08 Elementary, Exponential Functions
\(e^z\)¶
\[ \begin{aligned} |e^z| &= e^x \\ e^{z + (2k \pi) i} &= e^z \end{aligned} \]
\(\log z\)¶
\[ \begin{aligned} \log z &= \ln r + i \theta \\ &= \ln r + i[\text{Arg}(z) + 2k\pi] \\ \text{Log} z &= \ln r + i[\text{Arg}(z)] \end{aligned} \]
Complex Exponents¶
\[ \begin{aligned} z &= e^{\log z} \\ z_c &= e^{c \log z} \\ PV(z^c) &= e^{c \text{ Log} z} \end{aligned} \]