By L. I. Volkovyskii, G. L. Lunts, I. G. Aramanovich, Mathematics

ISBN-10: 0486669130

ISBN-13: 9780486669137

This accomplished assortment comprises over 1,500 difficulties at the thought of services of the advanced variable and covers approximately each department of classical functionality conception. will probably be of specified curiosity to practising engineers and researchers within the actual sciences, for huge recognition is given to these difficulties which illustrate the appliance of the speculation of capabilities of a fancy variable to difficulties facing the mechanics of continuing media and electric engineering.

The difficulties themselves variety in trouble from common to these encountered purely in additional complicated treatments.

The first 4 chapters contain complicated numbers and features of a fancy variable; conformal mappings attached with common features; supplementary geometrical questions and generalized analytic capabilities; and integrals and tool series.

Chapters V via VIII conceal the Laurent sequence, singularities of single-valued capabilities, and indispensable features; numerous sequence of services, parametric integrals, and countless items; residues and their purposes; integrals of the Cauchy type;and vital formulae of Poisson and Schwarz.

The ultimate 3 chapters speak about analytic continuation, singularities of many-valued personality, and Riemann Surfaces; conformal mappings; and purposes to mechanics and physics. solutions and ideas are grouped on the finish of the text.

Unabridged Dover (1991) republication of the version released by way of Pergamon Press, Oxford, England, 1965

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Extra info for A Collection of Problems on Complex Analysis

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Let, finally, r' and a' be the images of r and a in G'. Prove that n 7 ,(a') = n7 (a)c5, where c5 equals either +1 or -1, independently of r and a. In particular prove that if the mapping is continuously differentiable and the Jacobian J = au !!!.. - ~ !!!.. =I= 0 ax oy oy ax ' then the sign of c5 is identical with the sign of J.

1) Find the images of the circles and domains of the z-plane mentioned in problem 274 if the mapping function w(z) is given by the equation :+~=(~+~r-" (0<«5<2, w>O for z>l). Al'PINGS 276. ne with a. rg (w - l)/(w + 1) = fJ (0 < lfJI < :n) in such a. rg w' (ex>) = In problems 277-280 find the domains obtained from the mapping of the given domains by the functions indicated. 277. The disk lzl < 1; w = z/(z2 +1). 278. The semicircle lzl < 1, Imz > O; w = l/(z2 +1). 279. rg z < :n/n; w = 1/2 (z"+l/z").

II Fm. 8 48 PROBLEMS ON COMPLEX ANALYSIS 330. Map the plane with parallel cuts -a ~ x ~ a, y = n/2+kn (k = 0, ±1, ±2, ... ) onto the plane with cuts along the real axis segments [kn-b, kn+b] (k = 0, ±1, ±2, ... ; 0 < b < n/2). HINT. Make supplementary cuts along the imaginary axis, map one of the domains formed onto the upper half-plane and use the principle of symmetry. 331. Map the plane with cuts along the rays (-oo, -n/2], [n/2, +oo) and along the segments -a ~ y ~a, x = n/2+kn (k = 0, ± 1, ± 2, •..

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