Advanced Calculus fi..

Chapter 8 Functions of a Complex VariableBy the Corollary to Theorem 33 in Section 8.17, if f'(z) = 0 at some point ofthe domain. then the mapping cannot be one-to-one. One should therefore verify thatf '(2) # 0 in the domain as a first step. Even if this is satisfied, the mapping need notbe one-to-one, and one must apply additional tests. The following are useful testsfor one-to-one-ness:I. Explicit fortnula for the inverse function. If an explicit formula for theinverse function z = z(w) is available and one can show that, by this formula,there is at most one z in D, for each w, then the mapping must be one-to-one.For example, w = z' is one-to-one in the first quadrant of the z plane, for toeach w there is at most one square root fi in the quadrant. As 7 varies overD,, w varies over D,,: the upper half-plane v > 0, as will be seen.11. Analysis of level curves of u and v. One can verify one-to-one-ness and atthe same time obtain a very clear picture of the mapping by plotting the levelcurves: u(x, y) = cl, v(x, y) = c2. If for arbitrary choice of cl, c2 the lociu = cl, v = c2 intersect at most once in D,, then the mapping from DZ tothe uv plane is one-to-one.111. One-to-one-ness on the boundary. This is the test formulated in Theorem33 in Section 8.17: if f (z) is analytic in C plus interior D, and one-to-oneon C, then f (z) is one-to-one on D,. As one can show, it is sufficient thatf be continuous on C plus interior, and analytic on the interior; in variousimportant examples this is the case, analyticity being violated at one or morepoints on the boundary. One can also reason thus: if one can show that,8for every simple closed curve C' approximating the boundary sufficientlyclosely, f is one-to-one on C', then f must be one-to-one in D:; for if ffails to be one-to-one in D,, then f (21) = f (z2) for two points zl , zz ofD,. A curve C' sufficiently close to the boundary would enclose both 7 andz2, SO that the failure of one-to-one-ness would have to reveal itself on C'.Other natural extensions of the principle will be pointed out in the examplesto follow.IV. Real or imagitzary part maxin~a and rninima on the boundary. Let f (z)be analytic in domain D,, bounded by the simple closed curve C. Let u =Re [ f (z)] be continuous in D, plus C. Ifu hasprecisely one relative maximumand one relative minimuln on C, then w = f (2) is one-to-one in D,. Asimilar criterion holds for v = Im [ f (z)]. Furthermore, the same conclusionholds for mappings: u = u(x, y), v = v(x, y) by nonanalytic functions,provided the Jacobian J = a(u, v)/a(.u, y) is always positive (or alwaysnegative) in D;. For a proof see the paper by Kaplan listed at the end of thechapter.V. Let f (z) = u + iv be analytic in D; and let DZ be convex; that is, for eachpair of points zl and z2 of D,, let the line segment from zl to z2 lie in D,. Ifreal constants a, b can be found such thatthen f is one-to-one in D, (see Problem 15 below, following Section 8.21).