📂Complex Anaylsis

Conformal Mapping of a Semicircle onto Quadrants

Theorem ¹

The conformal mapping $\displaystyle w = f(z) = {{z - a} \over {z + a}}$ maps a semicircle onto a quadrant.

Explanation

$\displaystyle w = {{z - a} \over {z + a}}$ has no particular name, but it is a very important and frequently used function. In particular, let us directly compute and verify that $f(a) = 0$, $f(ai) = i$, $f(-a) = \infty$.