Conformal Maps

Conformal maps are maps of Complex Functions (namely comparisons of the 2d-graph of initial input complex numbers $z$ to that of the output complex numbers $w$. The change is called a complex map) in which the angle between functions in preserved in the output $w$-plane. The images show a conformal complex map. They can be seen as infinitesimally small scales of transformations done to the entire complex plane, instead of different regions in it. All complex differentiable maps have the property of conformity.

Computing Angles

To compute the angles between two functions, we compare their tangent vectors at a point $z\_0$. They are always conformal when $f^{\prime }(z)\ne 0$. The conformal maps of square (a) are shown in the image.

Properties

• Conformal maps can stretch bounded $z$ domains into unbounded $w$ domains.
• Every simply connected region in the $w$ plane can be viewed as rectangle under a conformal map, except the entire plane. (Because at a point where $f^{\prime }(z)=0$ the concept of conformality breaks down)
• $f^{\prime }(z)$ acts as the complex scaling and rotation term.