By A. I. Markushevich
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Extra info for Complex Numbers and Conformal Mappings
In special cases of functions z' = z + a or z' = cz the preservation of angles in the figures beine transformed directly follows from the fact that here we mean translation, homothetic transformation or rotation. It is remarkable that the same thing is observed in transformations by means of any rational functions of a complex variable as well as by many other more general and more complex functions of a complex variable called analytic functions. But the scope of the book does not allow us to consider the latter.
42 transformation (1), then apply transformation (2) to what we shall obtain and, finally, to the result of the second operation, we shall apply transformation (3). Recall that we found, in Sec. 30, that the figure shown on the left-hand side of Fig. 38 (and it coincides with the figure 57 in Fig. 42) is transformed by means of the function z- 1 Zl = Z+1 (i. e. the function (1)) into the figure shown in Fig. 38, right. The latter figure is bounded by a straight line passing through the point 0 and forming an angle
36a), then the corresponding straight ray also forms an angle cx with the positive direction of the real axis; in other words, the argument z" is equal to cx. Since point z is on the arc of the circle PLE passing through points a and c and the angle between the tangent PTt to that circle and the direction caU is equal to p +
Complex Numbers and Conformal Mappings by A. I. Markushevich