Holomorphic function

A holomorphic function $f: \mathbb{C} \to \mathbb{C}$ is a differentiable complex function. That is, just as in the real case, $f$ is holomorphic at $z$ if $\lim_{h\to 0} \frac{f(z+h)-f(z)}{h}$ exists. This is much stronger than in the real case since we must allow $h$ to approach zero from any direction in the complex plane.

Usually, we speak of functions as holomorphic on (open) sets, rather than at points, for when we consider the behavior of a function at a point, we prefer to consider it in the context of the points nearby.

Cauchy-Riemann Equations

We can obtain an equivalent definition if we break $f$ and $z$ into real and imaginary components.

Specifically, let $u, v : \mathbb{R \times R \to R}$ be definted by $u(x,y) = \text{Re}\,f(x+iy), \qquad v(x,y) = \text{Im}\,f(x+iy) .$ If $z = x+iy$ , then $f(z) = u(x,y) + i v(x,y).$

It turns out that we can express the idea " $f$ is holomorphic" entirely in terms of partial derivatives of $u$ and $v$ .

Theorem. Let $D$ be an open, connected subset of $\mathbb{C}$ . Let us abbreviate $x = \text{Re}\, z$ and $y = \text{Im}\, z$ . Then the function $f$ is holomorphic on $D$ if and only if all the partial derivatives of $u$ and $v$ with respect to $x$ and $y$ are continuous on $D$ , and the following system holds for every point $z \in D$ : $\begin{align*} \frac{\partial u}{\partial x} &= \frac{\partial v}{\partial y} ,\\ \frac{\partial u}{\partial y} &= -\frac{\partial v}{\partial x}. \end{align*}$ These equations are called the Cauchy-Riemann Equations.

For convenience, we may abbreviate $\frac{\partial f}{\partial x} = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x}, \qquad \frac{\partial f}{\partial y} = \frac{\partial u}{\partial y} + i \frac{\partial v}{\partial y} .$ With this abuse of notation, we may rewrite the Cauchy-Riemann equations thus: $\frac{\partial f}{\partial y} = i \frac{\partial f}{\partial x} .$

Proof of theorem. First, suppose that $f$ is complex-differentiable at $z$ . Then at $z$ ,

\begin{align*}
\frac{\partial f}{\partial y} = \lim_{h\to 0} \frac{f(z+ih)-f(z)}{h}
&= i \cdot \lim_{h\to 0} \frac{f(z+ih) - f(z)}{ih} \\
&= i \cdot f'(z) \\
&= i \cdot \lim_{h\to 0} \frac{f(z+h)-f(z)}{h} 
= i \cdot \frac{\partial f}{\partial x} . (Error compiling LaTeX. Unknown error_msg)

Breaking $f$ into real and imaginary components, we see $\frac{\partial u}{\partial y} + i \frac{\partial v}{\partial y} = \frac{\partial f}{\partial y} = i \frac{\partial f}{\partial x} = -\frac{\partial v}{\partial x} + i \frac{\partial u}{\partial y}.$ Setting real and imaginary components equal, we obtain the Cauchy-Riemann equations. It follows from the Cauchy Integral Formula that the second derivative of $f$ exists at $z$ ; thus the derivative of $f$ is continuous at $z$ , and so are the partial derivatives of $u$ and $v$ .

Now, suppose the Cauchy-Riemann equations hold a point $z$ , and that the partial derivatives of $u$ and $v$ exist and are continuous in a neighborhood of $z$ . Let $h = h_1 + i h_2$ be an arbitrarily small complex number, with $h_1, h_2 \in \mathbb{R}$ . Then $\begin{align*} \frac{f(z + h) - f(z)}{h} &= \frac{f(z+h_1+ih_2)-f(z+h_1)}{h_1+ih_2} + \frac{f(z+h_1)-f(z)}{h_1+ih_2} \\ &\approx \frac{ih_2}{h_1+ih_2} \frac{\partial f}{\partial y}(z+h_1) + \frac{h_1}{h_1 + ih_2} \frac{\partial f}{\partial x}(z) \\ &\approx \frac{ih_2}{h_1+ih_2} \frac{\partial f}{\partial y}(z) + \frac{h_1}{h_1 + ih_2} \frac{\partial f}{\partial x}(z) , \end{align*}$ with the first approximation from the definition of the partial derivatives and the second from the continuity of the partial derivatives. We may force $h$ to be small enough that both approximations are arbitrarily accurate. Now, by the Cauchy-Riemann equations, $\frac{i h_2}{h_1+ih_2} \frac{\partial f}{\partial y}(z) + \frac{h_1}{h_1 + ih_2} \frac{\partial f}{\partial x}(z) = \frac{\partial f}{\partial x} (z) .$ Therefore $\lim_{h\to 0} \frac{f(z+h)-f(z)}{h} = \frac{\partial f}{\partial x} (z) .$ In particular, the limit exists, so $f$ is differentiable at $z$ . Since $z$ was arbitrary, it follows that $f$ is differentiable everywhere in $D$ . $\blacksquare$

Analytic Functions

A related notion to that of homolorphicity is that of analyticity. A function $f:\mathbb{C}\to\mathbb{C}$ is said to be analytic at $z$ if $f$ has a convergent power series expansion on some neighborhood of $z$ . Amazingly, it turns out that a function is holomorphic at $z$ if and only if it is analytic at $z$ .

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Holomorphic function

Cauchy-Riemann Equations

Analytic Functions