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 w_{1}=p_{1}z+q_{1}

and w_{2}=p_{2}w_{1}+q_{2}

then w_{2}=p_{2}p_{1}z+p_{2}q_{1}+q_{2}

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

w_{1}=cz+d (L)

w_{2}=1/w_{1} (N)

w_{3}=p+qw_{2} (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

**More**

Click here for pdf booklet for Möbius transformations lessons. Click here for exercises from the textbook, and answers.

Click here for worked answers to questions in textbook and all recent FP2 exam questions on this topic