integral requires a choice of path. Fortunately, if the integrand is holomorphic, then the value of the integral is unchanged by deforming the path (while holding the endpoints fixed), and in a simply connected region (a region with "no holes"), any path from to inside can be continuously deformed inside into any other. All this leads to the following: 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 of form an angle (in the sense that the tangent lines to the curves at form an angle ), then the images of the two curves form the same angle at . Since a branch of is holomorphic, and since its derivative is never 0, it defines a conformal map. For example, the principal branch , viewed as a mapping from to the horizontal strip defined by , 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 to , where is the real log of 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. The associated Riemann surface Construction The various branches of cannot be glued to give a single continuous function because two branches may give different values at a point where both are defined. Compare, for example, the principal branch on with imaginary part in and the branch on whose imaginary part lies in . These agree on the upper half plane, but not on the lower half plane. So it makes sense to glue the domains of these branches only along the copies of the upper half plane. The resulting glued domain is connected, but it has two copies of the lower half plane. Those two copies can be visualized as two levels of a parking garage, and one can get from the level of the lower half plane up to the level of the lower half plane by