By Vladimir Bogachev
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Extra info for E Measure Theory
We say that lim f (z) = ∞ z→ζ if for all M > 0, there exists a δ > 0 such that 0 < |z − ζ| < δ ⇒ |f (z)| > M . (b) Let α ∈ C, and let f be deﬁned in |z| > M for some M > 0 (we say that f is deﬁned in a deleted neighborhood of ∞ in C). We say lim f (z) = α z→∞ provided lim f z→0 1 z = α. (c) The above deﬁnes the concept of continuous maps between sets in C. (d) A function f is holomorphic (has a power series expansion) at ∞ if and only if g(z) = f 1z is holomorphic (has a power series expansion) at z = 0.
Sn−1 z n−1 ) + sn z n . Now ∞ f (z) = lim Sn (z) = (1 − z) n→∞ sn z n . n=0 36 3. POWER SERIES Given > 0, choose N ∈ Z>0 such that |sn | < for n > N. Then ∞ N |f (z)| ≤ |1 − z| |sn | |z|n sn z n + n=N +1 n=0 N ≤ |1 − z| sn z n=0 n |z|N +1 + 1 − |z| N ≤ |1 − z| sn z n + M. n=0 Thus we conclude that limz→1 f (z) = 0. 21. Observe that we have not needed or used polar coordinates in our formal development thus far. 3. The exponential function, the logarithm function, and some complex trigonometric functions In this section we use power series to develop several functions.
The set H(D) is referred to as the set of holomorphic functions on D. We will see in Chapter 5 that this terminology is consistent with our earlier deﬁnition of a holomorphic function on D. 50. Assume that D is a domain in C. The set H(D) is an integral domain and an algebra over C. Its units are the functions that never vanish. 51. Let D be a domain in C. A function f : D → C is meromorphic if it is locally4 the ratio of two functions having power series expansions (with the denominator not identically zero).
E Measure Theory by Vladimir Bogachev