Preliminary

Complex Number & Plane

Basic Concepts

Assume \(i^2=-1\), then \(\mathbb{C}:= \{z: z=x+iy, \quad x,y\in \mathbb{R} \}\). From definition we know there exsits a bijection bwtween \(\mathbb{C}\) and \(\mathbb{R}^2\).

It is natural to define operations like addition and multiplication. Its modulus is consistent with norm-2 in \(\mathbb{R}^2\):

\[ |z|=(x^2+y^2)^{\frac{1}{2}}. \]

which satisfies three conditions for being a metric. So from functional analysis point of view, it is a metrix space.

Now we have some introduction specially for complex number. Define a complex conjugete of \(z\) as \(\overline{z}=x-iy.\) Easy to see that

\[ \begin{equation} \text{Re}(z)=\frac{z+\overline{z}}{2}, \quad \text{Im}(z)=\frac{z-\overline{z}}{2i}\label{variables-x-y-z-zbar} \end{equation} \]

Since \(z\overline{z}=|z|^2\), for \(z\neq 0\), we have

\[ \frac{1}{z}=\frac{\overline{z}}{|z|^2}. \]

We also have its formula in terms of polar coordinate. That is,

\[ z=re^{i\theta}, \]

where \(r=|z|\) and \(\theta\) is argument of \(z\).

Convergence

Definition of Convergence

Assume sequence \(\{z_n\}_{n\geq 1}\subset \mathbb{C}\), \(w\in \mathbb{C}\). \(\{z_n\}_{n\geq 1}\) is said to converge to \(w\), if

\[ \lim_{n\rightarrow \infty}|z_n-w|=0. \]

With conclusion from multi-variable functions, we could easily see that \(\{z_n\}_{n\geq 1}\) converges to \(w\) iff the corresponding sequence of points \(\{(x_n,y_n)\}_{n \geq 1}\) in the complex plane converges to the point \((x_w,y_w)\) that corresponds to \(w\), also iff the sequence of real and imaginary parts of \(\{z_n\}_{n\geq 1}\) converge to those of \(w\), respectively.

Here we simply introduce Cauchy Sewquence, which is almost similar to \(\mathbb{R}\).

Cauchy Sequence

A sequence of \(\{z_n\}_{n\geq 1}\) is said to be a Cauchy Sequence, if

\[ |z_n-z_m|\rightarrow 0,\quad n,m\rightarrow \infty. \]

Or, \(\forall \varepsilon>0\), \(\exists N>0\), s.t \(|z_n-z_m|<\varepsilon\) whenever \(n,m>N\).

Completeness of \(\mathbb{C}\)

\(\mathbb{C}\) is complete.

Sets in the complex plane

Here most definitions are of the same as in Sets of Real Function, only we have to change \(\mathbb{R}^n\) into \(\mathbb{C}\).

Now we introduce some special definitions for complex number.

Definitions

(i) Assume \(\Omega\in \mathbb{C}\). If \(\Omega\) is bounded, we define its diameter by

\[ \text{diam}(\Omega)=\sup_{z,w\in \Omega}|z-w|. \]

(ii) An open set \(\Omega\in \mathbb{C}\) is said to be connected, if it is not possible to find two disjoint non-empty open sets \(\Omega_1\), \(\Omega_2\) such that

\[ \Omega=\Omega_1\cup\Omega_2. \]

Or we have another equivalent definition of connectedness: \(\forall x,y\in \Omega\), \(\exists\) curve \(\gamma\), its image is completely contained in \(\Omega\).

A connected open set in \(\mathbb{C}\) will be called a region.

Functions on the complex plane

Basic Concepts

Similar to \(\mathbb{R}\), we could have definition of continuity for a function defined on \(\mathbb{C}\). And a continuous function defined on compact set could assess its maximum and minimum.

Holomorphic Functions

Definition of Holomorphic function

Assume \(\Omega\subset \mathbb{C}\) is an open set, and \(f\) is a complex-valued function on \(\Omega\). \(f\) is holomorphic (or regular, complex differentiable) at point \(z_0\in \Omega\), if the quotient

\[ \begin{align} \frac{f(z_0+h)-f(z_0)}{h}\label{derivative-f-complex} \end{align} \]

converges when \(h\rightarrow 0\). Here notice that \(h\in \mathbb{C}\). If the above formula has limit, we call it the derivative of \(f\) ar \(z_0\), denoted as \(f'(z_0)\).

The function \(f\) is said to be holomorphic on \(\Omega\), if \(f\) is holomorphic at every point of \(\Omega\). For \(f\) defined on a closed set \(\Omega'\), we say \(f\) is holomorphic on \(\Omega'\), if f is holomorphic on some open set containing \(\Omega'\). If \(f\) is holomorphic on \(\mathbb{C}\), we call \(f\) is entire.

Notice that definition of holomorphic function is resemble to definition of real differentiable function, but we will see that the former has much stronger properties than the latter.

Quotes from textbook

A holomorphic function of one complex variable will satisfy much stronger properties than a differentiable function of one real variable.

For example, a holomorphic function will actually be infinitely many times complex differentiable, that is, the existence of the first derivative will guarantee the existence of derivatives of any order. This is in contrast with functions of one real variable, since there are differentiable functions that do not have two derivatives.

In fact more is true: every holomorphic function is analytic, in the sense that it has a power series expansion near every point (power series will be discussed in the next section), and for this reason we also use the term analytic as a synonym for holomorphic.

Again, this is in contrast with the fact that there are indefinitely differentiable functions of one real variable that cannot be expanded in a power series. (See Exercise 23.)

Example.

(i) Function \(f(z)=z\) is holomorphic on any open set in \(\mathbb{C}\) with \(f'(z)=1\).

(ii) Any polynomial function

\[ p(z)=a_0+a_1z+\cdots+a_nz^n \]

is holomorphic in the entire complex plane with

\[ p'(z)=a_1+2a_2z^2+\cdots+na_nz^{n-1}. \]

(iii) Function \(f(z)=\frac{1}{z}\) is holomorphic on \(\mathbb{C}-\{0\}\) with \(f'(z)=-\frac{1}{z^2}\).

(iv) Function \(f(z)=\overline{z}\) is not holomorphic, since its quotient

\[ \frac{f(z_0+h)-f(z_0)}{h}=\frac{\overline{h}}{h} \]

differ when \(h\rightarrow 0\) from different rays.

Similar to Taylor's expansion in \(\mathbb{R}\), we have

\[ f(z_0+h)=f(z_0)+f'(z_0)h+o(h). \]

Properties of holomorphic function

If \(f,g\) are holomorphic in \(\Omega\), then

(i) \(f+g\) is holomorphic in \(\Omega\), and \((f+g)'=f'+g'\).

(ii) \(fg\) is holomorphic in \(\Omega\), and \((fg)'=f'g+fg'\).

(iii) If \(g(z_0)\neq 0\), then \(f/g\) is holomorphic at \(z_0\), and

\[ (f/g)'=\frac{f'g-fg'}{g^2}. \]

(iv) Chain rule. If \(f:\Omega\rightarrow U\) and \(g:U\rightarrow \mathbb{C}\), then

\[ (g\circ f)'(z)=g'(f(z))\cdot f'(z),\quad \forall z\in \Omega. \]

Relationship between real & imaginary parts

To distinguish, we have the follwoing definition of the so-called differentiability.

Definitions of differentiability

(i) Differentiability of real function. Assume \(f(z)=u(x,y)+iv(x,y)\) is defined on \(\Omega\), \(z_0=x_0+iy_0\in \Omega\). \(f\) is said to be real differentiable, if binary real functions \(u(x,y), v(x,y)\) is differentiable at \((x_0,y_0)\).

Recall that a binary function \(f\) is differentiable at \((x_0,y_0)\), could implies \(f\) if continuous at \((x_0,y_0)\) and has partial derivatives from all directions.

(ii) Differantiability of Vector function \(\pmb{f}: \mathbb{R}^n\rightarrow \mathbb{R}^m\) is said to be differentiable at \(\pmb{x}_0\), if there exists matrix (or linear transformation) \(\pmb{A}_{m\times n}(\pmb{x_0})\) independent of \(\Delta \pmb{x}\), such that

\[ \Delta\pmb{y}=\pmb{f}(\pmb{x}+\Delta\pmb{x})-\pmb{f}(\pmb{x})=\pmb{A}\Delta \pmb{x}+o(\Delta\pmb{x}). \]

Here \(\pmb{A}\) is denoted as Jacobi matrix \(J\).

(iii) Differentiability of complex function \(f\). We could use from a vector function \(\mathbb{R}^2\rightarrow \mathbb{R}^2\). That is, there exsits a complex number \(w\), such that

\[ f(z_0+h)=f(z_0)+wh+o(h), \quad h\in \mathbb{C}. \]

Compared to Jacobi matrix expressed by \((u,v)\)

\[ J=\left[\begin{array}{cc} \frac{\partial u}{\partial x}& \frac{\partial u}{\partial y}\\ \frac{\partial v}{\partial x}& \frac{\partial v}{\partial y} \end{array}\right], \]

how to connect \(w=f'(z_0)\in \mathbb{C}\) and matrix \(J\in \mathbb{R}^{2\times 2}\)? We consider two special derivatives, real and pure imaginary ones.

Deduction for Cauchy-Riemann Equations

In definition of \(f'(z_0)\), we let \(h=h_1+ih_2\), where \(h_1,h_2\in mathbb{R}\), in order to change the dimension of denominator of formulation \(\ref{derivative-f-complex}\).

(i) \(h_2=0\), then \(h=h_1\in \mathbb{R}\), this is just a real derivative of complex function \(f\), i.e.

\[ f'(z_0)=\frac{\partial f}{\partial x}=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x} \]

is still a complex number.

(ii) \(h_1=0\), then \(h=ih_2\) is a pure imaginary number. So this gives

\[ f'(z_0)=\frac{1}{i}\frac{\partial f}{\partial y}=-i\frac{\partial u}{\partial y}+\frac{\partial v}{\partial y} \]

is also still a complex number.

Combine (i) and (ii) and let real and imaginary parts equals, we have

\[ \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y},\quad \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial y} \]

Deduction 2

Recall from formulation \(\ref{variables-x-y-z-zbar}\), we could express \(x,y\) using \(z,\overline{z}\). Then we could deduce the relationship of derivatives between \(x,y\) and \(x,\overline{z}\).

Notice that

\[ f(x,y)=f\left(\frac{z+\overline{z}}{2}, -i\frac{z-\overline{z}}{2}\right), \]

so

\[ \begin{align*} \frac{\partial f}{\partial z}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial z}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial z}=\frac{1}{2}\left(\frac{\partial f}{\partial x}-i\frac{\partial f}{\partial y}\right),\\ \frac{\partial f}{\partial \overline{z}}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial \overline{z}}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial \overline{z}}=\frac{1}{2}\left(\frac{\partial f}{\partial x}+i\frac{\partial f}{\partial y}\right). \end{align*} \]

So we could define operator for \(z\) and \(\overline{z}\)

\[ \frac{\partial }{\partial z}=\frac{1}{2}\left(\frac{\partial }{\partial x}-i\frac{\partial }{\partial y}\right),\quad \frac{\partial }{\partial \overline{z}}=\left(\frac{\partial }{\partial x}+i\frac{\partial }{\partial y}\right). \]

The more specific relationship could be expressed in the following theorem.

Holomorphic implies relationship of real & imaginary parts

Assume \(f\) is holomorphic at \(z_0\), then

\[ \frac{\partial f}{\partial \overline{z}}=0 \]

and

\[ f'(z_0)=\frac{\partial f}{\partial z}(z_0)=2\frac{\partial u}{\partial z}(z_0). \]

and for the corresponding vector function \(\pmb{F}: \mathbb{R}^2\rightarrow \mathbb{R}^2\) of complex function \(f\), \(\pmb{F}\) is differentiable, and for \((x_0,y_0)\in \mathbb{R}^2\) and \(z_0=x_0+iy_0\in \mathbb{C}\),

\[ \text{det}J_{F}(x_0,y_0)=|f'(z_0)|^2. \]

If we consider reversibly, we could also use Cauchy-Riemann equations to deduce holomorphism of \(f\).

Reverse version