1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

10. l • BASIC PROPE RTIES OF CONFORMAL MAPPI NGS 397

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Figure 10.2 The analytic mapping w = f ( z) is conformal at the point zo, where
f' (zo) f= O.

• EXAMPLE 10 .1 Show that the mapping w = f (z) = cosz is conformal at
  the points z 1 = i, z2 = 1, and z3 = 7r + i, and determine the angle of rotation
  given by a = Argf' (z) at the given points.

Solution Because f^1 (z) = - sinz, we conclude that the mapping w = cosz is

conformal at alJ points except z = n'lr, where n is an integer. Calculation reveals

that

f' (i) = - sin(i) = -isinhl, /' (1) = - sinl , and

f' (7r + i) =-sin (7r + i) = isinh 1.