§4.15 Graphics

Figure 4.15.7 illustrates the conformal mapping of the strip onto the whole $w$-plane cut along the real axis from $-\mathrm{\infty }$ to $-1$ and $1$ to $\mathrm{\infty }$, where $w=\sin z$ and $z=\arcsin w$ (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 $(0,\mathrm{\infty })$.

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.

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

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

The corresponding surfaces for $\arccos \left(x+\mathrm{i}y\right)$, $\mathrm{arccot}\left(x+\mathrm{i}y\right)$, $\mathrm{arcsec}\left(x+\mathrm{i}y\right)$ 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).