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4 Elementary FunctionsTrigonometric Functions

§4.15 Graphics


§4.15(i) Real Arguments

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Figure 4.15.1: sinx and cosx. Magnify
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Figure 4.15.2: Arcsinx and Arccosx. Principal values are shown with thickened lines. Magnify
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Figure 4.15.3: tanx and cotx. Magnify
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Figure 4.15.4: arctanx and arccotx. Only principal values are shown. arccotx is discontinuous at x=0. Magnify
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Figure 4.15.5: cscx and secx. Magnify
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Figure 4.15.6: arccscx and arcsecx. Only principal values are shown. (Both functions are complex when -1<x<1.) Magnify

§4.15(ii) Complex Arguments: Conformal Maps

Figure 4.15.7 illustrates the conformal mapping of the strip -12π<z<12π onto the whole w-plane cut along the real axis from - to -1 and 1 to , where w=sinz and z=arcsinw (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 ellipses in the w-plane with foci at w=±1, and lines parallel to the imaginary axis in the z-plane map onto rectangular hyperbolas confocal with the ellipses. 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 12π 12π+ir 12π-ir ir -ir -12π+ir -12π-ir -12π
w 0 1 coshr+i0 coshr-i0 isinhr -isinhr -coshr+i0 -coshr-i0 -1
Figure 4.15.7: Conformal mapping of sine and inverse sine. w=sinz, z=arcsinw. Magnify

§4.15(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 4.15.8: sin(x+iy). Magnify
Figure 4.15.9: arcsin(x+iy) (principal value). There are branch cuts along the real axis from - to -1 and 1 to . Magnify
Figure 4.15.10: tan(x+iy). Magnify
Figure 4.15.11: arctan(x+iy) (principal value). There are branch cuts along the imaginary axis from -i to -i and i to i. Magnify
Figure 4.15.12: csc(x+iy). Magnify
Figure 4.15.13: arccsc(x+iy) (principal value). There is a branch cut along the real axis from -1 to 1. Magnify

The corresponding surfaces for cos(x+iy), cot(x+iy), and sec(x+iy) are similar. In consequence of the identities

4.15.1 cos(x+iy) =sin(x+12π+iy),
4.15.2 cot(x+iy) =-tan(x+12π+iy),
4.15.3 sec(x+iy) =csc(x+12π+iy),

they can be obtained by translating the surfaces shown in Figures 4.15.8, 4.15.10, 4.15.12 by -12π parallel to the x-axis, and adjusting the phase coloring in the case of Figure 4.15.10.

The corresponding surfaces for arccos(x+iy), arccot(x+iy), arcsec(x+iy) can be visualized from Figures 4.15.9, 4.15.11, 4.15.13 with the aid of equations (4.23.16)–(4.23.18).