$f(z) = z^2 = (x+iy)^2 = x^2 - y^2 + 2ixy$. Here:

• $u(x, y) = x^2 - y^2$
• $v(x, y) = 2xy$

The function $f$ maps the complex plane to itself, squaring the modulus and doubling the argument.

$f(z) = e^z = e^{x + iy} = e^x(\cos y + i\sin y)$. Thus:

• $u(x, y) = e^x \cos y$
• $v(x, y) = e^x \sin y$

The exponential function is periodic in the imaginary direction with period $2\pi i$.

Every polynomial $p(z) = a_0 + a_1 z + \cdots + a_n z^n$ is continuous on $\mathbb{C}$. This follows from continuity of addition and multiplication: if $f$ and $g$ are continuous, so are $f+g$ and $fg$.

$f(z) = \bar{z} = x - iy$ is continuous everywhere. However, as we will see, it is not analytic (differentiable in the complex sense) anywhere.

Consider $f(z) = \frac{z^2 - 1}{z - 1}$ for $z \neq 1$. Then

$\lim_{z \to 1} f(z) = \lim_{z \to 1} \frac{(z-1)(z+1)}{z-1} = \lim_{z \to 1} (z+1) = 2.$

This holds regardless of how $z$ approaches $1$ in the complex plane.

Let $f(z) = \frac{\bar{z}}{z}$ for $z \neq 0$. If $z = re^{i\theta}$, then $\bar{z} = re^{-i\theta}$, so

$f(z) = \frac{re^{-i\theta}}{re^{i\theta}} = e^{-2i\theta}.$

As $z \to 0$ along different rays (different values of $\theta$), $f(z)$ takes different values (all points on the unit circle). Thus $\lim_{z \to 0} f(z)$ does not exist.

$p(z) = a_0 + a_1 z + a_2 z^2 + \cdots + a_n z^n$

Polynomials are continuous and differentiable everywhere. They are the simplest examples of entire functions (functions analytic on all of $\mathbb{C}$).

$f(z) = \frac{p(z)}{q(z)}$

where $p$ and $q$ are polynomials. The function $f$ is defined and analytic on $\mathbb{C} \setminus \{\text{zeros of } q\}$.

$e^z = e^{x+iy} = e^x(\cos y + i\sin y)$

Properties:

• $e^{z+w} = e^z e^w$
• $|e^z| = e^x$ and $\arg(e^z) = y$ (mod $2\pi$)
• $e^z$ is periodic with period $2\pi i$: $e^{z + 2\pi i} = e^z$
• $e^z$ is entire and never zero

$\cos z = \frac{e^{iz} + e^{-iz}}{2}, \quad \sin z = \frac{e^{iz} - e^{-iz}}{2i}$

Unlike in real analysis, $\sin z$ and $\cos z$ are unbounded on $\mathbb{C}$. For instance, $\cosh(y) = |\sin(iy)| \to \infty$ as $y \to \infty$.

The complex logarithm is the inverse of $e^z$. If $z = re^{i\theta}$, then

$\log z = \ln r + i\theta$

where $\theta$ is determined up to integer multiples of $2\pi$. To make $\log z$ single-valued, we choose a branch cut (commonly the negative real axis) and restrict the argument to $(-\pi, \pi]$.

$f(z) = z^2$ doubles the argument and squares the modulus:

• The right half-plane $\{\text{Re}(z) > 0\}$ maps onto $\mathbb{C} \setminus \mathbb{R}_{\leq 0}$ (the complex plane minus the non-positive real axis).
• The unit circle $|z| = 1$ maps to itself, but each point is visited twice.
• The ray $\arg(z) = \pi/4$ maps to the ray $\arg(z) = \pi/2$ (the positive imaginary axis).

A Möbius transformation is a function of the form

$f(z) = \frac{az + b}{cz + d}$

where $a, b, c, d \in \mathbb{C}$ and $ad - bc \neq 0$. These are the only conformal automorphisms of the Riemann sphere $\mathbb{C} \cup \{\infty\}$. They map circles to circles (including lines as circles of infinite radius).