|This article/section deals with mathematical concepts appropriate for a student in late high school or early university.|
The Taylor series of a function is useful for approximating a mathematical function near to some particular point. For a function f(x), the Taylor series about the point x0 is
where each of the derivatives is to be evaluated at x = x0. If as the series converges, then it is exact. Otherwise, it can be used as an approximation. Often, Taylor series are performed around x0 = 0, 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 yi,
- = cosy + isiny
By the power laws then all complex numbers have an exponential,
ex + yi = ex(cosy + isiny).