Complex logarithm
The complex logarithm as a conformal map

Any holomorphic map f: U → C {\displaystyle f\colon U\to \mathbb {C} } satisfying f ′ (z) ≠ 0 {\displaystyle f'(z)\neq 0} for all z ∈ U {\displaystyle z\in U} is a conformal map, which means that if two curves passing through a point a {\displaystyle a} of U {\displaystyle U} form an angle α {\displaystyle \alpha } (in the sense that the tangent lines to the curves at a {\displaystyle a} form an angle α {\displaystyle \alpha }), then the images of the two curves form the same angle α {\displaystyle \alpha } at f (a) {\displaystyle f(a)}.Since a branch of log z {\displaystyle \log z} is holomorphic, and since its derivative 1 / z {\displaystyle 1/z} is never 0, it defines a conformal map.

For example, the principal branch w = Log z {\displaystyle w=\operatorname {Log} z}, viewed as a mapping from C − R ≤ 0 {\displaystyle \mathbb {C} -\mathbb {R} _{\leq 0}} to the horizontal strip defined by | Im z | < π {\displaystyle \left|\operatorname {Im} z\right|<\pi }, has the following properties, which are direct consequences of the formula in terms of polar form:

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.
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