Abstract
The first section of this chapter introduces the complex plane, fixes notation, and discusses some useful concepts from real analysis. Some readers may initially choose to skim this section. The second section contains the definition and elementary properties of the class of holomorphic functions—the basic object of our study.
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Notes
- 1.
The reader may want to consult J. R. Munkres Topology (Second Edition), Dover, 2000, or J. L. Kelley, General Topology, Springer-Verlag, 1975 as well as definitions in Chap. 4.
- 2.
In general X { condition} and {x ∈ X; { condition}} will describe the set of all x in X that satisfy the indicated condition.
- 3.
With these operations \((\mathbb{C}, +,\cdot )\) is a field.
- 4.
The number π will be defined rigorously in Definition 3.34. Trigonometric functions will be introduced in the next chapter where some of their properties, including addition formulae, will be developed. For the moment, polar coordinates should not be used in proofs.
- 5.
Exercises can be found at the end of each chapter and are numbered by chapter, so that Exercise 2.7 is to be found at the end of Chap. 2.
- 6.
LHS (RHS) are standard abbreviations for left (right) hand side and will be used throughout this book.
References
Ahlfors, L.V.: Complex Analysis, 3rd edn. McGraw-Hill, New York (1979)
Bak, J., Newman, D.J.: Complex Analysis. Springer, Berlin (1982)
Berenstein, C.A., Gay, R.: Complex variables, an introduction. In: Graduate Texts in Mathematics, vol. 125. Springer, Berlin (1991)
Boas, R.P.: Invitation to Complex Analysis. Random House, New York (1987)
Cartan, H.: Elementary Theory of Analytic Functions of One or Several Complex Variables. Addison-Wesley, Reading (1963)
Churchill, R.V., Brown, J.W.: Complex Analysis and Applications, 5th edn. McGraw-Hill, New York (1990)
Conway, J.B.: Functions of One Complex Variable, 2nd edn. Springer, Berlin (1978)
Derrick, W.R.: Complex Analysis and Applications, 2nd edn. Wadsworth International Group, New York (1982)
Fisher, S.D.: Complex Variables, 2nd edn. Dover Publications, New York (1999)
Freitag, E., Busam, R.: Complex Analysis. Springer, Berlin (2005)
Greene, R.E., Krantz, S.G.: Function Theory of one Complex Variable. John Wiley & Sons Inc., New York (1997)
Heins, M.: Complex Function Theory. Academic Press, New York (1968)
Hille, E.: Analytic Function Theory, vol. I. Blaisdell, New York (1959)
Hille, E.: Analytic Function Theory, vol. II. Blaisdell, Waltham (1962)
Hörmander, L.: An Introduction to Complex Analysis in Several Variables. Van Nostrand, Princeton (1966)
Knopp, K.: Theory of Functions I. Elements of the General Theory of Analytic Functions. Dover Publications, New York (1945)
Knopp, K.: Theory of Functions II. Applications and Continuation of the General Theory. Dover Publications, New York (1947)
Knopp, K.: Problem Book in the Theory of Functions: Problems in the Elementary Theory of Functions, , vol. 1. Dover Publications, New York (1948) (Translated by Lipman Bers)
Knopp, K.: Elements of the Theory of Functions. Dover Publications Inc., New York (1953) (Translated by Frederick Bagemihl)
Knopp, K.: Problem Book in the Theory of Functions: Problems in the Advanced Theory of Functions, vol. II. Dover Publications, New York, NY (1953) (Translated by F. Bagemihl)
Lang, S.: Complex analysis, 4th edn. In: Graduate Texts in Mathematics, vol. 103. Springer, Berlin (1999)
Lax, P.D., Zalcman, L.: Complex Proofs of Real Theorem. American Mathematical Society University Lecture Series, AMS, New York (2012)
Marsden, J.E.: Basic Complex Analysis. W. H. Freeman and Company, New York (1973)
Narasimhan, R.: Complex analysis in One Variable. Verlag, Birkha̋user (1985)
Needham, T.: Visual Complex Analysis. Oxford University Press, Oxford (2004)
Nevanlinna, R., Paatero, V.: Introduction to Complex Analysis. Addison-Wesley, New York (1964)
Palka, B.: An Introduction to Complex Function Theory. Springer, Berlin (1991)
Remmert, R.: Theory of Complex Functions. Springer, Berlin (1991) (Translated by R. B. Burckel)
Roy, R.: Sources in the Development of Mathematics. Cambridge University Press, Cambridge (2011)
Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987)
Sandifer, C.E.: The Early Mathematics of Leonhard Euler. The Mathematical Association of America, Washington (2007)
Shakarchi, R.: Problems and Solutions for Complex Analysis. Springer, Berlin (1999)
Silverman, R.A.: Complex Analysis with Applications. Prentice-Hall (1974)
Stein, E.M., Shakarchi, R.: Complex Analysis. Princeton Lectures in Analysis. Princeton University Press, Princeton (2003)
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Rodríguez, R.E., Kra, I., Gilman, J.P. (2013). Foundations. In: Complex Analysis. Graduate Texts in Mathematics, vol 245. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7323-8_2
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