Abstract Nonsense

Crushing one theorem at a time

Loci of Holomorphic Functions and the Inverse Function Theorem (Pt. II)


Point of Post: This is a continuation of this post.

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The proof the of the holomorphic IFT follows from the following lemma, which is a slight generalization of the Argument Principle:

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Lemma(Generalized Argument Principle): Let \Omega\subseteq\mathbb{C} be a domain and let f,g\in\mathcal{O}(\Omega). Let \Gamma be a simple closed curve in \Omega and suppose that f has no zeroes on \Gamma. Then,

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\displaystyle \frac{1}{2\pi i}\int_\Gamma g(\zeta)\frac{f'(\zeta)}{f(\zeta)}d\zeta=\sum_{j=1}^{n}g(a_j)

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where a_1,\cdots,a_n are the roots of f in the interior of \Gamma.

Proof: This is done by merely applying the residue Residue theorem. Namely, we know that:

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\displaystyle \frac{1}{2\pi i}\int_\Gamma g(\zeta)\frac{f'(\zeta)}{f(\zeta)}=\sum_{\alpha}\text{Res}\left(g\frac{f'}{f},\alpha\right)

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where the sum runs over the distinct roots of f in the interior of \Gamma. We see though that if \alpha is a root of degree m at f then f(z)=(z-\alpha)^mh(z) with h\in\mathcal{O}(\Omega) and h(\alpha)\ne 0. We see then that

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\displaystyle g(z)\frac{f'(z)}{f(z)}=g(z)\frac{h(z)}{h'(z)}+g(z)\frac{m}{z-\alpha}

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From this it’s evident that

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\displaystyle \text{Res}\left(g\frac{f'}{f},\alpha\right)=mg(\alpha)

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from where the conclusion follows. \blacksquare

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The operative part of this theorem is the following:

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Corollary: Let \Omega\subseteq\mathbb{C} be open and f\in\mathcal{O}(\Omega). Suppose that \Gamma is a simple closed curve in \Omega such that f has one zero \alpha, which is simple, in the interior of \Gamma and no roots on \Gamma.Then,

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\displaystyle \alpha=\frac{1}{2\pi i}\int_\Gamma \zeta\frac{f'(\zeta)}{f(\zeta)}\;d\zeta

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Now, we can pretty easily prove the holomorphic IFT:

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Proof(Holomorphic IFT): Now, since F_v is continuous (remember, Hartog’s theorem!) there exists an \varepsilon>0 such that |w|,|v|<\varepsilon_1 implies that F_v(w,v)\ne 0. Moreover, since F_v(0,0)\ne 0 there exists a holomorphic function g(y) not vanishing at the origin such that F(0,v)=vg(v). Thus, we see that there is some \varepsilon_2>0 such that F(0,v) has only the simple zero (0,0) as a root for |yv<\varepsilon_2. Let \varepsilon=\text{min}\{\varepsilon_1,\varepsilon_2\} and define

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\displaystyle \eta(x)=\frac{1}{2\pi i}\int_{|v|=\varepsilon}\frac{F_v(w,v)}{F(w,v)}\; dv

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Now, the Argument Principle tells us that \eta is a continuous mapping D_\varepsilon(0)\to\mathbb{Z}. By construction \eta(0)=1, and thus we see that \eta is identically 1 on D_\varepsilon(0). Thus, for each w\in D_\varepsilon(0) there exists a unique solution to F(w,v)=0, call this v value \varphi(w). By the corollary above, we see that

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\displaystyle \varphi(w)=\int_{|y|=\varepsilon}v\frac{F_v(w,v)}{F(w,v)}\;dv

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and is thus holomorphic. In particular, we see that \varphi is a holomorphic mapping D_\varepsilon(0)\to\mathbb{C} such that on the polydisc D_\varepsilon(0)\times D_\varepsilon(0) one has that (w,\varphi(w)) is the unique solution to F(w,v)=0. \blacksquare

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Remark: Note that choosing v as the non-zero partial played no role and the above is true if we require that F_w(0,0)\ne0.

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Now, from this we can prove that loci of certain holomorphic functions are actually Riemann surface. In particular, like in the case of smooth manifolds we define a regular value of a holomorphic map f:U\to\mathbb{C} where U\subseteq\mathbb{C}^2 to be a point y\in\mathbb{C} such that for every p\in f^{-1}(y) we have that (f_w(p),f_v(p))\ne(0,0). We call a function f\in\mathcal{O}(U) non-singular if 0 is a regular value for f. We then have the following theorem:

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Theorem: Let U\subseteq\mathbb{C}^2 be open and f\in\mathcal{O}(U) be non-singular. Then, Z(f) is a complex manifold of dimension one.

Proof: Let p=(w_o,v_0)\in Z(f) be arbitrary. By assumption, one of f_w(p),f_v(p)\ne 0. Thus, the holomorphic IFT tells us that there exists a neighborhood W_p of w_0 in \mathbb{C} and a holomorphic function g_p:W_p\to\mathbb{C} such that \pi_{i_p}^{-1}(W_p) is just the graph of g_p over W_p, where i_p\in\{1,2\} depends on which partial is nonzero. We claim that \left\{(\pi_{i_p}^{-1}(W_p),\pi_1)\right\} is an atlas for Z(f). Indeed, it’s clear that this is a topological atlas, and so it suffices to check that the elements of the atlas are compatible. To this end, suppose that we have two points p_1,p_2\in Z(f) such that \pi_{i_{p_1}}^{-1}(W_{p_1})\cap\pi_{i_{p_2}}^{-1}(W_{p_2}) contains a point p. Now, if i_{p_1}=i_{p_2}, say they are both equal to 1, then obviously the transition map just projects down and lifts back up–in other words, is the identity. Now, if i_{p_1}\ne i_{p_2}, say i_{p_j}=j then one can verify fairly easily that the overlap map is just g_p(z), which is holomorphic. The conclusion follows. \blacksquare

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References: 

[1] Varolin, Dror. Riemann Surfaces by Way of Complex Analytic Geometry. Providence, RI: American Mathematical Society, 2011. Print.

[2] Miranda, Rick. Algebraic Curves and Riemann Surfaces. Providence, RI: American Mathematical Society, 1995. Print.

[3] Forster, Otto. Lectures on Riemann Surfaces. New York: Springer-Verlag, 1981. Print.

[4] Conway, John B. Functions of One Complex Variable. New York: Springer-Verlag, 1978. Print.

[5] Gong, Sheng. Concise Complex Analysis. Singapore: World Scientific, 2001. Print

[6] Hörmander, Lars. An Introduction to Complex Analysis in Several Variables. Princeton, NJ: Van Nostrand, 1966. Print.

[7] Krantz, Steven G. Function Theory of Several Complex Variables. New York: Wiley, 1982. Print.

[8] Ohsawa, T. Analysis of Several Complex Variables. Providence, RI: American Mathematical Society, 2002. Print.

[9] Ebeling, Wolfgang. Functions of Several Complex Variables and Their Singularities. Providence, RI: American Mathematical Society, 2007. Print.

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October 3, 2012 - Posted by | Complex Analysis, Riemann Surfaces | , , , , , ,

1 Comment »

  1. […] Loci of Holomorphic Functions and the Inverse Function Theorem (Pt. III) Point of Post: This is a continuation of this post. […]

    Pingback by Loci of Holomorphic Functions and the Inverse Function Theorem (Pt. III) « Abstract Nonsense | October 3, 2012 | Reply


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