formula to do an analytic continuation. Borel summation is also treated. These are used in recovering a Minkowski space integral from a Euclidean integral in Feynman path integration. The following chapter on asymptotic expansions and summable series has material that bears on renormalization, though it's hard to see -they (physicists) jump fast and operate on faith it seems. As it's my habit to at least give a tidbit from the text, I choose a necessary part of the proof of analytic continuation by Borel integral that I just mentioned-it only involves high school geometry, pretty and simple. This is at the bottom footnote of p. 141. Here's the lead up. We start with a circle of radius r where the function is analytic within circle. Next we draw lines from the center of the circle to each of the singularities in the plane. Next draw the perpendicular to the line through the singularity at the singularity for each of them. These perpendiculars form a polygon surrounding the circle. Next take any point P within the polygon and form a circle with OP as diameter where O is the center of our original circle of analyticity. There can be no singularity on or within this circle on OP. Why? Any inscribed angle with O and P as endpoints has vertex or inscribed angle of 90 degrees since OP cuts 180 degrees. If there's a singularity on the circle-P must be on a side of the polygon. If there's a singularity inside the circle, extend the line from O to the singularity till it intersects the circle and draw the line from this point to P (it's perpendicular of course). By our original construction there's a perpendicular through the singularity which we see is parallel to that line through P. This puts P outside the polygon contrary to original choice-it's on a line parallel to a side of the polygon and outside it. The polygon is convex of course by the way we constructed it from the circle-it's the intersection of convex sets, half-planes, and we've just shown P to lie in the complement of one of these half-planes,i.e., P is definitely outside. Next you do analysis proper. Weren't these guys great? Some pages may be partly cut off or even missing in your copy but there are free PDF's available online if you need to remedy or copy. Still giving it a 5 for content and for making it available again. There was a text by G.N.Watson titled "Complex Integration and Cauchy's Theorem" meant to supplement this text. It gives rigorous proofs of the Jordan Curve Theorem and in depth treatment of branch points and the related notion of winding numbers. You'll recognize ideas used here as being topological like homotopy. This text makes rigorous ideas that needed to be assumed in Whittaker and Watson. These ideas were new at the time. This text is available again by Forgotten Books. A modern text which puts some emphasis on power series methods is the one by Serge Lang.P.S. You'll find two exercises on contour integration at the end of the chapter on Riemann's Zeta function (p.280 in mine) which constitute the main parts of Hardy's Theorem on its zeroes on the critical line. These effectively prove the theorem. Later results on the stubborn Riemann Hypothesis have really hinged on sharpening Hardy's arguments. I guess though Whittaker and Watson is old its ideas put forth in the development of complex analysis are still current.
The Course of Modern Analysis has much theory with formulae, but it should be use only as a reference book and not for a formal course. The derivations of the formulae are incomplete in most cases with many steps left out which make it difficult to follow. Some formulae do not have derivation but are left for to user to work out. I have used this book as reference, and I appreciate its availability. .
Whittaker & Watson is one of the most important books on analysis. Every mathematician should read it.However, it was written over a hundred years ago and there is, at the time of this writing, no public domain digitalization of the text. So instead this printing features has scans of the original work. The text is therefore smudgy and the font is too small. The result is that this printing is hard to read without a magnifier. At best, its only marginally better than reading the scan on a tablet. Some pages even have letters cropped out!
This volume is divided into 2 parts. Part I is titled The Processes of Analysis, and part II is titled The Trancendental Functions. The topics covered in the first part are standard and offer preparation for the Second Part.More modern Analysis Texts do not offer as much content on Special Functions as is presented in Part II. Much modern treatment of the functions covered in Part II seems to have been relocated to Mathematical Physics.This volume is a worthy addition to a library as a reference work for transcendental functions. The title Modern must be related to the time of publication, not today.
This book streamed the modern developments in complex analysis and its relevant areas such as complex differential equation and elliptic functions. The notions are described in a very clear way which should be a good guide for the beginners!
I am still reading this book, and I imagine it will take longer to do so than Amazon cares to wait. I really enjoy the nitty-gritty of infinite series and references to the greats at the end of the nineteenth century as they developed the concepts of converence. I would imagine that a math student taking a course in analysis would profit by reading it, along with people who need to know about special functions.
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