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: