holomorphic mapping of curve and tangent

Let D be a domain of the complex plane and the functionf:D  be holomorphic.  Then for each point z of D there is a corresponding point  w=f(z);  we think that z and w both lie in their own complex planes, z-plane and w-plane.

Since f is continuousMathworldPlanetmathPlanetmath in D, if z draws a continuous curve γ in D then its image point w also draws a continuous curve γw.  Let z0 and z0+Δz be two points on γ and w0 and w0+Δw their image points on γw.

We suppose still that the curve γ has a tangent lineMathworldPlanetmath at the point z0 and that the value of the derivativePlanetmathPlanetmath f has in z0 a nonzero value

f(z0)=ϱeiω. (1)

If the slope angles of the secant linesMathworldPlanetmath(z0,z0+Δz)  and  (w0,w0+Δw)  are α and αw, then we have


and the difference quotient of f has the form


Let now  Δz0.  Then the point z0+Δz tends on the curve γ to z0 and


This implies, by (1), that

limΔz0kwk=ϱ. (2)

From this we infer, because  ϱ0  that, up to a multiple of 2π,

limΔz0(αw-α)=ω. (3)

But the limit of α is the slope angle φ of the tangentPlanetmathPlanetmath of γ at z0.  Hence (3) implies that

φw=limΔz0αw=φ+ω. (4)

Accordingly, we have the

Theorem 1.  If a curve γ has a tangent line in a point z0 where the derivative f does not vanish, then the image curve f(γ) also has in the corresponding point w0 a certain tangent line with a direction obtained by rotating the tangent of γ by the angle


If the curve γ is smooth, then also γw is smooth, and it follows easily from (2) the corresponding limit equation between the arc lengthsMathworldPlanetmath:

limΔz0sws=|f(z0)|. (5)


If we have besides γ another curve γ emanating from z0 with its tangent, the mapping f from D in z-plane to w-plane gives two curves and their tangents emanating from w0.  Thus we have two equations (4):


By subtracting we obtain

φw-φw=φ-φ, (6)

whence we have the

Theorem 2.  The mapping created by the holomorphic function f preserves the magnitude of the angle between two curves in any point z where  f(z)0.  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 f were not holomorphic but antiholomorphic.

Title holomorphic mapping of curve and tangent
Canonical name HolomorphicMappingOfCurveAndTangent
Date of creation 2013-03-22 18:42:19
Last modified on 2013-03-22 18:42:19
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Topic
Classification msc 53A30
Classification msc 30E20
Defines directly conformal