By Gustave MALECOTT
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Use Morera's theorem to obtain this same result. Show more generally that any series of analytic functions 2//, uniformly convergent over a common region, converges to an analytic function / within that region. Show also that all derivatives of / may be obtained by Finally prove that if each // has the form 42 Functions of a Complex Variable: Theory and Technique and if then This result is called Weierstrass's double-series theorem; it provides a generalization of the result in Sec. 1-3 concerning the substitution of one series into another.
Obtain a version of Cauchy's integral theorem valid for the situation in which /(z) is analytic everywhere outside some closed curve C, with f(z) —> some constant a as z —-> <». Relate this result to the transformation f = 1/2. 6. Show, by the method used to derive Eq. 2-13, that, if $(2) is any function continuous on C (but not necessarily analytic there or anywhere else), and if a function f(z) is defined for z inside C by then f(z) is analytic inside C. Invent an example to illustrate the fact that, as an interior point z approaches a boundary point 20, f(z) need not approach
A particularly simple transformation is the general linear one w = az -+- 6, where a and b are fixed (complex) constants. To obtain the image point w •of a given point z, it is clear that one must rotate the point z around the origin by an amount arg a, increase its distance from the origin in the ratio |oj/l, translate it to the right by an amount Re b, and translate it upward Complex Numbers and Their Elementary Properties 19 by an amount Im 6. Thus this transformation corresponds to a uniform rotation, linear stretching, and uniform translation of the z plane.