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Handbook of Complex Variables by Steven G. Krantz

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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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.

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