8.3: Aside- The Meaning of "Infinity" for Complex Numbers
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When talking about z=∞, 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/∞=0 and 1/0=∞.
The complex ∞ behaves differently from the familiar concept of infinity associated with real numbers. For real numbers, positive infinity (+∞) is distinct from negative infinity (−∞). 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 ∞.
From this discussion, we can see why zp is has a branch point at z=∞. For any finite and nonzero z, we can write z=reiθ, where r is a positive number. The zp operation then yields a set of complex numbers of the form rpeipθ×{root of unity}. For each number in this set, the magnitude goes to infinity as r→∞. In this limit, the argument (i.e., the choice of root of unity) becomes irrelevant, and the result is simply ∞.
By similar reasoning, one can prove that ln(z) has branch points at z=0 and z=∞. This is left as an exercise.