Analytic Functions

A function $f$ is said to be analytic at $z_0$ iff it is differentiable throughout a neighbourhood of $z_0$. Aka. regular functions, holomorphic functions and monogenic functions.

Examples

• Polynomial functions of $z$ (analytic everywhere)
• Functions with a converging Taylor series for all $z$ (analytic everywhere)
  • $\sin z$
  • $\cos z$
  • $e^z$

Non-examples

Function	Note
$\lvert z \rvert ^2$	Differentiable only at $z=0$.
$\overline{z}$	Nowhere differentiable. Derivative taken on the real and imaginary axes are different.
$\text{Re}(z)$	Similar to above.
$z+\overline{z}$	Similar to above.
$\text{Im}(z)$	Similar to above.
$z-\overline{z}$	Similar to above.

Analytic implies differentiable

$$f \text{ is analytic at } z_0 \implies f \text{ is differentiable at } z_0$$