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Although there may be some question of whether the implicit or the explicit form of one-to-one and invertible mappings should be called Möbius transformations, that is the name given to this class of transformations. Mappings often form a group. Since the associative law for their composition is a given, the items to be checked are closure, invertibility and the existence of the identity.
To check closure, consider the composite of two maps. First, suppose we have
t |
= |
![$\displaystyle \frac{Aw+B}{Cw+D}$](img54.gif) |
(29) |
w |
= |
![$\displaystyle \frac{az+b}{cz+d}$](img55.gif) |
(30) |
which gives, on substituting,
t |
= |
![$\displaystyle \frac{A\frac{az+b}{cz+d}+B}{C\frac{az+b}{cz+d}+D},$](img56.gif) |
(31) |
and, on simplifying,
t |
= |
![$\displaystyle \frac{(Aa+Bc)z+(Ab+Bd)}{(Ca+Dc)z+(Cb+Dd)},$](img57.gif) |
(32) |
which mimics the matrix product
From this it is apparent that the
unimodular matrices form a representation of the group of Möbius transformations; in particular the identity is represented by the unit matrix,
z |
= |
![$\displaystyle \frac{1z+0}{0z+1},$](img60.gif) |
(33) |
and the inverse transformation by the inverse matrix.
Next: eigenvalues and eigenvectors of
Up: Functions of a complex
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Microcomputadoras
2001-04-05