2.1 Functions

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1. Functions
2. Real and Imaginary Parts
3. Visualizing Functions
4. Continuity

Functions

Our main objects of study in this course are complex-valued functions $f$ where the inputs come from a domain $D\subset \mathbb{C}$. We will mainly be interested in differentiating and integrating such functions.

Figure 1: A function defined on a domain $D$ maps each complex number $z\in D$ to a complex number $f(z)\in \mathbb{C}$.

Example

Often, functions are defined by formulae.

1. $f(z)=z^2$ defines a function $f:\mathbb{C}\to \mathbb{C}$.
2. $f(z)={\displaystyle \frac{1}{z}}$ defines a function $f:\mathbb{C}\setminus \{0\}\to \mathbb{C}$.
3. $f(z)={\displaystyle \frac{z}{1+z^2}}$ defines a function $f:\mathbb{C}\setminus \{i,-i\}\to \mathbb{C}$.

More complicated functions such as trigonometric and exponential functions will be formally defined by power series later on in the course. A complex version of the natural logarithm will have to wait until we have discussed integration!

Real and Imaginary Parts

Given a function $f:D\to \mathbb{C}$ we can think of $D$ as a subset of ${\mathbb{R}}^2$ and of $\mathbb{C}$ as the whole plane ${\mathbb{R}}^2$. Writing $z=x+iy$ and $$f(z)=f(x+iy)=u(x,y)+iv(x,y)$$ lets us think of $f$ as being made up of two real-valued functions $u,v:D\to \mathbb{R}$.

Example

What are the real and imaginary parts of $f(z)=z^2$? Writing $z=x+iy$ we have $$f(x+iy)=(x+iy{)}^2=x^2-y^2+2xyi$$ so $u(x,y)=x^2-y^2$ and $v(x,y)=2xy$.

Visualizing Functions

We cannot easily draw the graph of a function $f:D\to \mathbb{C}$. We conclude this section with a brief discussion of two alternative methods for visualizing such a function.

The first method is to plot the level curves of the real and imaginary parts $u$ and $v$ of $f:D\to \mathbb{C}$. Recall that the level curves of a function $h:D\to \mathbb{R}$ are the curves $h(x,y)=k$ for different values of $k$. We plot level curves in the domain of the function. If the input $z$ is nudged along a level curve of $u$ the real part of the output will not change. Similarly, if the input is nudged along a level curve of $v$ then the imaginary part of the output will not change.

Figure 2: The level curves of the real and imaginary parts $u(x,y)=x^2-y^2$ and $v(x,y)=2xy$ of $f(z)=z^2$ respectively.

The second method is to think of $f$ as a vector field on $D$. Indeed, for every $z\in D$ the output $f(z)$ can be thought of as a vector in ${\mathbb{R}}^2$. Placing the tail of this vector at the point $z\in D$ lets us think of $f$ as a vector field on $D$.

Figure 3: The function $f(z)=z^2$ as the vector field $⟨x^2-y^2,2xy⟩$ on ${\mathbb{R}}^2$.

Continuity

Continuity of functions of a complex variable is much like continuity of functions of a real variable. In this section we define what it means for a function $f:D\to \mathbb{C}$ on a domain to be continuous.

Definition (Continuity of a Function)

Fix a domain $D\subset \mathbb{C}$. A function $f:D\to \mathbb{C}$ is continuous at a point $c\in D$ if $$\underset{z\to c}{lim}f(z)=f(c)$$ holds.

The limit above is formally defined as follows.

Definition (Limits)

Fix a domain $D\subset \mathbb{C}$ and $f:D\to \mathbb{C}$. Fix also $c\in D$. We say that $$\underset{z\to c}{lim}f(z)=\ell$$ or that $f(z)$ tends to $\ell$ as $z$ tends to $c$, or that the limit of $f(z)$ exists as $z\to c$ if, for all $ϵ>0$, there exists $\delta >0$ such that if $z\in D$ and $0<|z-c|<\delta$ then $|f(z)-\ell |<ϵ$.

That is, $f(z)\to \ell$ as $z\to c$ means that if $z$ is very close and not equal to $c$ then $f(z)$ is very close to $\ell$. Note that in this definition we do not need to know the value of $f(c)$.

Example

Let $f:\mathbb{C}\to \mathbb{C}$ be defined by $$f(z)=\{\begin{array}{ll}1& z\ne 0\\ 0& z=0\end{array}$$ so that $$\underset{z\to 0}{lim}f(z)=1$$ but ${\displaystyle \underset{z\to 0}{lim}f(z)\ne f(0)}$. Thus $f$ is not continuous at 0.

Continuous functions of a complex variable obey the same rules as continuous functions of a real variable. Therefore we can use the same strategies for testing and evaluating limits in complex analysis as we used in real analysis.

Lemma (Limit Laws)

Suppose that ${\displaystyle \underset{z\to c}{lim}f(z)=\ell }$ and ${\displaystyle \underset{z\to c}{lim}g(z)=m}$.

1. ${\displaystyle \underset{z\to c}{lim}f(z)+g(z)=\ell +m}$
2. ${\displaystyle \underset{z\to c}{lim}af(z)=a\ell }$ for any $a\in \mathbb{C}$
3. ${\displaystyle \underset{z\to c}{lim}f(z)g(z)=\ell m}$
4. ${\displaystyle \underset{z\to c}{lim}f(z)/g(z)=\ell /m}$ if $m\ne 0$

Lemma (Polynomial Limits)

If $$f(z)=a_n{z}^n+a_{n-1}{z}^{n-1}+\cdots +a_{1}z+a_0$$ then $f$ is continuous at every $c\in \mathbb{C}$.

Lemma (Rational Limits)

If $f,g$ are polynomials and $g(c)\ne 0$ then $$\underset{z\to c}{lim}{\displaystyle \frac{f(z)}{g(z)}}={\displaystyle \frac{f(c)}{g(c)}}$$