Complex Logarithm - The Complex Logarithm As A Conformal Map

The Complex Logarithm As A Conformal Map

Any holomorphic map satisfying for all is a conformal map, which means that if two curves passing through a point a of U form an angle α (in the sense that the tangent lines to the curves at a form an angle α), then the images of the two curves form the same angle α at f(a). Since a branch of log z is holomorphic, and since its derivative 1/z is never 0, it defines a conformal map.

For example, the principal branch w = Log z, viewed as a mapping from to the horizontal strip defined by |Im z| < π, has the following properties, which are direct consequences of the formula in terms of polar form:

• Circles in the z-plane centered at 0 are mapped to vertical segments in the w-plane connecting a − πi to a + πi, where a is a real number depending on the radius of the circle.
• Rays emanating from 0 in the z-plane are mapped to horizontal lines in the w-plane.

Each circle and ray in the z-plane as above meet at a right angle. Their images under Log are a vertical segment and a horizontal line (respectively) in the w-plane, and these too meet at a right angle. This is an illustration of the conformal property of Log.