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21: 1.10 Functions of a Complex Variable

… ►A cut domain is one from which the points on finitely many nonintersecting simple contours (§1.9(iii)) have been removed. Each contour is called a cut. A cut neighborhood is formed by deleting a ray emanating from the center. … … ►(a) By introducing appropriate cuts from the branch points and restricting $F(z)$ to be single-valued in the cut plane (or domain). …

22: 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). … ►

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Figure 4.15.9: $\arcsin \left(x+\mathrm{i}y\right)$ (principal value). There are branch cuts along the real axis from $-\mathrm{\infty }$ to $-1$ and $1$ to $\mathrm{\infty }$. Magnify 3D Help

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Figure 4.15.11: $\arctan \left(x+\mathrm{i}y\right)$ (principal value). There are branch cuts along the imaginary axis from $-\mathrm{i}\mathrm{\infty }$ to $-\mathrm{i}$ and $\mathrm{i}$ to $\mathrm{i}\mathrm{\infty }$. Magnify 3D Help

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Figure 4.15.13: $\mathrm{arccsc}\left(x+\mathrm{i}y\right)$ (principal value). There is a branch cut along the real axis from $-1$ to $1$. Magnify 3D Help

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23: 6.3 Graphics

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Figure 6.3.3: $|E_1\left(x+\mathrm{i}y\right)|$, $-4\le x\le 4$, $-4\le y\le 4$. …There is a cut along the negative real axis. … Magnify 3D Help

24: 10.42 Zeros

… ►The distribution of the zeros of $K_n\left(nz\right)$ in the sector $-\frac{3}{2}\pi \le \mathrm{ph}z\le \frac{1}{2}\pi$ in the cases $n=1,5,10$ is obtained on rotating Figures 10.21.2, 10.21.4, 10.21.6, respectively, through an angle $-\frac{1}{2}\pi$ so that in each case the cut lies along the positive imaginary axis. …

25: 11.3 Graphics

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Figure 11.3.8: $\left|{𝐊}_0\left(x+\mathrm{i}y\right)\right|$ (principal value) for $-8\le x\le 8$ and $-3\le y\le 3$. There is a cut along the negative real axis. Magnify 3D Help

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Figure 11.3.9: $|{𝐇}_{\frac{1}{2}}\left(x+\mathrm{i}y\right)|$ (principal value) for $-8\le x\le 8$ and $-3\le y\le 3$. There is a cut along the negative real axis. Magnify 3D Help

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Figure 11.3.10: $|{𝐊}_{\frac{1}{2}}\left(x+\mathrm{i}y\right)|$ (principal value) for $-8\le x\le 8$ and $-3\le y\le 3$. There is a cut along the negative real axis. Magnify 3D Help

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Figure 11.3.12: $\left|{𝐊}_1\left(x+\mathrm{i}y\right)\right|$ (principal value) for $-8\le x\le 8$ and $-3\le y\le 3$. There is a cut along the negative real axis. Magnify 3D Help

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Figure 11.3.19: $|{𝐌}_{-\frac{1}{2}}\left(x+\mathrm{i}y\right)|$ (principal value) for $-3\le x\le 3$ and $-3\le y\le 3$. There is a cut along the negative real axis. Magnify 3D Help

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26: 10.3 Graphics

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Figure 10.3.10: $H_0^{(1)}\left(x+\mathrm{i}y\right)$, $-10\le x\le 5$, $-2.8\le y\le 4$. …There is a cut along the negative real axis. Magnify 3D Help

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Figure 10.3.12: $H_1^{(1)}\left(x+\mathrm{i}y\right)$, $-10\le x\le 5$, $-2.8\le y\le 4$. …There is a cut along the negative real axis. Magnify 3D Help

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Figure 10.3.14: $H_5^{(1)}\left(x+\mathrm{i}y\right)$, $-20\le x\le 10$, $-4\le y\le 4$. …There is a cut along the negative real axis. Magnify 3D Help

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Figure 10.3.15: $J_{5.5}\left(x+\mathrm{i}y\right)$, $-10\le x\le 10$, $-4\le y\le 4$. …There is a cut along the negative real axis. Magnify 3D Help

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Figure 10.3.16: $H_{5.5}^{(1)}\left(x+\mathrm{i}y\right)$, $-20\le x\le 10$, $-4\le y\le 4$. …There is a cut along the negative real axis. Magnify 3D Help

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27: 10.2 Definitions

… ►The principal branch of $J_{\nu }\left(z\right)$ corresponds to the principal value of ${(\frac{1}{2}z)}^{\nu }$ (§4.2(iv)) and is analytic in the $z$-plane cut along the interval $(-\mathrm{\infty },0]$. … ►The principal branch corresponds to the principal branches of $J_{\pm \nu }\left(z\right)$ in (10.2.3) and (10.2.4), with a cut in the $z$-plane along the interval $(-\mathrm{\infty },0]$. ►Except in the case of $J_{\pm n}\left(z\right)$, the principal branches of $J_{\nu }\left(z\right)$ and $Y_{\nu }\left(z\right)$ are two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$; compare §4.2(i). … ►The principal branches correspond to principal values of the square roots in (10.2.5) and (10.2.6), again with a cut in the $z$-plane along the interval $(-\mathrm{\infty },0]$. ►The principal branches of $H_{\nu }^{(1)}\left(z\right)$ and $H_{\nu }^{(2)}\left(z\right)$ are two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$. …

28: 14.23 Values on the Cut

§14.23 Values on the Cut

… ►If cuts are introduced along the intervals $(-\mathrm{\infty },-1]$ and $[1,\mathrm{\infty })$, then (14.23.4) and (14.23.6) could be used to extend the definitions of ${𝖯}_{\nu }^{\mu }\left(x\right)$ and ${𝖰}_{\nu }^{\mu }\left(x\right)$ to complex $x$. …

29: 19.3 Graphics

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Figure 19.3.7: $K\left(k\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. There is a branch cut where . Magnify 3D Help

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Figure 19.3.8: $E\left(k\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. There is a branch cut where . Magnify 3D Help

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Figure 19.3.9: $\mathrm{\Re }\left(K\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. …On the branch cut ($k^2\ge 1$) it is infinite at $k^2=1$, and has the value $K(1/k)/k$ when $k^2>1$. Magnify 3D Help

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Figure 19.3.10: $\mathrm{\Im }\left(K\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. …On the upper edge of the branch cut ($k^2\ge 1$) it has the value $K\left(k^{\prime }\right)$ if $k^2>1$, and $\frac{1}{4}\pi$ if $k^2=1$. Magnify 3D Help

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Figure 19.3.12: $\mathrm{\Im }\left(E\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. …On the upper edge of the branch cut ($k^2>1$) it has the (negative) value $K\left(k^{\prime }\right)-E\left(k^{\prime }\right)$, with limit 0 as $k^2\to 1+$. Magnify 3D Help

30: 14.21 Definitions and Basic Properties

… ►When $z$ is complex $P_{\nu }^{\pm \mu }\left(z\right)$, $Q_{\nu }^{\mu }\left(z\right)$, and ${𝑸}_{\nu }^{\mu }\left(z\right)$ are defined by (14.3.6)–(14.3.10) with $x$ replaced by $z$: the principal branches are obtained by taking the principal values of all the multivalued functions appearing in these representations when $z\in (1,\mathrm{\infty })$, and by continuity elsewhere in the $z$-plane with a cut along the interval $(-\mathrm{\infty },1]$; compare §4.2(i). … ►Many of the properties stated in preceding sections extend immediately from the $x$-interval $(1,\mathrm{\infty })$ to the cut $z$-plane $ℂ\backslash (-\mathrm{\infty },1]$. …