By Robert B. Ash (Auth.)

ISBN-10: 0120652501

ISBN-13: 9780120652501

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2(c). Thus 9(b) = φ(ο) + 2-nm, θ(ά) = φ(α) + Ιπνη, and the result follows. 3 Definition Let y be a closed curve, and z0 a point not in y*. Denote by y + w the curve y(t) + w, a < / < b. Let Θ be a continuous argument of y — z 0 . The zWex of z0 with respect to y, denoted by «(y, z0), is denned as —^^·. 2, w(y, z0) is well defined. Intuitively, η(γ, ζ0) is the net number of revolutions of y(f), a < / < è, about the point z0 ; for this reason, the term winding number is often used for the index. It follows from the above definition that η(γ, ζ0) = n(y + w, z0 + w).

Since | e~if | = ev, so does v. | PROBLEMS 1. Give an example of a nonconstant analytic function/on an open connected set U such that / has a limit point of zeros at a point outside of U. 2. 9, for continuous rather than analytic functions. What can be said about the relative strength of the statements? The proof in the text shows that (a) implies (b) implies (c) implies (d), but, for example, does (b) imply (a) ? ) 45 APPLICATIONS 3. Let / and g be analytic at z 0 , and not identically 0 in any neigh- f(z) borhood of z 0 .

_ _ Now if i^ is not contained in U, let z0 G # , — U. Then since z0 £ U, it cannot be in y*, hence [z z , z0] does not intersect y*. 6 that n(y9 zt) = n(y, z0). But n(y, z0) = 0 by hypothesis, so that y is equivalent to a cycle σ = Σ «(y, ζ^) δ^-,, where all j=1 _ Rj C C/. 63 CAUCHY'S THEOREM Thus fis actually defined on σ*, and the equivalence of y and σ yields f f(z)dz= J v f«(y,z,) f i But SdRjf(z)dz J f{z)dz. 11. 3; that is, if y is a closed path (or cycle) in U such that J r / ( z ) dz = 0 for every/analytic on £/, then w(y, z) = 0 for every z £ £/.

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Complex Variables by Robert B. Ash (Auth.)


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