Table of Common Complex Taylor Series

# 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 $f$.

Function | Taylor Series | Valid on |
---|---|---|

$\displaystyle{\frac{1}{1 - z}}$ | $\displaystyle{\sum_{n=0}^{\infty} z^n = 1 + z + z^2 + ...}$ | $\mid z \mid < 1$ |

$e^z$ | $\displaystyle{\sum_{n=0}^{\infty} \frac{1}{n!} z^n = 1 + z + \frac{z^2}{2!} + ...}$ | $\mathbb{C}$ |

$\sin z$ | $\displaystyle{\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n + 1)!} z^{2n + 1} = z - \frac{z^3}{3!} + \frac{z^5}{5!} - ...}$ | $\mathbb{C}$ |

$\cos z$ | $\displaystyle{\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} z^{2n} = 1 - \frac{z^2}{2!} + \frac{z^4}{4!} - ...}$ | $\mathbb{C}$ |

$\log(1 + z)$ | $\displaystyle{\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} z^n = z - \frac{z^2}{2} + \frac{z^3}{3} - ...}$ | $\mid z \mid < 1$ |