Call the operation of inversion (w=1/z) N
and the operations of linear equations w=pz+q, for various p and q, L’s (with complex numbers those equations do translation, rotation, enlargement)
Then we want to prove:
ONE: that every combination of N and L’s is a Möbius transformation w=(az+b)/(cz+d), for some a, b, c, d
TWO: that every Möbius transformation z=(az+b)/(cz+d) is some combination of N and L’s
In other words, “Möbius transformations” and “combinations of N and L’s” are the same thing.
Proof of CLAIM ONE
If you repeat N any number of times you get either N or the operation of leaving everything as it is, or multiplication by 1, or w=z
If you repeat L’s any number of times you just get new L’s
because if w1=p1z+q1
which is also an L
And if you do N or L to a Möbius transformation w=(az+b)/(cz+d) you just get another Möbius transformation with a different a, b, c, d
So as long as we can show NL and LN both give Möbius transformations, we’re done
NL gives something like w=1/(pz+q), which is a Möbius transformation
LN gives something like w=(p/z)+q, which is equivalent to w=(qz+p)/z, which is a Möbius transformation ∎
Proof of CLAIM TWO
If w=(az+b)/(cz+d) and c=0, then the transformation is obviously an L
If c≠0, w=[(a/c)(cz+d)+(bc-ad)/c]/[cz+d]=(a/c)+[(bc-ad)/c]/[cz+d]
If bc=ad, then the transformation is just w=a/c, i.e. collapsing everything to a point, so usually we say a Möbius transformation is w=(az+b)/(cz+d) with bc≠ad
If bc≠ad, write p=a/c and q=(bc-ad)/c
then w = p + q/(cz+d)
which we can get from
w3=p+qw2 (L) ∎
Since the inverse of every L (other than just w=q, collapsing everything to a point, which we’ve agreed to exclude) is another L
and the inverse of N is N
it follows that every Möbius transformation has an inverse, which is just another Möbius transformation.
In other words, Möbius transformations have a whole world of their own. Do two, twenty, two million Möbius transformations, and you end up with another Möbius transformation. Invert (“reverse”) a Möbius transformation, and you get another Möbius transformation.
Another way of showing this: multiplication of Möbius transformations is exactly like multiplication of matrices: https://www.futurelearn.com/courses/maths-power-laws/0/steps/12147
Click here for pdf booklet for Möbius transformations lessons. Click here for exercises from the textbook, and answers.
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