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  1. Consider a line on the Riemann sphere $\{z\in\hat{\mathbb C}\mid |z| = |z-2|\}$, and its image by a map $z\mapsto 1/z$, namely, $\{z\mid |1/z| = |1/z-2|\}$. Suppose you want to know whether 0 is included in the latter (which is in fact a circle). Is it correct that you plug $\infty$ in and get $|\infty| = |\infty - 2|$ by using the algebraic rule $1/0 = \infty$, and then have $+\infty=+\infty$ and answer the question yes?
  2. Consider a Moebius transformation $f:z\mapsto \frac{az+b}{cz+d}$, where $c \ne 0$. To get its value at $\infty$, you cannot use an algebraic rule like the one above, since you get $\infty/\infty$. Does $f(\infty)$ means $\lim_{z\rightarrow\infty}\frac{az+b}{cz+d}$?
  3. In general, to evaluate a fraction on the extended complex plane, which should be used, algebraic rules as in 1., or analytic methods as in 2.?
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    1. Not rigorous, but intuitively correct. You can use that kind of heuristic calculation to determine the answer, and then justify it. 2. Sure. Any Moebious transform is holomorphic mapping from $\hat{\mathbb{C}}$ onto itself, thus in particular it is continuous. Thus you can take limit $z \to \infty$ to calculate the value of $f(\infty)$. 3. Both is applicable, if that yields a 'calculable' result. For example, in 2, if we write $$f(z) = \frac{a + b/z}{c + d/z},$$ algebraic rule $1/\infty = 0$ also gives the correct answer. For rational functions, both methods are equivalent.2012-03-10
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    So in terms of which are fractions defined?2012-03-10

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