About texts for Math. 185 and Math. H185
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by Prof. W. Kahan, Spring semester 2008
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* The best Math. text I know from which to learn *
* about analytic functions of a complex variable: *
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"Problems and Theorems in Analysis" by G. Plya &
G. Szeg, 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.
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Less Expensive Texts:
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"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:
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"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
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"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:
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"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 & Szeg 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:
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"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.