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A Collection of Problems on Complex Analysis by L. I. Volkovyskii, G. L. Lunts, I. G. Aramanovich,

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

This finished assortment includes over 1,500 difficulties at the thought of features of the advanced variable and covers approximately each department of classical functionality thought. will probably be of specified curiosity to working towards engineers and researchers within the actual sciences, for enormous cognizance is given to these difficulties which illustrate the applying of the idea of capabilities of a fancy variable to difficulties facing the mechanics of continuing media and electric engineering.

The difficulties themselves diversity in hassle from common to these encountered basically in additional complex treatments.

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

Chapters V via VIII hide the Laurent sequence, singularities of single-valued features, and crucial features; numerous sequence of capabilities, parametric integrals, and limitless items; residues and their purposes; integrals of the Cauchy type;and crucial formulae of Poisson and Schwarz.

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

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

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A Collection of Problems on Complex Analysis

This complete assortment includes over 1,500 difficulties at the idea of features of the advanced variable and covers approximately each department of classical functionality idea. will probably be of designated curiosity to practising engineers and researchers within the actual sciences, for substantial cognizance is given to these difficulties which illustrate the appliance of the idea of capabilities of a fancy variable to difficulties facing the mechanics of continuing media and electric engineering.

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1) Give examples of domains posseBBing boundary points but not poBSessing exterior points. (2) Give examples of simply connected domains for which the set of exterior points is not a domain. runs through the set 848. Th~ number (! = inf iz-Z:I, where E is known as the distance from the point z to the set E. Prove that if Eis a closed set a point z' e E can be found such that iz-z'I = e. 349. The number (! =inf iz1 -z1 1, where z1 and za run through the sets E 1, E1 , respectively is known as the distance between these sets.

253. The plane with a cut along the segment [z1' z1]. 254. The plane with cuts along the rays (-oo, -R], [R, oo) (R>O). 255. ';['he plane with a cut along the ray in the first quadrant which issues from the point i and is parallel to the straight line y=x. 256. The half-plane Imz>O with a cut along the segment [O, ih], h>O. 257. The half-plane lmz>O with a cut from ih to oo along the positive imaginary semi-axis (h>O). 258. The disk lzl

Z2, (a is a real number); z (3) w = (l-z>9. 340. Prove that in the mapping w = f(z) the curvature of the image of the circle izl = r is expressed by the formula f"(z)] l+Re [z7fz) k= jzj'(z)j 50 PROBLEMS ON COMPLEX ANALYSIS 341. Prove that an analytic function /(z) maps the circle Jzl = r onto a convex curve when and only when a [n ]= l+Re [zf"f'(z)(z)] ;;,:: O a

)t. 342. Prove that the disk Jzl < r is mapped by the analytic function f (z) onto a starlike domain when and only when a~ arg/(z) = Re[ z;;z(;>];;::: 0 for all q, (z =re'"'>· 343.

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