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Physics LibreTexts

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.


This page titled 8.3: Aside- The Meaning of "Infinity" for Complex Numbers is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Y. D. Chong via source content that was edited to the style and standards of the LibreTexts platform.

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