Abstract
This chapter is devoted to some immediate consequences of the fundamental result for the Cauchy theory, Theorem 4.61, of the last chapter. Although the chapter is very short, it includes proofs of many of the implications of the fundamental theorem in complex function theory (Theorem 1.1). We point out that these relatively compact proofs of a host of major theorems result from the work put into Chap. 4 and earlier chapters.
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Notes
- 1.
We are following a course outlined by J. D. Dixon, A brief proof of Cauchy’s integral formula, Proc. Amer. Math. Soc. 29 (1971), 625–626.
- 2.
For a proof see the appendix to Ch. IX of J. Dieudonné, Foundations of Modern Analysis, Pure and Applied Mathematics, vol. X, Academic Press, 1960 or Chap. 10 of J. R. Munkres, Topology (Second Edition), Dover, 2000.
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Rodríguez, R.E., Kra, I., Gilman, J.P. (2013). The Cauchy Theory: Key Consequences. In: Complex Analysis. Graduate Texts in Mathematics, vol 245. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7323-8_5
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