# Courses in Math - Taylor's Series - Definition

A Taylor's series means a series expansion of an analytic function.

## Taylor's Series - Definition

When function

**f(z)** is analytic in the neighbourhood of a point

**a** embedded in the complex space it can be uniquely represented by the power series

In that way we end up with

**the Taylor series**:

The function is called analytical at point

**a** if it is differentiable everywhere around that point (i. e. at each point within the circle of small enough radius

**r** and centered at the

**a** point).

Sometimes could be useful to have a truncated Taylor series e. g. in

the Taylor Collocation Method. Let's consider the example of

**f(t)** function approximated by a truncated Taylor series:

where

**beta** is a parameter in [0, 1] and

**dt** denotes the time increment

**(t**_{n+1} - t_{n}). The upper

**dot** means the first time derivative.

Last update: April 22, 2020

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