Analytic Functions

A function $f$ is analytic at $z_0$ iff it is differentiable at every point of some neighborhood of $z_0$. $f$ is analytic on an open set $G$ iff it is analytic at every point of $G$.

If $f$ is analytic on an open set $G$, $f'$ is continuous on $G$.

Let $u$ and $v$ be real-valued functions with continuous first-order partial derivatives on an open set $U \subseteq \mathbb{C}$, and define $f(z) = u(x,y)+iv(x,y)$, $z = x+iy$. Then $f$ is analytic in $U$ iff $u$ and $v$ satisfy the Cauchy-Riemann equations throughout $U$.

Properties of Analytic Functions

Let $f$ be analytic on a domain $D$.

• $f'$ is analytic on $D$. So $f$ is infinitely differentiable on $D$.
• $\operatorname{Re} f$ and $\operatorname{Im} f$ are harmonic on $D$.
• If $f'(z) = 0$ throughout $D$, then $f$ is constant on $D$.
• If $|f|$ is constant on $D$, then $f$ is constant on $D$.
• If $\operatorname{Re} f$ is constant on $D$, or $\operatorname{Im} f$ is constant on $D$, then $f$ is constant on $D$.
• If $f$ is not constant on $D$, its zeros are isolated.

Algebra of Analytic Functions

Let $f$ and $g$ be analytic in a domain $D$.

• $f+g$, $f-g$, and $fg$ are analytic in $D$.
• $f/g$ is analytic in $D$ provided $g$ does not vanish at any point of $D$.
• If $f$ is analytic in $D$ and $g$ is analytic on a domain containing the image of $D$ under $f$, then $g \circ f$ is analytic in $D$, and

Entire Function

$f$ is an entire function iff $f$ is analytic at every point of $\mathbb{C}$.

Examples:

• Every polynomial
• $e^z$
• $f(z) = \cosh z$

Singular Point

$z_0$ is a singular point (singularity) of $f$ iff $f$ fails to be analytic at $z_0$ and $f$ is analytic at some point in every neighborhood of $z_0$.

Isolated Singularity

$z_0$ is an isolated singularity of $f$ iff $f$ is analytic on some punctured disc $D_r(z_0) \setminus \{z_0\}$ but not analytic at $z_0$. Equivalently, $z_0$ is a singular point and some punctured disc around $z_0$ contains no other singular point of $f$.

Examples:

• $z = 0$ is an isolated singularity of $f(z) = 1/z$.
• $z = i$ and $z = -i$ are isolated singularities of $f(z) = 1/(z^2+1)$.
• Every $z = n\pi$, $n \in \mathbb{Z}$, is an isolated singularity of $f(z) = \cot z$.

Non-Isolated Singularity

$z_0$ is a non-isolated singularity of $f$ iff $z_0$ is a singular point and every punctured disc $D_r(z_0) \setminus \{z_0\}$ contains at least one other singular point of $f$.

This happens when singular points accumulate at $z_0$, or when $f$ is singular along a whole curve or region through $z_0$.

Examples:

• $z = 0$ is a non-isolated singularity of $f(z) = 1/\sin(1/z)$. The singular points $z = 1/(n\pi)$, $n \in \mathbb{Z} \setminus \{0\}$, accumulate at $0$, so no punctured disc around $0$ is free of them.
• $z = 0$ is a non-isolated singularity of $f(z) = \tan(1/z)$, with the singular points $z = 2/((2n+1)\pi)$ accumulating at $0$.
• $z = 0$ is a non-isolated singularity of $f(z) = 1/(e^{1/z}-1)$, with the singular points $z = 1/(2\pi i n)$ accumulating at $0$.
• Every point of the non-positive real axis is a non-isolated singularity of the principal logarithm $\operatorname{Log} z$, since $\operatorname{Log}$ is discontinuous at every point of that ray.