This chapter explores the consequence of a remarkable geometric property of holomorphic functions. Look again at Figures 3.1 and 3.2 on page 42. For arbitrary k and l the lines u = k and v = l in the w-plane are of course mutually perpendicular, and visually at least it seems that the corresponding hyperbolic curves x2 — y2 = k and 2xy = l in the z-plane are also perpendicular. Again, the lines x = k and y = l are mutually perpendicular, and it appears also that the corresponding parabolic curves in the w-plane are also perpendicular. These observations are in fact mathematically correct (see Exercise 11.1), and are instances of a general theorem to be proved shortly. First, however, we need to develop a little more of the theory of the parametric representation of curves that was introduced in Section 5.2.
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