Abstract Nonsense

Crushing one theorem at a time

Complex Differentiable and Holmorphic Functions (Pt. III)


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

Holomorphic Functions

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We now discuss the notion of holomorphic functions which is where the Cauchy-Riemann equations of the introduction come into play.

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Let \Omega\subseteq\mathbb{C} be open. Let f:\Omega\to\mathbb{C} be a C^1 function in the real sense. Then, we define the Wirtinger’s Derivatives \displaystyle \frac{\partial f}{\partial z} and \displaystyle \frac{\partial f}{\partial \bar{z}} as follows

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\displaystyle \frac{\partial f}{\partial z}=\frac{1}{2}\left(\frac{\partial f}{\partial x}-i\frac{\partial f}{\partial y}\right)\quad\quad \frac{\partial f}{\partial \bar{z}}=\frac{1}{2}\left(\frac{\partial f}{\partial x}+i\frac{\partial f}{\partial y}\right)

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We say that f is holomorphic on \Omega if it’s C^1 and \displaystyle \frac{\partial f}{\partial \bar{z}}=0 as functions on \Omega. As was noted in the motivation if f=u+iv this equation is equivalent to the Cauchy-Riemann equations 

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\begin{cases}\displaystyle \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y} \\ \text{ }\\ \displaystyle \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x} \end{cases}

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One can verify using standard multivariable calculus that the Wirtinger derivatives satisfy the following equations

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Theorem: Let \Omega\subseteq\mathbb{C} be open. Suppose that f,g:\Omega\to\mathbb{C} are C^1 and a,b\in\mathbb{C}  then

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\begin{aligned}&\mathbf{(1)}\quad \frac{\partial}{\partial z}(af+bg)=a\frac{\partial f}{\partial z}+b\frac{\partial g}{\partial z}\\ &\mathbf{(2)}\quad \frac{\partial}{\partial z}(fg)=\frac{\partial f}{\partial z}g+\frac{\partial g}{\partial z}f\end{aligned}

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and the same holds with z replaced with \bar{z}.

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From this we may easily deduce that the set of holmorphic functions on \Omega is a \mathbb{C}-algebra which we shall denote either \mathcal{O}(\Omega) or \text{Hol}(\Omega).

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Being holomorphic is a very strong condition (we shall see that it’s as strong as being complex differentiable). For example, we have the following theorem:

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Theorem: Let \Omega\subseteq\mathbb{C} be open. If f\in\mathcal{O}(\Omega) and \text{Re}(f) or \text{Im}(f) is constant then f is constant. 

Proof: Write f=u+iv. The proof is the same for the case of \text{Im}(f)=\text{constant} so we only do the first case. We know that u is constant which tells us that \displaystyle \frac{\partial u}{\partial x}=\frac{\partial u}{\partial y}=0, but by the Cauchy-Riemann equations this tells us that \displaystyle \frac{\partial v}{\partial x}=\frac{\partial v}{\partial y}=0 and so v is constant as well. \blacksquare

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The obvious fact that we’d like to prove is that being holomorphic on \Omega is the same thing as being complex differentiable on \Omega. Unfortunately, due to a historical mistake, we can’t quite do this yet. What is this historical mistake? We have defined here that a holomorphic function is just a C^1 function which satisfies the Cauchy-Riemann equations. Of course though, we needn’t have that \displaystyle \frac{\partial f}{\partial x} and \displaystyle \frac{\partial f}{\partial y} be continuous to state that f satisfies the Cauchy-Riemann equations. We could call a function which satisfies the Cauchy-Riemann equations preholomorphic and then we’d have that the holomorphic functions are precisely the preholomorphic C^1 functions. In fact, in some books (viz. [3]) define holomorphic to mean preholomorphic. The inclusion of C^1 in our definition is purely historical and, for all intents and purposes, the correct definition (a view which should become clear when we start discussing contour integrals).

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Now, it’s easy to see that f being differentiable at z_0\in\Omega is equivalent to the existence of complex numbers \alpha,\beta\in\mathbb{C} (secretly just \displaystyle \frac{\partial f}{\partial z}(z_0) and \displaystyle \frac{\partial f}{\partial y}(z_0)) such that f(z)-f(z_0)=\alpha x+\beta y+\eta(z)(z-z_0) for some function \eta(z) with \displaystyle \lim_{z\to z_0}\eta(z)=0. One can then verify the identity

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\displaystyle \frac{f(z)-z_0)}{z-z_0}=\frac{\partial f}{\partial z}(z_0)+\frac{\partial f}{\partial\bar{z}}(z_0)\frac{\overline{z-z_0}}{z-z_0}+\eta(z)

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Since \displaystyle \lim_{z\to z_0}\frac{\overline{z-z_0}}{z-z_0} does not exist, it’s clear that the limit on the left hand side is going to exist if and only if \displaystyle \frac{\partial f}{\partial \bar{z}}=0 and in which case \displaystyle f'(z_0)=\frac{\partial f}{\partial z}(z_0). Thus, we obtain the following theorem:

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Theorem: Let \Omega\subseteq\mathbb{C} be open. Then, f is differentiable on \Omega if and only if f is preholomorphic on \Omega in which case \displaystyle f' =\frac{\partial f}{\partial z}.

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Of course, this tells us immediately that

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Corollary: Let \Omega\subseteq\mathbb{C} be open. Then, f\in\mathcal{O}(\Omega) implies that f is differentiable on \Omega and \displaystyle f' =\frac{\partial f}{\partial z}.

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Now, it’s not at all clear that the converse is true. There is no reason to believe at this moment that every complex differentiable function is C^1(\Omega). The important fact though is that this is true:

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Theorem(Goursat): Let \Omega\subseteq\mathbb{C} be open. Then, f is complex differentiable on \Omega if and only if f\in\mathcal{O}(\Omega) and moreover \displaystyle f'= \frac{\partial f}{\partial z}.

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We shall prove this theorem eventually but it is currently out of our reach, and so we defer it. The important thing to keep in mind is that while we will be working expressly with holomorphic (opposed to preholomorphic) functions in the coming posts, this is the same thing as working with just complex differentiable functions.

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References:

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[1] Greene, Robert Everist, and Steven George Krantz. Function Theory of One Complex Variable. Providence: American Mathematical Society, 2006. Print.

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

[3] Rudin, W. Real and Complex Analysis. New York,NY: McGraw-Hill, 1988. Print.

[4]  Ahlfors, Lars V. Complex Analysis; an Introduction to the Theory of Analytic Functions of One Complex Variable. New York: McGraw-Hill, 1966. Print.

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May 1, 2012 - Posted by | Uncategorized | , , , , , , , , , , , , ,

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