Taylor series

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 $x_{0}$ is

$f(x)=f(x_{0})+(x-x_{0}){\frac {df(x_{0})}{dx}}+{\frac {(x-x_{0})^{2}}{2!}}{\frac {d^{2}f(x_{0})}{dx^{2}}}+\ldots +{\frac {(x-x_{0})^{N}}{N!}}{\frac {d^{N}f(x_{0})}{dx^{N}}}$

where each of the derivatives is to be evaluated at $x=x_{0}$. If as $N\rightarrow \infty$ the series converges, then it is exact. Otherwise, it can be used as an approximation. Often, Taylor series are performed around $x_{0}=0$, in which case they are sometimes also known as a Maclaurin series.

Examples of common Taylor series

$e^{x}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\ldots$

$\sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-\ldots$

$\cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\ldots$

Extensions of the Exponential Function

Consider the exponential of imaginary number $yi$,

$e^{yi}=1+yi+{\frac {(yi)^{2}}{2!}}+{\frac {(yi)^{3}}{3!}}+{\frac {(yi)^{4}}{4!}}+{\frac {(yi)^{5}}{5!}}+\ldots$

$=1+yi-{\frac {y^{2}}{2!}}-i{\frac {y^{3}}{3!}}+{\frac {y^{4}}{4!}}+i{\frac {y^{5}}{5!}}-\ldots$

$=(1-{\frac {y^{2}}{2!}}+{\frac {y^{4}}{4!}}-\ldots )+i(y-{\frac {y^{3}}{3!}}+{\frac {y^{5}}{5!}}-\ldots )$

$=\cos y+i\sin y$

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

$e^{x+yi}=e^{x}(\cos y+i\sin y)$.