§4.15 Graphics

§4.15(i) Real Arguments Figure 4.15.1: sin⁡x and cos⁡x. Magnify Figure 4.15.3: tan⁡x and cot⁡x. Magnify Figure 4.15.5: csc⁡x and sec⁡x. Magnify

§4.15(ii) Complex Arguments: Conformal Maps

Figure 4.15.7 illustrates the conformal mapping of the strip $-\tfrac{1}{2}\pi<\Re z<\tfrac{1}{2}\pi$ onto the whole $w$-plane cut along the real axis from $-\infty$ to $-1$ and $1$ to $\infty$, where $w=\sin z$ and $z=\operatorname{arcsin}w$ (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 ellipses in the $w$-plane with foci at $w=\pm 1$, 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 $(0,\infty)$. Figure 4.15.7: Conformal mapping of sine and inverse sine. w=sin⁡z, z=arcsin⁡w. 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+i⁢y). Magnify 3D Help Figure 4.15.10: tan⁡(x+i⁢y). Magnify 3D Help Figure 4.15.12: csc⁡(x+i⁢y). Magnify 3D Help

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

 4.15.1 $\displaystyle\cos\left(x+iy\right)$ $\displaystyle=\sin\left(x+\tfrac{1}{2}\pi+iy\right),$ 4.15.2 $\displaystyle\cot\left(x+iy\right)$ $\displaystyle=-\tan\left(x+\tfrac{1}{2}\pi+iy\right),$ 4.15.3 $\displaystyle\sec\left(x+iy\right)$ $\displaystyle=\csc\left(x+\tfrac{1}{2}\pi+iy\right),$

they can be obtained by translating the surfaces shown in Figures 4.15.8, 4.15.10, 4.15.12 by $-\tfrac{1}{2}\pi$ parallel to the $x$-axis, and adjusting the phase coloring in the case of Figure 4.15.10.

The corresponding surfaces for $\operatorname{arccos}\left(x+iy\right)$, $\operatorname{arccot}\left(x+iy\right)$, $\operatorname{arcsec}\left(x+iy\right)$ 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).