Functions Of One Complex Variable Conway Djvu 13 !EXCLUSIVE!
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The theory of functions of a single complex variable is not the same as the theory of functions of one variable. There is no single distinction between a function on the real line and a function on the complex plane, but there are two distinct kinds of complex functions, holomorphic and algebraic. Algebraic functions are generally not holomorphic. Holomorphic functions are meromorphic and regular (analytic and non-singular); they are locally one-to-one and onto, and they are defined on domains, which are open and connected subsets of the complex plane. On the other hand, all rational functions are holomorphic, as are all polynomial functions. A simple example of a non-holomorphic function is the function on the complex plane that is equal to -1 everywhere except at the point zero; when restricted to the real line it is the non-trivial function that is equal to -1 everywhere except at the point zero. Here it is important to stress that the definition "...they are locally one-to-one and onto, and they are defined on domains" is in order to make sure that when the local one-to-one and onto properties fail for the real line, then these properties cannot be expected to hold for the corresponding domain that one takes over to the complex plane.
One of the differences between the theory of functions and functions of a single complex variable is the Cauchy transform. A function f on the open unit disk, D, is a Cauchy transform of a function, h, on the real line if it satisfies the following conditions: d2c66b5586
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