Functions of a Complex Variable

Let $f(z)$ be a complex function of a complex variable $z$, i.e., for every $z$, $f(z)$ is a complex number. Note that a real number is also considered as a complex number and, hence, $f(z)$ could have a zero imaginary part. Examples of functions include $f(z) = |z|, f(z) = \mbox{arg}(z), f(z) = z^n, f(z) = \exp(z)$, etc.

The exponential functions can be interpreted using Euler's identity as follows.

$f(z) = \exp(z) = e^xe^{jy} = e^x\cos y + je^x\sin y$

The real part of $f(z)$ is plotted as a function of the real part of $z$, namely $x$ for the case $x<0$ in Figure.

The logarithm of a complex number $\ln z$ can be also interpreted using Euler's identity as

$\ln(z) = \ln\left(r e^{j (\theta + 2 k \pi) } \right) = \ln r + j (\theta + 2 k \pi)$

It can be seen from the above expression that $\ln z$is not a function of $z$. However, if we set $k=0$ in the above expression, then we get what is called the principal value of $\ln z$, denoted by Ln $z$, which is a function.

Similarly, it is important to realize that for any integer $n$, the $n$th power of a complex number is a function of the complex number, i.e., for every complex number $z$, there is only one complex number $z^n$. However, for an integer $n$, the $n$th root of a complex number is not uniquely defined and hence, is not a function. Often, one may take the root corresponding to $k=0$ in the previous section as the default root and hence the principal value. Then, the principal value becomes a function. This is similar to square roots of positive real numbers being defined as the positive numbers. There are interesting examples where careless use of just the principal value as the $n$th root can lead to fallacious arguments.