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I'm wondering how to do this question. I believe the question wants to decompose the following statements together from the text book to find the Mobius function
Thank you in advance (:
2026-03-28 16:57:05.1774717025
Demonstrate the statement in section 3.I.3 on the decomposition of the Möbius transformations!!!
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They basically gave you the solution. Let $f, g, h, i$ are the functions in that order.
Then notice what $i(h(g(f(z))))$ is:
$i(h(g(f(z)))) = i(h(g(z+d/c))) = i(h(\frac{1}{z + \frac{d}{c}})) = i(h(\frac{c}{cz+d})) = i(-\frac{ad-bc}{c^2}\frac{c}{cz+d}) = i(-\frac{ad - bc}{c(cz+d)}) = \frac{a}{c} - \frac{ad-bc}{c(cz+d)} = \frac{az+b}{cz+d}$.
So $M = ihgf(z)$ as desired.