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[tex] f(z) = \frac{az+b}{cz+d} [/tex]

and [tex] ad \ne bc [/tex]

it is a one to one function how i can find a mobius transformation that take the real line into the unit circle

I read it in the net

[tex] f(z) = \frac{z - i}{z+i} [/tex]

and i checked it, it takes the real line into the unit circle, but there is a properties of the mobius transformation as the book said it is a combination of translation, inversion, rotation, dilation.

My question is how to find such map, or if we have the real line what first we have to do inversion,rotation,translation, ? to get the circle.

Thanks