Suppose we have a functionsends any pointin the plane to another pointWe can define smooth curves that pass throughThe functionis conformal if the angle between any two paths throughis unchanged by the transformation. This is shown below. The angle betweenandis the same as the angle betweenand
A function is conformal if it is conformal on its domain. In fact, any analytic function is conformal at any pointfor which
Proof:higher order terms.
higher order terms.
Subtract these two to givehigher order terms. The tangent vectorsandare both rotated bybut the angle between them is unchanged. In general of course entire regions are mapped. Some examples are shown below. The grid lines in the plane are transformed onto the curves. Since the angle between any tow grid lines is a right angle, so is the angle between any two curves.