Preliminaries to Complex Analysis

Preliminaries

Holomorphicity is the complex analytic version of differentiability.

Definition 370.1(Holomorphicity).

$f:\mathbb{C}\to \mathbb{C}$ is said to be holomorphic at a point $z\in \mathbb{C}$ if

$$\underset{h\to 0}{\lim }\frac{f(z+h)-f(z)}{h}$$

exists. The limit is denoted by $f^{\prime }(z_0)$ when it exists, and is called the derivative of $f$ at $z_0$. $f$ is said to be holomorphic in open $\Omega \subseteq \mathbb{C}$ if $f$ is holomorphic at every point in $\Omega$.

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$f$ is holomorphic at $z_0\in \Omega$ iff there exists a complex number $f^{\prime }(z_0)$ such that

$$\begin{array}{r}f(z_0+h)=f(z_0)+f^{\prime }(z_0)h+h\psi (h),\end{array}$$E1

where ${\lim }_{h\to 0}\psi (h)=0$.

Some trivialities:

Proposition 370.2(Proposition).

1. If $f$ is holomorphic at $z_0$, $f$ is continuous at $z_0$.
2. If $f$ and $g$ are holomorphic in $\Omega$, then
3. $f+g$ is holomorphic in $\Omega$ and $(f+g{)}^{\prime }=f^{\prime }+g^{\prime }$.
4. $fg$ is holomorphic in $\Omega$ and $(fg{)}^{\prime }=f^{\prime }g+f^{\prime }$.
5. If $g(z_0)\ne 0$, then $f/g$ is holomorphic at $z_0$ and $(f/g{)}^{\prime }=(f^{\prime }g-f{g}^{\prime })/g^2$.
6. If $f:\Omega \to U$ and $g:U\to \mathbb{C}$ are holomorphic, the chain rule holds: $(g\circ f{)}^{\prime }=g^{\prime }(f(z))f^{\prime }(z)$ for all $z\in \Omega$.

Proofs are routine.

Holomorphic functions have some really neat properties:

1. Every holomorphic function is infinitely many times complex differentiable (Cor 377.6).
2. It’s better: Every holomorphic function is analytic (Thm 377.8)! Holomorphic and analytic are used synonymously. Recall that for functions of real variables, analytic functions form a strict subset of smooth functions!

Complex functions as maps ${\mathbb{R}}^2\to {\mathbb{R}}^2$

For $f:\mathbb{C}\to \mathbb{C}$ with $f=u+iv$, define $F:{\mathbb{R}}^2\to {\mathbb{R}}^2$ by $F(x,y)=(u(x,y),v(x,y))$. For $z=x+iy$, we may write $f(x,y)$ in place of $f(z)$ (treating $f$ as a function ${\mathbb{R}}^2\to \mathbb{C}$).

Contrast Def 1 for the derivative of $f$ with Def 185.1 for the derivative of $F$. Clearly, complex differentiability differs significantly from the usual notion of multivariable real differentiability.

Example 370.3(Example).

The function $f(z)=\overline{z}$ is not holomorphic. Indeed, we have

$$\begin{array}{r}\frac{f(z_0+h)-f(z_0)}{h}=\frac{\overline{h}}{h}\end{array}$$

which has no limit as $h\to 0$. However, when seen as a function from ${\mathbb{R}}^2\to {\mathbb{R}}^2$, the function $(x,y)\mapsto (x,-y)$ is clearly differentiable (indefinitely, even).

There is, however, a connection between $f^{\prime }$ (the complex derivative) and $F^{\prime }$ (the total derivative): for $z_0=z_0+i{y}_0$, we have

$$\begin{array}{rl}{f}^{\prime }(z_0)& =\underset{h_1\to 0}{\lim }\frac{f(x_0+h_1,y_0)-f(x_0,y_0)}{h_1}\\ & =\frac{\partial f}{\partial x}(z_0)\\ & \\ f^{\prime }(z_0)& =\underset{h_2\to 0}{\lim }\frac{f(x_0,y_0+h_2)-f(x_0,y_0)}{i{h}_2}\\ & =\frac{1}{i}\frac{\partial f}{\partial y}(z_0).\end{array}$$

Therefore, if $f$ is holomorphic, we have

$$\begin{array}{r}\frac{\partial f}{\partial x}=\frac{1}{i}\frac{\partial f}{\partial y}.\end{array}$$

Writing $f=u+iv$ and separating real and imaginary parts, we find that the partials of $u$ and $v$ exist, and they satisfy what are called the Cauchy-Riemann equations:

$$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\text{and}\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}.$$E2

Definition 370.4(Definition).

$$\begin{array}{r}\frac{\partial }{\partial z}:=\frac{1}{2}\left(\frac{\partial }{\partial x}+\frac{1}{i}\frac{\partial }{\partial y}\right)\text{and}\frac{\partial }{\partial \overline{z}}:=\frac{1}{2}\left(\frac{\partial }{\partial x}-\frac{1}{i}\frac{\partial }{\partial y}\right).\end{array}$$

Proposition 370.5(Proposition).

If $f=u+iv$ is holomorphic at $z_0$, then

$$\begin{array}{r}\frac{\partial f}{\partial \overline{z}}(z_0)=0\text{and}{f}^{\prime }(z_0)=\frac{\partial f}{\partial z}(z_0)=2\frac{\partial u}{\partial z}(z_0).\end{array}$$

Also, the real-variate function $F(x,y)=(u(x,y),v(x,y))$ is differentiable, and

$$\begin{array}{r}\det F^{\prime }(x_0,y_0)=|f^{\prime }(z_0){|}^2.\end{array}$$

Proof.

The first two results are clear. Let $H=(h_1,h_2)$ and $h=h_1+i{h}_2$. Using (E2),

$$\begin{array}{r}{F}^{\prime }(x_0,y_0)(H)=\left(\frac{\partial u}{\partial x}-i\frac{\partial u}{\partial y}\right)(h_1+i{h}_2)=f^{\prime }(z_0)h,\end{array}$$

where a complex number has been identified with the pair of real and imaginary parts. Using (E1),

$$\begin{array}{rl}& f(z_0+h)=f(z_0)+f^{\prime }(z_0)h+h\psi (h)\\ & ⟹F(x_0+h_1,x_0+h_2)=F(x_0,y_0)+F^{\prime }(x_0,y_0)(h_1,h_2)+|h|\underset{\to 0\text{ as }h\to 0}{\underbrace{\left(\frac{h_1{\psi }_1(h)-h_2{\psi }_2(h)}{|h|},\frac{h_1{\psi }_2(h)+h_2{\psi }_1(h)}{|h|}\right)}}.\end{array}$$

Thus, $F$ is differentiable. The last result is a routine application of (E2).□

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Remark 370.6(Remark).

In Prp 5, $f$ being holomorphic at $z_0$ guarantees that the total derivative of $F$ exists at $t_0$, but does not imply that the partials are continuous (so Thm 188.3 cannot be used in the proof). However, $f$ being holomorphic in a neighborhood of $z_0$ does imply the continuity of partials.

What follows is an attempt at a “converse” of Prp 5 (It is actually an iff characterization, but we do not have the tools to prove the reverse implication yet).

Theorem 370.7(Theorem).

Suppose $f=u+iv$ is a complex-valued funciton defined on an open set $\Omega$. If $u,v\in C^1$and satisfy the Cauchy-Riemann equations on $\Omega$, then $f$ is holomorphic on $\Omega$ and $f^{\prime }(z)=\frac{\partial f}{\partial z}$.

Power series

Recall Thm 153.5 and Rmk 153.6.

The complex analog of Thm 154.2:

Theorem 370.8(Stein & Shakarchi (2003) 1.2.6).

The power series $f(z)={\sum }_{n=0}^{\infty }{a}_n{z}^n$ defines a holomorphic function in its disc of convergence. The derivative of $f$ is also a power series obtained by differentiating term by term the series for $f$. Moreover, $f^{\prime }$ has the same radius of convergence as $f$.

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This tells us that an analytic function on $\Omega$ is also holomorphic on $\Omega$. We will see (Thm 377.8) that the converse is true, too.

Integration along curves

Definition 370.9(Definition).

Given a smooth curve $\gamma$ in $\mathbb{C}$ parameterized by $z:[a,b]\to \mathbb{C}$, and $f$ a continuous function on $\gamma$, we define the integral of $f$ along $\gamma$ by

$${\int }_{\gamma }f(z)dz:={\int }_a^{b}f(z(t))z^{\prime }(t)dt.$$

The length of $\gamma$ is defined to be

$${\int }_a^b|z^{\prime }(t)|dt.$$

It is easily shown that these definitions are independent of the parameterization $z$.

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Proposition 370.10(Proposition).

Let $f$ be continuous and $\gamma$ be piecewise smooth.

$$\begin{array}{rl}1.& {\int }_{\gamma }(\alpha f(z)+\beta g(z))dz=\alpha {\int }_{\gamma }f(z)dz+\beta {\int }_{\gamma }g(z)dz.\\ & \\ 2.& {\int }_{\gamma }f(z)dz=-{\int }_{{\gamma }^{-}}f(z)dz.\\ & \\ 3.& \left|{\int }_{\gamma }f(z)dz\right|⩽\underset{z\in \gamma }{\sup }|f(z)|\cdot \text{length}(\gamma ).\end{array}$$

Lemma 370.11(Lemma).

Let $F$ be holomorphic in $\Omega$, and $z:[a,b]\to \mathbb{C}$ be smooth. Then,

$$\frac{d}{dt}F(z(t))=F^{\prime }(z(t))z^{\prime }(t).$$

Proof.

When $F=u+iv$ is seen as a function form ${\mathbb{R}}^2$ to ${\mathbb{R}}^2$ and $z=z_1+i{z}_2$ as a function from $[a,b]$ to ${\mathbb{R}}^2$, we can use the multivariable chain rule to write

$$\begin{array}{rl}\frac{d}{dt}F(z(t))& =\left[\begin{array}{cc}\frac{\partial u}{\partial x}(z(t))& \frac{\partial u}{\partial y}(z(t))\\ \frac{\partial v}{\partial x}(z(t))& \frac{\partial v}{\partial y}(z(t))\end{array}\right]\left[\begin{array}{c}{z}_1^{\prime }(t)\\ z_2^{\prime }(t)\end{array}\right]\end{array}$$

Using (E2), we can see that the RHS above is equal to

$$\left(\frac{\partial u}{\partial x}(z(t))+\frac{1}{i}\frac{\partial u}{\partial y}(z(t))\right)(z_1^{\prime }(t)+i{z}_2^{\prime }(t)),$$

which is equal to $F^{\prime }(z(t))z^{\prime }(t)$ by Prp 5.□

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Theorem 370.12(Theorem).

If a continuous function $f$ has a primitive $F$ in $\Omega$, and $\gamma$ is a curve in $\Omega$ that begins at $w_1$ and ends at $w_2$, then

$${\int }_{\gamma }f(z)dz=F(w_1)-F(w_2).$$

Proof.

If $\gamma$ is smooth, the proof is a simple application of Lem 11 and the fundamental theorem of calculus. Indeed, if $z(t):[a,b]\to \mathbb{C}$ is a parameterization for $\gamma$, then $z(a)=w_1$ and $z(b)=w_2$, and we have

$$\begin{array}{rl}{\int }_{\gamma }f(z)dz& ={\int }_a^{b}f(z(t))z^{\prime }(t)dt\\ & ={\int }_a^b{F}^{\prime }(z(t))z^{\prime }(t)dt\\ & ={\int }_a^b\frac{d}{dt}F(z(t))dt\\ & =F(z(b))-F(z(a)).\end{array}$$

If $\gamma$ is piecewise smooth, then we get a telescoping sum, with the end result unchanged.□

Explore connections to Thm 323.13.

Corollary 370.13(Corollary).

If $\gamma$ is a closed curve in an open set $\Omega$, and $f$ is continuous and has a primitive in $\Omega$, then

$${\int }_{\gamma }f(z)dz=0.$$

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Corollary 370.14(Corollary).

If $f$ is holomorphic in a (connected) region $\Omega$ and $f^{\prime }=0$, then $f$ is constant.

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References

Stein, E. M., & Shakarchi, R. (2003). Complex Analysis. Princeton University Press.