# Complex Variables (Dover Books on Mathematics)

## Francis J. Flanigan

Language: English

Pages: 384

ISBN: 0486613887

Format: PDF / Kindle (mobi) / ePub

*Complex Variables*does not follow conventional outlines of course material. One reviewer noting its originality wrote: "A standard text is often preferred [to a superior text like this] because the professor knows the order of topics and the problems, and doesn't really have to pay attention to the text. He can go to class without preparation." Not so here — Dr. Flanigan treats this most important field of contemporary mathematics in a most unusual way. While all the material for an advanced undergraduate or first-year graduate course is covered, discussion of complex algebra is delayed for 100 pages, until harmonic functions have been analyzed from a real variable viewpoint. Students who have forgotten or never dealt with this material will find it useful for the subsequent functions. In addition, analytic functions are defined in a way which simplifies the subsequent theory. Contents include: Calculus in the Plane, Harmonic Functions in the Plane, Complex Numbers and Complex Functions, Integrals of Analytic Functions, Analytic Functions and Power Series, Singular Points and Laurent Series, The Residue Theorem and the Argument Principle, and Analytic Functions as Conformal Mappings.

Those familiar with mathematics texts will note the fine illustrations throughout and large number of problems offered at the chapter ends. An answer section is provided. Students weary of plodding mathematical prose will find Professor Flanigan's style as refreshing and stimulating as his approach.

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following hypotheses: (i) The interior of is contained in Ω; (ii) Ω is simply connected (no holes); (iii) p dx + q dy is exact in Ω; (iv) p, and py = qx throughout Ω. (a) Only hypothesis (iii) is sufficient to guarantee that the integral in question vanishes. Give a full justification of this statement, including proofs and counterexamples. (b) Verify that either geometric hypothesis (i) or (ii), taken with either condition (iii) or (iv), together suffice to guarantee that the integral

term by term (using the Laplacian in polar coordinates), and verify that each term (/R)n (An cos n + Bn sin n) is harmonic. This raises obvious convergence questions, which we will not treat here. 6. We add that it is possible to obtain the Poisson Integral Representation for u by manipulating with the sines and cosines in the series for u given above. In fact, the integral sign in the Poisson formula is provided by the An and Bn, which are definite integrals. 7. We emphasize, however, that the

interior includes the point and is contained entirely in Ω. Since f is analytic, the derivative f′() also exists. How do we compute it in terms of the integral formula? Just as in the case of Green’s III and the Poisson Integral Formula, our approach is straightforward calculus: We differentiate under the integral sign. We have Note that this last integral is finite because is not on the curve and the integrand is continuous in z on . To justify the crucial third equality in the chain here,

few sections, you should be able to prove quite simply the following uniqueness assertion. ASSERTION Let f and g be analytic in a domain Ω and let be a Jordan curve inside Ω whose interior is also inside Ω. If f(z) = g(z) for all z on the curve , then f(z) = g(z) for all z in the interior of as well. Note that it suffices to prove that the function f − g vanishes throughout the interior of if it vanishes on . Refer to Figure 4.13. Figure 4.13 Question Is it true that f = g throughout

such that T(zk) = wk for k = 0, 1, 2. Let us prove the uniqueness first. If both T and T1 have the stated property, then T−1(T1(zk)) = zk, k = 0, 1, 2. Since the composition of T−1 and T1 has three fixed points, it must be the identity (Property 7) whence T = T1. To prove existence, we first observe that it suffices to prove that there is a transformation T1 that takes any triple z0, z1, z2 of distinct points to the three points 0, 1, . For if we can do this, then we can also construct a