Advanced Calculus fi..

592 Advanced Calculus, Fifth EditionIf f (z) is analytic, such a mapping has an additional property, that of beingconformal. A mapping from D, to D, is termed conformal if to each pair of curvesin Dz intersecting at angle a there corresponds a pair of curves in D, intersectingat angle a. The mapping is termed conformal and sense-preserving if the angles areequal and have the same sign, as illustrated in Fig. 8.23.THEOREM 35 Let w = f (z) be analytic in the domain D;, and map D, in a oneto-onefashion on a domain D,. If f'(z) # 0 in D,, then f (z) is conformal andsense-preserving.Proof. Let z(t) = x(t) + iy(t) be parametric equations of a smooth curve throughzo in DZ. By proper choice of parameter (e.g., by using arc length) we can ensurethat the tangent vectordz dx dy--- -+idtdt dtis not 0 at zb. The given curve corresponds to a curve w = w(t) in the w plane, withtangent vectordw dwdz----dt dz dtas in Section 8.2. Hencedw dw dzarg - = arg - + arg -.dt dz drThis equation asserts that, at wo = f (zo), the argument of the tangent vector differsfrom that of dzldt by the angle arg f '(zo), which is independent of the particularcurve chosen through zo. Accordingly, as the direction of the curve through zo isvaried, the direction of the corresponding curve through wo must vary through thesame angle (in magnitude and sign). The theorem is thus established.Conversely, it can be shown that all conformal and sense-preserving maps w =f (2) are given by analytic functions; more explicitly, if u and v have continuousfirst partial derivatives in D, and J = a(u, v)/a(x, y) # 0 in D,, then the fact thatthe mapping by w = u + i v = f (z) is conformal and sense-preserving implies thatux = v,, uy = -v,, so that w is analytic (see Section 2.3 of the book by Ahlforslisted at the end of the chapter). From this geometric characterization it is also clearthat the inverse of a one-to-one conformal, sense-preserving mapping has the sameproperty and is itselfanalytic. (See Problem 9 following Section 8.7.)Iff '(zo) = 0 at a point zo of D,, then arg f '(zo) has no meaning and the argumentabove breaks down. One can in fact show that conformality breaks down at zo; indeed,the transformation is not one-to-one in any neighborhood of zo by the Corollary toTheorem 33 in Section 8.17; see also Problem 14 following Section 8.21.In practice, the term "conformal" is used loosely to mean "conformal and sensepreserving";that will be done here. It should be noted that a reflection, such as themapping w = Z, is conformal but sense-reversing.Tests for one-to-one-ness. For the applications of conformal mapping, it is crucialthat the mapping be one-to-one in the domain chosen. In most examples, the mappingwill also be defined and continuous on the boundary of the domain; failure of oneto-one-nesson the boundary is less serious.