# §4.15 Graphics

## §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 $(0,\infty)$.

## §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.

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).