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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.

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