holomorphic mapping of curve and tangent
Let be a domain of the complex plane and the function be holomorphic. Then for each point of there is a corresponding point ; we think that and both lie in their own complex planes, -plane and -plane.
Since is continuous in , if draws a continuous curve in then its image point also draws a continuous curve . Let and be two points on and and their image points on .
and the difference quotient of has the form
Let now . Then the point tends on the curve to and
This implies, by (1), that
From this we infer, because that, up to a multiple of ,
But the limit of is the slope angle of the tangent of at . Hence (3) implies that
Accordingly, we have the
Theorem 1. If a curve has a tangent line in a point where the derivative does not vanish, then the image curve also has in the corresponding point a certain tangent line with a direction obtained by rotating the tangent of by the angle
If the curve is smooth, then also is smooth, and it follows easily from (2) the corresponding limit equation between the arc lengths:
If we have besides another curve emanating from with its tangent, the mapping from in -plane to -plane gives two curves and their tangents emanating from . Thus we have two equations (4):
By subtracting we obtain
whence we have the
Theorem 2. The mapping created by the holomorphic function preserves the magnitude of the angle between two curves in any point where . The equation (6) tells also that the orientation of the angle is preserved.
The facts in Theorem 2 are expressed so that the mapping is directly conformal. If the orientation were reversed the mapping were called inversely conformal; in this case were not holomorphic but antiholomorphic.
|Title||holomorphic mapping of curve and tangent|
|Date of creation||2013-03-22 18:42:19|
|Last modified on||2013-03-22 18:42:19|
|Last modified by||pahio (2872)|