|This article/section deals with mathematical concepts appropriate for late high school or early college.|
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
Extensions of the Exponential Function
Consider the exponential of imaginary number ,
By the power laws then all complex numbers have an exponential,