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33 lines
1.8 KiB
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33 lines
1.8 KiB
Markdown
---
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lastmod: "2021-09-12T22:47:44.0000000+01:00"
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author: patrick
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categories:
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- stack-exchange
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comments: true
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date: "2016-12-31T00:00:00Z"
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title: What does Mathematica mean by ComplexInfinity?
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summary: Answering the question, "Why does WolframAlpha say that a quantity is ComplexInfinity?".
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---
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*This is my answer to the same [question posed on the Mathematics Stack Exchange](https://math.stackexchange.com/q/2078754/259262). It is therefore licenced under [CC-BY-SA 3.0](https://creativecommons.org/licenses/by-sa/3.0/).*
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# Question
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When entered into [Wolfram|Alpha](https://www.wolframalpha.com/), \\(\infty^{\infty}\\) results in "complex infinity".
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Why?
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# Answer
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WA's `ComplexInfinity` is the same as Mathematica's: it represents a complex "number" which has infinite magnitude but unknown or nonexistent phase.
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One can use `DirectedInfinity` to specify the phase of an infinite quantity, if it approaches infinity in a certain direction.
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The standard `Infinity` is the special case of phase `0`.
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Note that `Infinity` is different from `Indeterminate` (which would be the output of e.g. `0/0`).
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Some elucidating examples:
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* `0/0` returns `Indeterminate`, since (for instance) the limit may be approached as \\(\frac{1/n}{1/n}\\) or \\(\frac{2/n}{2/n}\\), resulting in two different real numbers.
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* `1/0` returns `ComplexInfinity`, since (for instance) the limit may be approached as \\(\frac{1}{-1/n}\\) or as \\(\frac{1}{1/n}\\), but every possible way of approaching the limit gives an infinite answer.
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* `Abs[1/0]` returns `Infinity`, since the limit is guaranteed to be infinite and approached along the real line in the positive direction.
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In your particular example, you get `ComplexInfinity` because the infinite limit may be approached as (e.g.) \\(n^n\\) or as \\(n^{n+i}\\).
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