values on the cut

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11: 8.19 Generalized Exponential Integral

… ►When the path of integration excludes the origin and does not cross the negative real axis (8.19.2) defines the principal value of $E_p\left(z\right)$, and unless indicated otherwise in the DLMF principal values are assumed. … ►In Figures 8.19.2–8.19.5, height corresponds to the absolute value of the function and color to the phase. … ►

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Figure 8.19.2: $E_{\frac{1}{2}}\left(x+\mathrm{i}y\right)$, $-4\le x\le 4$, $-4\le y\le 4$. Principal value. … Magnify 3D Help

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Figure 8.19.3: $E_1\left(x+\mathrm{i}y\right)$, $-4\le x\le 4$, $-4\le y\le 4$. Principal value. … Magnify 3D Help

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§8.19(iii) Special Values

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12: 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 $(-\mathrm{\infty },0]$. … ►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 $\mathrm{ph}z=\pm \pi$. …

13: 1.10 Functions of a Complex Variable

… ►(a) By introducing appropriate cuts from the branch points and restricting $F(z)$ to be single-valued in the cut plane (or domain). …

14: 25.12 Polylogarithms

… ►Other notations and names for ${\mathrm{Li}}_2\left(z\right)$ include $S_2(z)$ (Kölbig et al. (1970)), Spence function $\mathrm{Sp}(z)$ (’t Hooft and Veltman (1979)), and ${\mathrm{L}}_2(z)$ (Maximon (2003)). … ►The principal branch has a cut along the interval $[1,\mathrm{\infty })$ and agrees with (25.12.1) when $|z|\le 1$; see also §4.2(i). … ►

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Figure 25.12.2: Absolute value of the dilogarithm function $\left|{\mathrm{Li}}_2\left(x+\mathrm{i}y\right)\right|$, $-20\le x\le 20$, $-20\le y\le 20$. Principal value. There is a cut along the real axis from 1 to $\mathrm{\infty }$. Magnify 3D Help

… ►For other values of $z$, ${\mathrm{Li}}_s\left(z\right)$ is defined by analytic continuation. …

15: 15.6 Integral Representations

… ►In (15.6.2) the point $1/z$ lies outside the integration contour, $t^{b-1}$ and ${(t-1)}^{c-b-1}$ assume their principal values where the contour cuts the interval $(1,\mathrm{\infty })$, and ${(1-zt)}^a=1$ at $t=0$. …

16: 19.3 Graphics

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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 19.3.11: $\mathrm{\Re }\left(E\left(k\right)\right)$ 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 branch cut ($k^2>1$) it has the value $kE\left(1/k\right)+(k_{}^{\prime }{}_{}{}^2/k)K\left(1/k\right)$, with limit 1 as $k^2\to 1+$. 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

17: Philip J. Davis

… ►The surface color map can be changed from height-based to phase-based for complex valued functions, and density plots can be generated through strategic scaling. Moreover, a cutting plane feature allows users to track curves of intersection produced as a moving plane cuts through the function surface. …

18: 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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19: 28.12 Definitions and Basic Properties

… ►(28.12.10) is not valid for cuts on the real axis in the $q$-plane for special complex values of $\nu$; but it remains valid for small $q$; compare §28.7. …

20: 6.2 Definitions and Interrelations

… ►As in the case of the logarithm (§4.2(i)) there is a cut along the interval $(-\mathrm{\infty },0]$ and the principal value is two-valued on $(-\mathrm{\infty },0)$. …