By Steven G. Krantz

This publication is written to be a handy reference for the operating scientist, scholar, or engineer who must recognize and use uncomplicated recommendations in complicated research. it's not a publication of mathematical thought. it truly is in its place a ebook of mathematical perform. the entire uncomplicated rules of advanced research, in addition to many usual applica tions, are handled. when you consider that we're not constructing idea and proofs, we've not been obliged to comply to a strict logical ordering of subject matters. as a substitute, themes were prepared for ease of reference, in order that cognate issues look in a single position. Required heritage for interpreting the textual content is minimum: an exceptional floor ing in (real variable) calculus will suffice. notwithstanding, the reader who will get greatest software from the ebook should be that reader who has had a path in advanced research at it slow in his existence. This booklet is a convenient com pendium of all simple evidence approximately complicated variable thought. however it isn't a textbook, and anyone will be tough placed to pastime to profit the topic through examining this book.

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**Additional resources for Handbook of Complex Variables**

**Example text**

7): The Fundamental Theorem of Algebra: Let p{z) be a nonconstant (holomorphic) polynomial. Then p has a root. That is, there exists an a E C such that p{ a) = o. Proof: Suppose not. Then g{z) = l/p{z) is entire. Also when Izi -+ 00, then Ip{z)1 -+ +00. Thus 1/lp{z)1 -+ 0 as Izi -+ 00; hence g is bounded. By Liouville's Theorem, g is constant, hence p is constant. Contradiction. o If, in the theorem, p has degree k 2': 1, then let al denote the root provided by the Fundamental Theorem. By the Euclidean algorithm (see [HUN]), we may divide z - al into p with no remainder to obtain p{z) = (z - ad· Pl{Z).

A) -a+ z ax 8y 1 =~( au_8v)+ ax ay i(av+au). 2) 14 Chapter 1. 3) a a az z =1. -z=o, az If a Cl function f satisfies ofloz == 0 on an open set U, then f does not depend on z (but it does depend on z). If instead f satisfies oflaz == 0 on an open set U, then f does not depend on z (but it does depend on z). 4. 4 Definition of Holomorphic Function Functions f that satisfy (olaz)f == 0 are the main concern of complex analysis. 1 ) at every point of U. 2). 2) Of course the far right-hand side cannot be identically zero unless each of its real and imaginary parts is identically zero.

Complex Line Integrals First we need a suitable notion of limit. The definition is in complete analogy with the usual definition in calculus: Let U ~ C be open, P E U, and 9 : U \ {P} --+ C a function. 1) z--+P if for any E > 0 there is a 8> 0 such that when z E U and 0 < then Ig(z) - £1 < E. 2 Iz - PI < 8, Continuity In a similar fashion, if j is a complex-valued function on an open set U and P E U, then we say that j is continuous at P if limz--+p j(z) = j(P). 3 The Complex Derivative Now let j be a function on the open set U in C and consider, in analogy with one variable calculus, the difference quotient j(z) - j(P) z-P for P '" z E U.