Mobius transformation $0$ to $2$, $-2i$ to $0$ and $i$ to $\frac32$.

Question

I want to determine the Mobius tranformation mapping $0$ to $2$, $-2i$ to $0$ and $i$ to $\frac32$.

I don't know how to do this at all. I am fairly sure what I want is a composition of easier mappings, but having no examples I don't know what to do.

Also, how do I plot this shift? Will it still be 2D? or Can I have a 3D mapping representing the change.

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Also a side comment on a good reference text for this would be nice(but isn't the question here as it is opinionated.)

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Should I be getting the $a,b,c,d$ in terms of one another? Also are mobius transforms unique?

Answer 1

A Mobius tranformation has the form $$f(z) = \frac{az+b}{cz+d},\qquad ad-bc\ne 0.$$ Apply the hypothesis: $$\frac{a0+b}{c0+d} = f(0) = 2,$$ $$\cdots$$ And solve the system of equations (How? Three equations and four unknowns!)