Taylor series

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The Taylor series of a function is useful for approximating a mathematical function near to some particular point. For a function , the Taylor series about the point  is

where each of the derivatives is to be evaluated at . If as  the series converges, then it is exact. Otherwise, it can be used as an approximation. Often, Taylor series are performed around , in which case they are sometimes also known as a Maclaurin series.

## Examples of common Taylor series[edit]

### Extensions of the Exponential Function[edit]

Consider the exponential of imaginary number ,

By the power laws then all complex numbers have an exponential,