MHB Mobius transformation satisfying certain properties

[kalish1]

I'm having some trouble showing that a Mobius transformation $F$ maps $0$ to $\infty$ and $\infty$ to $0$ iff $F(z)=dz^{-1}$ for some $d \in \mathbb{C}.$ Mainly with the "only if" part. Do I need to use pictures?

This is Exercise $23$ in Section $3.3$ of Conway's *Functions of One Complex Variable*.

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[Opalg]

I'm having some trouble showing that a Mobius transformation $F$ maps $0$ to $\infty$ and $\infty$ to $0$ iff $F(z)=dz^{-1}$ for some $d \in \mathbb{C}.$ Mainly with the "only if" part.

The Möbius transformation has the form $F(z) = \dfrac{pz+q}{rz+s}$. If $s\ne0$ then $f(0) = \dfrac qs$, which is not equal to $\infty$. Therefore we must have $s=0$, and $F(z) = \dfrac{pz+q}{rz} = \dfrac pr + \dfrac q{rz}$. For this to take $\infty$ to $0$ we must have $p=0$. Therefore $F(z) = \dfrac{q}{rz}$, which is of the form that you want.