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The Cauchy Theory

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The Cauchy Theory ISOLATED SINGULARITIES AND MEROMORPHIC FUNCTIONS TSOGTGEREL GANTUMUR Contents 1. Laurent series1 2. Isolated singularities2 3. Residues and indices4 4. The argument principle7 5. Mapping properties of holomorphic functions9 6. The Riemann sphere and meromorphic functions 10 1. Laurent series Let Dr;R = fz 2 C : r < jzj < Rg be an annulus centered at 0 with 0 ≤ r < R ≤ 1, and consider a function f 2 O(Dr;R). Then for r < ρ < σ < R and ζ 2 Dρ,σ the Cauchy integral formula gives 1 Z f(z)dz 1 Z f(z)dz f(ζ) = − : (1) 2πi @Dσ z − ζ 2πi @Dρ z − ζ Proceeding as in the proof of the Cauchy-Taylor theorem, the first term on the right hand side can be written as 1 1 X Z f(z)dz f +(ζ) := ζn ; (2) 2πi zn+1 n=0 @Dσ + + with the series converging in Dσ, so in particular f 2 O(Dσ). Note that f does not depend on σ, as long as r < σ < R, because the integrals Z f(z)dz n+1 ; (3) @Dσ z are independent of σ 2 (r; R). Moreover, since σ < R can be arbitrarily close to R, we + conclude that f 2 O(DR). Similarly, the last term in (1) can be rewritten as 1 ! 1 Z f(z)dz 1 Z X f(z)zn f −(ζ) = = dz; (4) 2πi ζ − z 2πi ζn+1 @Dρ @Dρ n=0 where we have used 1 1 1 1 1 z X zn = · = 1 + + ::: = : (5) ζ − z ζ 1 − z/ζ ζ ζ ζn+1 n=0 Each term in the series under integral in (4) can be estimated as n n f(z)z 1 ρ ≤ · · max jf(z)j; (6) ζn+1 jζj jζj jzj=ρ Date: April 16, 2015. 1 2 TSOGTGEREL GANTUMUR so as a function of z, the series converges uniformly in @Dρ, as long as jζj > ρ. Therefore we can interchange the integral with the sum, resulting in 1 1 1 X 1 Z 1 X Z f −(ζ) = f(z)zndz = ζ−n f(z)zn+1dz; jζj > ρ. (7) 2πi ζn+1 2πi n=0 @Dρ n=1 @Dρ Observe that the power series 1 1 X Z g(w) = wn f(z)zn+1dz; (8) 2πi n=1 @Dρ 1 converges for jwj < ρ , because the individual term of the series satisfies Z wn f(z)zn+1dz ≤ 2πρ(ρjwj)n · max jf(z)j; n ≥ 1; jzj=ρ @Dρ − 1 implying that g 2 O(D1/ρ). This means that f (ζ) = g( ζ ) is holomorphic in fjζj > ρg, and that the series (7) converges for jζj > ρ. Since the integrals Z f(z)zn+1dz; (9) @Dρ are independent of ρ 2 (r; R), the function f − does not depend on ρ 2 (r; R). Moreover, since − ρ > r can be arbitrarily close to r, we conclude that f 2 O(Dr;1). Finally, setting σ = ρ in (2), we can write the two series together as 1 1 X Z f(z)dz f(ζ) = f +(ζ) + f −(ζ) = ζn : 2πi zn+1 n=−∞ @Dρ This series is called the Laurent series of f, and the decomposition f = f + + f − is called the Laurent decomposition of f. In the Laurent decomposition, f − is said to be the principal part of f, and f + the regular part of f. We summarize this discussion in the following theorem. Theorem 1. Let A = Dr;R(c) be an annulus centered at c 2 C with 0 ≤ r < R ≤ 1. + − + Then any f 2 O(A) has a unique decomposition f = f + f such that f 2 O(DR(c)) and − − f 2 O(Dr;1(c)) with f (z) ! 0 as jzj ! 1. Proof. The existence of the Laurent decomposition has been demonstrated. For uniqueness, let f = g+ + g− be another such decomposition of f. Then we have f + − g+ = g− − f − in A, + + − − and so h defined as h = f − g in DR and h = g − f in Dr;1 is an entire function. But − − g − f goes to 0 at 1, hence by Liouville's theorem h ≡ 0. 2. Isolated singularities × We call the set Dr (c) = fz 2 C : 0 < jz − cj < rg the punctured disk centred at c with × × × × radius r. If c = 0, we simply write Dr = Dr (0), and if in addition, r = 1, then D = D1 (0). × Let f 2 O(Dr (c)) with some r > 0. Then we have the Laurent series expansion X n × f(z) = an(z − c) ; z 2 Dr (c): (10) n2Z The point c is called an isolated singularity of f. Now we introduce the notation Ord(f; c) = minfn 2 Z : an 6= 0g; (11) with Ord(f; c) = 1 if all an = 0, and Ord(f; c) = −∞ if there are infinitely many n < 0 with an 6= 0. Note that if Ord(f; c) ≥ 0 then upon defining f(c) = a0, one has f 2 O(Dr(c)). This ISOLATED SINGULARITIES 3 process of extending the definition of f to include c in the domain of f so that the resulting function is holomorphic, is called removing the singularity at c. In what follows, we will often implicitly assume that all singularities that can be removed have been removed, and so in particular function values at \removable singularity" points will be discussed without ever mentioning the process of removing those singularities. For example, note that Ord(f; c) > 0 is equivalent to saying that c is a zero of order k = Ord(f; c). We will see that one cannot remove the singularity at c if Ord(f; c) < 0. Definition 2. In this setting, c is called • a removable singularity if Ord(f; c) ≥ 0; • a pole of order N = −Ord(f; c) if 0 > Ord(f; c) > −∞; • an essential singularity if Ord(f; c) = −∞. We say c is a simple pole if Ord(f; c) = −1, and likewise a double pole if Ord(f; c) = −2. Sometimes very useful is the trivial fact that the three cases in the preceding definition are mutually exclusive and exhaust all possibilities. It is clear that if f has a removable singularity at c then f is bounded in a neighbourhood of c. The converse statement is also true, and called Riemann's removable singularity theorem, or Riemann's continuation theorem. × × Theorem 3 (Riemann 1851). If f 2 O(Dr (c)) and f is bounded in Dr (c), then c is a removable singularity. × Proof. Assume c = 0, and suppose that jfj ≤ M in Dr for some constant M 2 R. The Laurent series coefficients of f are given by 1 Z f(z)dz an = n+1 ; 2πi @Dρ z where ρ 2 (0; r). These can be estimated as 1 ja j ≤ · 2πρ · ρ−n−1M = Mρ−n ! 0 as ρ ! 0; for n < 0; n 2π which shows that an = 0 for all n < 0. We say that a function is holomorphic at a point if the function is holomorphic in an open neighbourhood of that point. × Theorem 4 (Characterization of poles). If f 2 O(Dr (c)) then the following are equivalent: (a) c is a pole of f of order N. (b) g(z) = (z − c)N f(z) is holomorphic at c with g(c) 6= 0. (c) 1=f is holomorphic at c and has a zero of order N at c. Proof. If (a) is satisfied then from the Laurent series expansion of f we have 1 N X n g(z) = (z − c) f(z) = an−N (z − c) ; with g(c) = a−N 6= 0; n=0 implying (b). 1 (z − c)N 1 If (b) is true, then we have = and g is holomorphic and nonzero at c, so f(z) g(z) f is holomorphic and has a zero of order N at c. 1 (z − c)N Now (c) implies that = with some h holomorphic and nonzero at c. Then f(z) h(z) by inverting this relation and expanding h in its Taylor series around c, we get (a). From (c) of this theorem it follows that jfj approaches 1 near the poles. The converse can also be proved with a little effort. 4 TSOGTGEREL GANTUMUR × Corollary 5. If f 2 O(Dr (c)) and jf(z)j ! 1 as z ! c then c is a pole of f. × × Proof. Firstly, we infer that f is zero-free on D" (c) for some " > 0. Hence 1=f 2 O(D" (c)) and since it is also bounded, by the removable singularity theorem we have 1=f 2 O(D"(c)). Moreover, 1=f is zero at c, which therefore is a pole of f by (c) of the preceding theorem. The following result is called the Casorati-Weierstrass theorem, although it appears that Charles Briot (1817{1882) and Claude Bouquet (1819{1885) published it in 1859. The proof by Felice Casorati (1835{1890) appeared in 1868, and then Weierstass proved it in 1876. In addition, Yulian Sokhotsky (1842{1927) included this theorem in his Master's thesis of 1868. × Theorem 6. Let f 2 O(Dr (c)) and let c be an essential singularity of f. Then for any × α 2 C, there is a sequence zn ! c such that f(zn) ! α. In other words, f(D" (c)) is dense in C for any " > 0. Proof. Suppose for the sake of contradiction that there are α 2 C and positive numbers δ, × and " such that jf(z) − αj ≥ δ for any z 2 D" (c).
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