Table of Common Complex Taylor Series
We will now provide a table of some of the Taylor series for some of the most common complex functions. All of the Taylor series below are centered at $z_0 = 0$ and we also provide the largest open domain for which the corresponding Taylor series converges to $f$.
$\displaystyle{\frac{1}{1 - z}}$ $\displaystyle{\sum_{n=0}^{\infty} z^n = 1 + z + z^2 + ...}$ $\mid z \mid < 1$
$e^z$ $\displaystyle{\sum_{n=0}^{\infty} \frac{1}{n!} z^n = 1 + z + \frac{z^2}{2!} + ...}$ $\mathbb{C}$
$\sin z$ $\displaystyle{\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n + 1)!} z^{2n + 1} = z - \frac{z^3}{3!} + \frac{z^5}{5!} - ...}$ $\mathbb{C}$
$\cos z$ $\displaystyle{\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} z^{2n} = 1 - \frac{z^2}{2!} + \frac{z^4}{4!} - ...}$ $\mathbb{C}$
$\log(1 + z)$ $\displaystyle{\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} z^n = z - \frac{z^2}{2} + \frac{z^3}{3} - ...}$ $\mid z \mid < 1$