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Showing content from https://dlmf.nist.gov/4.3 below:

§4.3 Graphics ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions

§4.3 Graphics Contents
  1. §4.3(i) Real Arguments
  2. §4.3(ii) Complex Arguments: Conformal Maps
  3. §4.3(iii) Complex Arguments: Surfaces
§4.3(i) Real Arguments Figure 4.3.1: ln ⁡ x and ex . Parallel tangent lines at ( 1 , 0 ) and ( 0 , 1 ) make evident the mirror symmetry across the line y = x , demonstrating the inverse relationship between the two functions. Magnify §4.3(ii) Complex Arguments: Conformal Maps

Figure 4.3.2 illustrates the conformal mapping of the strip − π < ℑ ⁡ z < π onto the whole w -plane cut along the negative real axis, where w = ez and z = ln ⁡ w (principal value). Corresponding points share the same letters, with bars signifying complex conjugates. Lines parallel to the real axis in the z -plane map onto rays in the w -plane, and lines parallel to the imaginary axis in the z -plane map onto circles centered at the origin in the w -plane. In the labeling of corresponding points r is a real parameter that can lie anywhere in the interval ( 0 , ∞ ) .

(i) z -plane (ii) w -plane Figure 4.3.2: Conformal mapping of exponential and logarithm. w = ez , z = ln ⁡ w . Magnify §4.3(iii) Complex Arguments: Surfaces

In the graphics shown in this subsection height corresponds to the absolute value of the function and color to the phase. See also About Color Map.

Figure 4.3.3: ln ⁡ ( x + i ⁢ y ) (principal value). There is a branch cut along the negative real axis. Magnify 3D Help

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