About texts for Math. 185 and Math. H185 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ by Prof. W. Kahan, Spring semester 2008 ***************************************************** * The best Math. text I know from which to learn * * about analytic functions of a complex variable: * ***************************************************** "Problems and Theorems in Analysis" by G. P—lya & G. Szegš, 2 vol. (1972 & 1976, Springer-Verlag, Berlin). Expensive unless bought used! The original editions in German cost me a lot less. A sequence of problems leads the reader through the subject with at least hints for their solutions at the back of the book. You can do it all yourself. No applications. ***************************************************** Less Expensive Texts: ~~~~~~~~~~~~~~~~~~~~~ "Notes on Complex Function Theory" by Donald Sarason (1994, Henry Helson, Berkeley) It's inexpensive, and covers the syllabus plus a little more, but it explores no applications. A recent second edition, "Complex Function Theory" published in 2007 by the Amer. Math. Soc., costs a bit more but includes more details and, in Appendices, background material. Applications have to be learned from some other text. Math. 185's syllabus is very much worth knowing. You could teach yourself the subject from these notes without attending my lectures nor anyone else's. If you do attend lectures, be sure to find out at each lecture's end what the next lecture will cover, if the instructor knows, and read about it before the next lecture. This will actually save time for you! If a text puzzles you try another; ask an instructor. If you procrastinate you will be no better off than most the rest of us and rather worse off than some. To see more about applications, consult almost any other text used for this course or with a title like "Complex ... for ... (engineers or physicists or ...)" There are vastly many such texts. For example: "Theory of Functions as applied to Engineering Problems" ed. by R. Rothe, F. Ollendorff & K. Pohlhausen (1933, reprinted in 1961 by Dover, N.Y.) A classic gem that surveys numerous early 20th century applications to fields and flows in continua. The translation from German is sometimes ponderous and old-fashioned. Recently added to my shelves: ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ "Complex Variables with Applications" by A.D. Wunsch (2d ed. 1994, 3rd 2005, Addison-Wesley, Reading MA) Applications scatter among the chapters' exercises. "A First Course in Complex Analysis" by D.G. Zill & P.D. Shanahan (2003, Jones & Bartlett, Sudbury MA) Each leisurely chapter ends with ample applications. "Complex Analysis for Mathematics and Engineering" by J.H. Mathews & R.W. Howell (5th ed., 2006, Jones & Bartlett, etc.) Easy reading for applications to early 20th century signal processing, electrostatics, fluid flow, integral transforms, and fractal chaos. "Applied Complex Analysis with Partial Differential Equations" by N.H. Asmar (2002, Prentice-Hall N.J.) A leisurely treatment of complex variables followed by their application to many of the early 20th century's solutions of boundary-value problems arising in physics and engineering of that era. ... and so on ... Browse through library shelves to find a text that you like if none mentioned here appeal to you. Some other texts instructors have used for Math. (H)185 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ "Basic Complex Analysis" by J.E. Marsden & M.J. Hoffman (2d ed. 1987 or 3d ed. 1999, W.H. Freeman, N.Y.) plus an Internet Supplement BCA_NS.pdf from Marsden's web page (Click on "Books") and a Student Guide from Freeman. A superb text, covers many important applications plus more than the syllabus for this course. Well worth buying second-hand as an alternative to Sarason's Notes or any other text for this course. "Complex Variables and Applications" by J.W. Brown & R.V. Churchill (6th ed. 1996, or 7th ed. 2004, McGraw-Hill, N.Y.) Discusses many applications; otherwise not a text I like. "Visual Complex Analysis" by T. Needham (1996, Oxford Univ. Press) Vivid explanations of the fundamentals. Other texts well worth a look: ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ "Elements of the Theory of Functions" and "Theory of Functions, part I" and "... part II" by K. Knopp (1945 - 1952, Dover, N.Y.) Astonishingly short texts though old-fashioned. "Functions of a Complex Variable with Applications" by E.G. Phillips (8th ed. 1957, Oliver & Boyd, Edinburgh) My generation used this tiny text to cram before an exam, but the book's small size is deceptive. "Introduction to Complex Analysis" by R. Nevanlinna & V. Paatero, 2d ed. (1969) reprinted in 2007 by the Amer. Math. Soc. This classic text, though translated sometimes awkwardly from the original German, covers well the syllabus of Math. 185 plus a thorough treatment of Elliptic Integrals, Euler's Gamma Function, and the Riemann Zeta Function. The text includes 320 problems each well worth trying. "The Theory of Functions of a Complex Variable" by A. Sveshnikov & A. Tikhonov (1971, Mir Publishers, Moscow). Another small but surprisingly comprehensive text with a wealth of applications each described well and succinctly, though the translation from Russian is awkward sometimes. "Complex Variables" by F.P. Greenleaf (1972, Saunders, Philadelphia) Its long chapter on applications explains several well. "Complex Variables" by G. Polya & G. Latta (1974 Wiley N.Y.) A leisurely treatment with many problems, none too hard, and references to solutions in the volumes by Polya & Szegš cited above. "Concrete Mathematics, a foundation for Computer Science" by R.L. Graham, D.E. Knuth & O. Patashnik (1989, Addison Wesley, Reading MA) This magnificent text's chapters 5 and 7 explore, as very few texts do, applications of Generating Functions to combinatorial problems. This text's citations alone can save years of research. Other more comprehensive texts suitable for reference: ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ "Classical Complex Analysis" by L-s. Hahn & B. Epstein (1996, Jones & Bartlett, Sudbury MA) Very readable. I reach for it often. "Functions of One Complex Variable" by John B. Conway (2d ed. 1978, Springer-Verlag, New York) Heavy reading at a graduate level. "Complex Analysis" by E.M. Stein & R. Shakarchi (2003, Princeton Univ. Press) Heavy reading at a graduate level, but covers some transcendental special functions important in Number Theory and Mathematical Physics, for which functions this book is a good source. "Applied and Computational Complex Analysis" by P. Henrici (3 vol., 1974, 1977 1986, Wiley N.Y.) A vast compendium of constructive methods, it can save months of research. "Dictionary of Conformal Representations" by H. Kober 2d ed. (1957, Dover, N.Y.) Many drawings, still tedious to produce on a computer, were produced by hand in the 1940s and still constitute the best organized collection of conformal maps of diverse regions useful to many engineers, scientists, architects, ... . "Theory of Functions of a Complex Variable" by A.I. Markushevich (3 vol. 1965-1967, Prentice-Hall, N.J.) Over 1000 pages of a leisurely treatment of everything a student should learn about complex analytic functions before proceeding to graduate study of mathematics. "Methods of Theoretical Physics" by P.M. Morse & H. Feshbach (2 vol. 1953, McGraw-Hill, N.Y.) 140 pges of Ch.4, vol. I, plus parts of ch. 10 & 11 in vol. II covered what a student should have learned about complex analytic functions before entering upon graduate study of theoretical physics when I was a student half a century ago. 2000 pages explain most of the mathematics applied during the 19th and early 20th centuries.