Analytic Functions

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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 $z$ $z$ (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

Written by

Sahithyan K.

September 29, 2026 1 min read

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