Type of bind: Paperback
Dewey Decimal Number: 515.9
EAN num: 9780486613888
ISBN number: 0486613887
Label: Dover Publications
Manufacturer: Dover Publications
Quantity: 1
Page Count: 364
Printing Date: February 01, 1983
Publishing house: Dover Publications
Sale Popularity Level: 729829
Studio: Dover Publications
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Product Description:
The text covers enough material for an advanced undergraduate or first-year graduate course. Contents include: Calculus in the Plane; Harmonic Functions in the Plane; Analytic Functions and Power series; Singular Points and Laurent Series; and much more. Many fine illustrations and numerous problems (with solutions). 1972 edition.
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Rated by buyers -
The other reviewers have already done a terrific job describing the content. I'll just add that while a profesional/accademic mathematician might find this book a bit informal, as an amateur I really appreciate its appeal to intuition and the author's tendency to review the meaning of terminology for a few of it's subsequent uses immediately following it's introduction. Not a difficult read, but a working knowledge of calculus for one variable is prerequisite. Seems to be out of print as of this date -- I've had good luck with used book purchases through Amazon marketplace from dealers with positive ratings better than 95%.
Rated by buyers -
Check this sentence from the preface:
"The Cauchy Integral Theorem is thereby an easy consequence of Green's Theorem and the Cauchy-Riemann equations. Goursat's remarkable deepening of the Integral Theorem is discussed, but is not proved."
Such an upfront motivation of physicality in Complex Variables or Analysis is more than a rare find, it can only be justly defined as heavensent. A gift from the gods! This miraculous text absolutely deserves its many 5 star reviews. (other readers should still figure out the previous text that has an even clearer presenation of the physical foundations than Flanigan [to think i gave it 3 stars!])
Perhaps the most Physically Intuitive text on Complex Variables Ever (here's the very first full paragraph): "We examine the the geography of the xy-plane. Some of this will be familiar from basic calculus (for example, distance between points), some may be new to you (for example, the important notion of 'domain'). We must also consider curves in the plane."
Rated by buyers -
Just like finding two solutions for the same problem gives additional insight, Flannigan was able to give me an additional insight to the whole subject of complex analysis with his approach that is drasticly different from any other book on the subject I know.
Used this book during Mathematics Ph.D. studies to prepare for a preliminary exam in complex analysis. The unorthodox approach helped me get another angle of the subject. In particular I would note the introduction of harmonic functions before analytic functions and using "real analysis" techniques to prove "complex analysis" theorems like the maximum principal and the Liouville theorem for harmonic functions. Before the number "i" is even introduced, you already know these theorems for analytic functions once you define them as a pair of harmonic ones.
The student friendly tone of the author was a blessed interchange from the standard graduate books like Ahlfors, and for a fraction of the cost, it makes a wonderful buy for a self study book for the complex Ph.D. exam.
I would not assign it as the course book for undergraduate students taking a very first course in complex analysis (which is what it is intedned for) though. It would be frustrating for a student to ponder through Green's theorem and real analysis material, which is by no means introductory, for 100 pages or so, when what he or she needs and/or wants to be doing is to deal with the algebra and geometry of complex numbers.
Overall, an awsome book if you already tasted the subject and want to get a better feel for it. If it's your very first time, stick with the traditional books.
Rated by buyers 
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The two basic facts about analytic functions are these: they satisfy the Cauchy-Riemann equations and they are conformal. These are two lanes of a two way street between complex function theory and potential theory. The Cauchy-Riemann equations imply that the real and imaginary part of the function are conjugate harmonic functions. Harmonic functions are functions that satisfy the Laplace equation, and they thus describe steady-state heat flows and such. So facts about heat flows translate into facts about analytic functions. For instance, if no heat is generated inside a circle then the temperature at some interior point will be some sort of smeared out average of the temperatures along the circumference, so the maximum temperature in the disc must be somewhere on the boundary. This carries over to analytic functions: the maximum of the modulus of an analytic function on a disc must be attained on the boundary, and, if the function is never zero we can invert it and find that the minimum of the modulus must be attained on the boundary. And from here we obtain a quick and easy proof of the fundamental theorem of algebra: if a polynomial is never zero the minimum of its modulus on a disc must be attained on the boundary, but as the disc is taken larger and larger, the modulus on the boundary of course goes to infinity. QED. Thus we have a sort of physical proof of a very formal mathematical theorem. And there's plenty more where that came from. Integrating along a closed loop sort of corresponds to integrating the heat flux across the boundary, and poles correspond to heat sources, so if there are no poles inside the loop the influx and the outflux will be equal and the integral will be zero, and in general the net flux will be determined by the strength of the sources (i.e. residues of the poles). All this because the Cauchy-Riemann equations turned analytic functions into physics. But we promised a two way street, although admittedly there is less traffic in the opposite direction (flows around obstacles could have evened the score but are omitted). The key here is that harmonic functions are conformally invariant, and analytic functions are conformal, so an analytic function applied to a harmonic function produces a new harmonic function. An indication of the usefulness of this fact is this: the Dirichlet problem for the disc is easily solved by the Poisson formula but remains hard for a general domain, but because any domain can be mapped to a circle by an analytic function we can, in principle, solve the general problem by simply mapping the circle solution to our new domain. In conclusion, we very much applaud the idea of a harmonic function approach to analytic functions, but we also feel that this book is a bit stiff and does not sufficiently exploit the power of the intuitive and geometric ideas involved; we strongly recommend Needham's wonderful book for these aspects.
Rated by buyers -
Flanigan's book starts at the beginning, and it covers some central aspects of complex function theory, elementary geometry, harmonic and analytic functions.
The central topics are (in this order) calculus and geometry of the plane, harmonic functions, complex numbers, integrals, power series and analytic functions, and the standard Cauchy-and residue theorems, ending with a brief chapter on conformal mappings.
The book was published very first in 1972, but reprinted since by Dover. It is suitable as a text or as a supplement in a standard course in complex function theory, late undergraduate level, or beginning graduate. While it contains the standard elements in such a course, we note that a systematic treatment of power series comes relatively late, in Chapter 5, beginning on page 194. Some readers might want to begin with that. Flanigan concludes with the Riemann mapping theorem.
Of other Dover titles on the same subject we recommend the books by Volkovyskii et al, Schwerdtfeger, and Silverman. Review by Palle Jorgensen, August 5, 2006.
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