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The fixed points of a Möbius transformation satisfy the requirement
z 
= 

(34) 
which translates into the quadratic equation
cz^{2} + (da) z  b 
= 
0 
(35) 
with roots
In case c=0 the quadratic term is absent and the single fixed point would be b/(da) unless (da) were zero. That alternative would leave z without constraint and require b to vanish as well as c. But then all points would be fixed and the transformation would be the identity. And if b did not vanish, the equation would read z=z+b which could not be satisfied unless z were ,
which might as well be considered admissible.
Another case of a single root occurs when the radicand is zero leaving (ad)/2c as the only fixed point. Otherewise there are always two distinct fixed points
Next: hyperbolic, parabolic, elliptic transformations
Up: Functions of a complex
Previous: Möbius transformations representable as
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