Abstract
We now turn from the study of a single holomorphic function to the study of collections of holomorphic functions. In the first section we will see that under the appropriate notion of convergence of a sequence of holomorphic functions, the limit function inherits several properties that the approximating functions have, such as being holomorphic. In the second section we show that the space of holomorphic functions on a domain can be given the structure of a complete metric space. We then apply these ideas and results to obtain, as an illustrative example, a series expansion for the cotangent function. In the fourth section we characterize the compact subsets of the space of holomorphic functions on a domain. This powerful characterization is used in Sect. 7.5, to study approximations of holomorphic functions and, in particular, to prove Runge’s theorem, which describes conditions under which a holomorphic function can be approximated by rational functions with prescribed poles. The characterization will also be used in Chap. 8 to prove the Riemann mapping theorem.
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- 1.
This method is often used in analysis.
- 2.
We follow a course outlined by S. Grabiner, A short proof of Runge’s theorem, Am. Math. Monthly 83 (1976), 807–808, and rely on arguments appearing in Conway’s book listed in our bibliography.
- 3.
The argument that follows also applies for z 0 = ∞.
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Rodríguez, R.E., Kra, I., Gilman, J.P. (2013). Sequences and Series of Holomorphic Functions. In: Complex Analysis. Graduate Texts in Mathematics, vol 245. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7323-8_7
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