📂Complex Anaylsis

Conformal Mapping of a Semicircle onto Quadrants

Theorem ¹

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

Explanation

$\displaystyle w = {{z - a} \over {z + a}}$ is a function without a specific name but is very important and frequently used. Be sure to directly calculate $f(a) = 0$, $f(ai) = i$, $f(-a) = \infty$ to verify.