Definition: 

Bilinear transformation: The transformation $w=\dfrac{az+b}{cz+d}$ where a,b,c,d are complex constants and $ad-bc\neq0$ is called as bilineat transformation.

Theorem:

Let the bilinear transformation be $w=\dfrac{az+b}{cz+d}$where a,b,c,d are complex constants and $ad-bc\neq0$

Dividing $az+b \ by\ \ cz+d$, we get

$w=\dfrac{a}{c}+\dfrac{b-\dfrac{ad}{c}}{cz+d}$

$w=\dfrac{a}{c}+\dfrac{b-ad}{c}\times\dfrac{1}{c(z+d/c)}$

$w=\dfrac{a}{c}+\dfrac{b-ad}{c^2}\times\dfrac{1}{\bigg(z+\dfrac{d}{c}\bigg)}$

Let $w_1=z+\dfrac {d}{c}$ which is a translation.

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