z = log r + iθ, where log r is the ordinary natural logarithm of a positive real number—the inverse of the real exponential. Why? It is intuitively clear from Fig. 5.7 that such a real logarithm function exists. In Fig. 5.7a we have the graph of r = ex. We just flip the axes over to get the graph of the inverse function x = log r, as in Fig. 5.7b. It is not so surprising that the real part of z = log w is just an ordinary real logarithm. What is somewhat more remarkable4 is that the imaginary part of z is just the angle θ that is the argument of the complex number w. This fact makes explicit
...more
