Koch R. Complex Variable Outline 2016 torrent

Textbook in PDF format

The following material is an outline of complex variable theory. It omits many proofs, but gives enough informal information that a good student could fill in the proofs on their own.
There are no exercises. Please consult other books.
Some authors prefer to write mathematics in a logical order in which results cannot be mentioned until the tools are available to prove them. An advantage of this style is that the reader always know where they stand because if a result is stated in the text, its proof is nearby. But there are disadvantages, best illustrated with an example. Almost all treatments of complex analysis extend ex; sin x; cos x; and ln x to functions of a complex variable ez; sin z; cos z; and Log(z). This happens early in the text, providing many examples of holomorphic functions, but the extensions can seem arbitrary, one choice out of many.
Much later on, these books prove the crucial fact that if a function of a real variable has a holomorphic extension, it is unique. These notes mention that fact as soon as extensions are de ned, and sketch a rough proof to be improved later on