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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. – Sangchul Lee Mar 10 '12 at 8:04
So in terms of which are fractions defined? – Pteromys Mar 10 '12 at 9:58
  1. "Is it correct that you plug $\infty$ in" -- no it's not. The arithmetic operations are defined on $\mathbb C$ only. Some expressions extend to $\infty$ by continuity. For example, $|z|=|z-2|$ can be rewritten as $|z|/|z-2|=1$. The function $|z|/|z-2|$ extends continuously to $\infty$ by letting it be $1$, which is $\lim_{z\to\infty } |z|/|z-2| = 1$. In this sense, $\infty$ satisfies the equation.

  2. Yes, $f(\infty)$ is evaluated as a limit, which is $a/c$.

  3. I do not consider $1/0 =\infty$ an algebraic rule. It's a stenographic way to write down $\lim_{z\to 0} 1/z = \infty$. But one can perform such "algebraic" manipulation and arrive at correct results, because they are backed by basic limit theorems.

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