8.3: Aside- The Meaning of "Infinity" for Complex Numbers
When talking about \(z = \infty\) , we are referring to something called complex infinity , which can be regarded as a complex number with infinite magnitude and undefined argument.
The fact that the argument is undefined may seem strange, but actually we already know of another complex number with this feature: \(z = 0\) has zero magnitude and undefined argument. These two special complex numbers are the reciprocals of each other: \(1/\infty = 0\) and \(1/0 = \infty\) .
The complex \(\infty\) behaves differently from the familiar concept of infinity associated with real numbers. For real numbers, positive infinity ( \(+\infty\) ) is distinct from negative infinity ( \(-\infty\) ). But this doesn’t hold for complex numbers, since complex numbers occupy a two-dimensional plane rather than a line. Thus, for complex numbers it does not make sense to define “positive infinity” and “negative infinity” as distinct entities. Instead, we work with a single complex \(\infty\) .
From this discussion, we can see why \(z^p\) is has a branch point at \(z = \infty\) . For any finite and nonzero \(z\) , we can write \(z = re^{i\theta}\) , where \(r\) is a positive number. The \(z^p\) operation then yields a set of complex numbers of the form \(r^p \, e^{ip\theta}\,\times\, \{\text{root of unity}\}\) . For each number in this set, the magnitude goes to infinity as \(r \rightarrow \infty\) . In this limit, the argument (i.e., the choice of root of unity) becomes irrelevant, and the result is simply \(\infty\) .
By similar reasoning, one can prove that \(\ln(z)\) has branch points at \(z = 0\) and \(z = \infty\) . This is left as an exercise.