By L. I. Volkovyskii, G. L. Lunts, I. G. Aramanovich, Mathematics
This accomplished assortment includes over 1,500 difficulties at the conception of services of the complicated variable and covers approximately each department of classical functionality conception. it will likely be of unique curiosity to working towards engineers and researchers within the actual sciences, for substantial awareness is given to these difficulties which illustrate the applying of the idea of features of a posh variable to difficulties facing the mechanics of continuing media and electric engineering.
The difficulties themselves diversity in hassle from effortless to these encountered merely in additional complicated treatments.
The first 4 chapters contain complicated numbers and features of a fancy variable; conformal mappings hooked up with effortless features; supplementary geometrical questions and generalized analytic features; and integrals and gear series.
Chapters V via VIII hide the Laurent sequence, singularities of single-valued capabilities, and vital services; numerous sequence of capabilities, parametric integrals, and endless items; residues and their purposes; integrals of the Cauchy type;and critical 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 resources for A Collection of Problems on Complex Analysis
Map onto the vertical strip 0 < Rew < 1: (1) The half-plane Re z > 0 with the disk lz-d/21 d/2 deleted; (2) The lune contained between the circles 167. The semicircle lzl > O; < lz-d1 /21 = d1 /2, lz-d2 /21 = d2 /2 (d1 < d2 ); (3) The exterior of the circles lz+d1 /21 = di/2, lz-d2/21 = d2 /2 so that w(d2 ) = 0. 172. Find the bilinear function which transforms the points -1, i, 1+i respectively into the points: (1) 0, 2i, 1-i; (2) i, oo, 1. 173. Find the bilinear function which transforms the points -1, oo, i respectively into the points : (1) i, 1, l+i; (2) oo, i, 1; (3) 0, oo, 1.
198. Prove that in the mapping of a disk onto a disk the linear transformation is uniquely determined by the specification of one interior and one boundary point. 199. The unit disk is mapped onto itself in such a way that the point z0 ~0 passes into the centre of the disk. Prove that then a unit semicircle is mapped onto a unit semicircle if and only if its ends lie on a diameter passing through the point z0 • 200. re O~>~n/2 of the unit circle is mapped onto the following arcs: (1) O~()~n/2; (2) O~()~n/3; (3) n/2~()~7n/6.
As a preliminary consider the case when the fixed points are 0 and oo. 206. Prove the following properties of the elliptic transformation: (1) Any circle orthogonal to the circles passing through two fixed points is transformed into itself with preservation of the direction of traversal. (2) A circular arc connecting the fixed points passes into a circular arc connecting the fixed points and making an angle IX with the first arc (IX= arg k). 207. (1) Prove that in the loxodromic transformation the properties (2) of the hyperbolic (see problem 205) and of the elliptic transformation (see problem 206) are preserved.
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