- §4.3(i) Real Arguments
- §4.3(ii) Complex Arguments: Conformal Maps
- §4.3(iii) Complex Arguments: Surfaces

Figure 4.3.2 illustrates the conformal mapping of the strip $$ onto the whole $w$-plane cut along the negative real axis, where $w={\mathrm{e}}^{z}$ and $z=\mathrm{ln}w$ (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 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 $\left(0,\mathrm{\infty}\right)$.

(i) $z$-plane | (ii) $w$-plane |

A | B | C | $\overline{\mathrm{C}}$ | D | $\overline{\mathrm{D}}$ | E | $\overline{\mathrm{E}}$ | F | |
---|---|---|---|---|---|---|---|---|---|

$z$ | 0 | $r$ | $r+\mathrm{i}\pi $ | $r-\mathrm{i}\pi $ | $\mathrm{i}\pi $ | $-\mathrm{i}\pi $ | $-r+\mathrm{i}\pi $ | $-r-\mathrm{i}\pi $ | $-r$ |

$w$ | 1 | ${\mathrm{e}}^{r}$ | $-{\mathrm{e}}^{r}+\mathrm{i}0$ | $-{\mathrm{e}}^{r}-\mathrm{i}0$ | $-1+\mathrm{i}0$ | $-1-\mathrm{i}0$ | $-{\mathrm{e}}^{-r}+\mathrm{i}0$ | $-{\mathrm{e}}^{-r}-\mathrm{i}0$ | ${\mathrm{e}}^{-r}$ |

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.