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4 Elementary FunctionsLogarithm, Exponential, Powers

§4.3 Graphics


§4.3(i) Real Arguments

See accompanying text
Figure 4.3.1: lnx 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<z<π onto the whole w-plane cut along the negative real axis, where w=ez and z=lnw (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,).

See accompanying text
(i) z-plane (ii) w-plane
A B C C¯ D D¯ E E¯ F
z 0 r r+iπ r-iπ iπ -iπ -r+iπ -r-iπ -r
w 1 er -er+i0 -er-i0 -1+i0 -1-i0 -e-r+i0 -e-r-i0 e-r
Figure 4.3.2: Conformal mapping of exponential and logarithm. w=ez, z=lnw. 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.

See accompanying text
Figure 4.3.3: ln(x+iy) (principal value). There is a branch cut along the negative real axis. Magnify 3D Help
See accompanying text
Figure 4.3.4: ex+iy. Magnify 3D Help