Mathematical Methods for Physics and Engineering - Matematica.NET

COMPLEX VARIABLESyz 1z 2sC ′ 1w 1w 2C 1C 2z 0θ 2 θ 1xw = g(z)φ 2w 0φ 1rC ′ 2Figure 24.3 Two curves C 1 and C 2 in the z-plane, which are mapped ontoC 1 ′ and C′ 2 in the w-plane.important properties are that, except at points at which g ′ (z), and hence h ′ (z), iszero or infinite:(i) continuous lines in the z-plane transform into continuous lines in thew-plane;(ii) the angle between two intersecting curves in the z-plane equals the anglebetween the corresponding curves in the w-plane;(iii) the magnification, as between the z-plane and the w-plane, of a small lineelement in the neighbourhood of any particular point is independent ofthe direction of the element;(iv) any analytic function of z transforms to an analytic function of w andvice versa.Result (i) is immediate, and results (ii) and (iii) can be justified by the followingargument. Let two curves C 1 and C 2 pass through the point z 0 in the z-planeand let z 1 and z 2 be two points on their respective tangents at z 0 , each a distanceρ from z 0 . The same prescription with w replacing z describes the transformedsituation; however, the transformed tangents may not be straight lines and thedistances of w 1 and w 2 from w 0 have not yet been shown to be equal. Thissituation is illustrated in figure 24.3.In the z-plane z 1 and z 2 are given byz 1 − z 0 = ρ exp iθ 1 and z 2 − z 0 = ρ exp iθ 2 .The corresponding descriptions in the w-plane arew 1 − w 0 = ρ 1 exp iφ 1 and w 2 − w 0 = ρ 2 exp iφ 2 .The angles θ i and φ i are clear from figure 24.3. The transformed angles φ i arethose made with the r-axis by the tangents to the transformed curves at their840