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Can somebody please expand upon the specific meaning of these two similar mathematical ideas and provide usage examples of each one? Thank you!

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Complex infinity is where Santa lives on the Riemann sphere. –  Random Variable Jun 14 '13 at 20:55
    
Apart from infinity and undefined, there is yet another related notion: indeterminate. –  Hagen von Eitzen Jun 14 '13 at 21:39
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up vote 10 down vote accepted

I don't think they are similar.

"Undefined" is something that one predicates of expressions. It means they don't refer to any mathematical object.

"Complex infinity", on the other hand, is itself a mathematical object. It's a point in the space $\mathbb C\cup\{\infty\}$, and there is such a thing as an open neighborhood of that point, as with any other point. One can say of a rational function, for example $(2x-3)/(x+5)$, that its value at $\infty$ is $2$ and its value at $-5$ is $\infty$.

To say that $f(z)$ approaches $\infty$ as $z$ approaches $a$, means that for any $R>0$, there exists $\delta>0$ such that $|f(z)|>R$ whenever $0<|z-a|<\delta$. That's one of the major ways in which $\infty$ comes up. An expression like $\lim\limits_{z\to\infty}\cos z$ is undefined; the limit doesn't exist.

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Thank you for the clarification. Your help is much appreciated. –  David Schilpp Jun 14 '13 at 20:57
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C∪{∞} has infinity in its definition. Infinity is undefined therefore complex infinity is undefined. If infinity were definable then it would be finite.

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