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Hi guys I need some help with this homework question!

Find a bilinear transformation $w = f(z)$ which maps the line $\{\Re z = 0\}$ to the circle $\{|w|=1\}$ and $\{\Re z > 0\}$ to $\{|w|< 1\}$.

Hence find a bilinear transformation $w = g(z)$ which maps the line $\{\Im z =\Re z\}$ to the circle $\{|w+i| = 1\}$ and $\{\Im z <\Re z\}$ to $\{|w+i| < 1\}$.

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    Does "bilinear" here mean "Mobius" or "linear fractional"?2012-03-15

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Hint 1: $f(z) = \frac{az+b}{cz+d}$. Try for $f(0) = 1$, $\lim_{z \to \infty} f(z) = -1$, and $f(i)=-i$.

Hint 2: $g = h_1 \circ f \circ h_2$ where $h_1$ is a translation and $h_2$ is a rotation.