By George F. Carrier;Max Krook;Carl E. Pearson
The e-book is a literal reprint of the 1966 version. This contains the font within which that variation was once published. it's going to have helped if the writer had recast the textual content in a extra glossy font, that present readers may locate more straightforward to learn. now not that it really is very unlikely with the unique presentation, yet there's a the reason is, fonts have replaced in fresh a long time. might be it is going to have extra to the price of this ebook to have performed so.
Having stated that, the textual content is a excitement to learn, no less than for this reviewer, who studied complicated research from Jerrold Marsden's "Basic complicated Analysis". The latter ebook is a extra casual presentation than this. the present e-book emphasises the mathematical rigour, whereas Marsden inquisitive about giving a practical figuring out of the way to discover contour integrals.
Compared to contemporary texts, Carrier's e-book is skimpy with diagrams. A extra beneficiant use of those could relief a few readers new to the topic.
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Additional info for Functions of a Complex Variable: Theory and Technique (Classics in Applied Mathematics)
Example text
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.