Complex Functions Review

# Complex Functions Review

We will now review some of the recent material regarding complex functions. This material is summarized in the following table:

Function | Notation | Definition | Important Properties |
---|---|---|---|

The Complex Exponential Function |
$e^z$ | If $z = x + yi \in \mathbb{C}$ then $e^z = e^x (\cos y + i \sin y)$ | 1) $e^z \neq 0$.2) $e^z = 1$ if and only if $z = 2k\pi i$ for some integer $k \in \mathbb{Z}$.3) $e^{z + w} = e^{z} \cdot e^{w}$.4) $\mid e^z \mid = e^x$.5) $e^0 = 1$, $e^{\frac{\pi}{2}i} = i$, $e^{\pi i} = -1$, and $e^{\frac{3\pi}{2}i} = -i$ |

The Complex Cosine and Sine Functions |
$\cos z$, $\sin z$ | $\displaystyle{\cos z = \frac{e^{iz} + e^{-iz}}{2}}$ and $\displaystyle{\sin z = \frac{e^{iz} - e^{-iz}}{2i}}$. | 1) $\sin^2 z + \cos^2 z = 1$.2) $\sin (z + w) = \sin z \cos w + \cos z \sin w$.3) $\cos (z + w) = \cos z \cos w - \sin z \sin w$.4) $\sin (-z) = - \sin (z)$.5) $\cos(-z) = \cos (z)$. |

The Complex Natural Logarithm Function |
$\log z$ (or $\ln z$) | $\log z = \log \mid z \mid + i \arg(z)$ for a choice of branch for the logarithm function. | a) $\log (zw) = \log (z) + \log (w)$ up to an integer multiple of $2\pi i$. |

Complex Power Functions |
$a^b$ | For $a \neq 0$, $a^b = e^{b \log a}$ for a choice of branch for the logarithm function. | a) For a fixed branch of the logarithm function, $a^{b + c} = a^b \cdot a^c$.b) If $\log (ab) = \log (a) + \log(b)$ then $(ab)^c = a^c \cdot b^c$. |

Some example problems regarding this material can be found on the following pages: