Determine the Mobius transform from $\infty$ to $0$, $0$ to $-i$ and $i$ to $\infty$
I have done the following:
1) $$\lim \limits_{z\to \infty} \frac{az+b}{cz+d}=0$$ $$\iff \lim_{z\to 0} \frac{a\frac{1}{z} + b}{c\frac{1}{z}+d}$$ $$=\lim_{z\to 0} \frac{a+bz}{c+dz}\implies \frac ac=0\implies a=0$$
2) $$\frac bd = -i \implies b=-id$$
3) $$\lim_{z\to i} \frac{az+b}{cz+d}=\infty$$ $$\iff \lim_{z\to i} \frac{cz+d}{az+b}=0$$ $$=\lim_{z\to i} \frac{cz+d}{b} \implies ci+d=0\implies ci=-d$$
Hence $a=0,b=b,c=-b,d=bi$
And I believe we obtain $$T(z) = \frac{b}{-bz+bi} = \frac{1}{-z+i}$$
Is this correct?