values on the cut

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1: 14.23 Values on the Cut

§14.23 Values on the Cut

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2: 4.37 Inverse Hyperbolic Functions

… ►The principal values (or principal branches) of the inverse $\sinh$, $\cosh$, and $\tanh$ are obtained by introducing cuts in the $z$-plane as indicated in Figure 4.37.1(i)-(iii), and requiring the integration paths in (4.37.1)–(4.37.3) not to cross these cuts. …Each is two-valued on the corresponding cut(s), and each is real on the part of the real axis that remains after deleting the intersections with the corresponding cuts. … ►

§4.37(iv) Logarithmic Forms

… ►For the corresponding results for $\mathrm{arccsch}z$, $\mathrm{arcsech}z$, and $\mathrm{arccoth}z$, use (4.37.7)–(4.37.9); compare §4.23(iv). …

3: 4.2 Definitions

… ►Most texts extend the definition of the principal value to include the branch cut …where $k$ is the excess of the number of times the path in (4.2.1) crosses the negative real axis in the positive sense over the number of times in the negative sense. … ►Consequently $\ln z$ is two-valued on the cut, and discontinuous across the cut. … ►This is an analytic function of $z$ on $ℂ\setminus (-\mathrm{\infty },0]$, and is two-valued and discontinuous on the cut shown in Figure 4.2.1, unless $a\in \mathbb{Z}$. …

4: 6.3 Graphics

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

5: 4.23 Inverse Trigonometric Functions

… ►The principal values (or principal branches) of the inverse sine, cosine, and tangent are obtained by introducing cuts in the $z$-plane as indicated in Figures 4.23.1(i) and 4.23.1(ii), and requiring the integration paths in (4.23.1)–(4.23.3) not to cross these cuts. …Each is two-valued on the corresponding cuts, and each is real on the part of the real axis that remains after deleting the intersections with the corresponding cuts. … ►

§4.23(iv) Logarithmic Forms

… ►Care needs to be taken on the cuts, for example, if then $1/(x+\mathrm{i}0)=(1/x)-\mathrm{i}0$. …

6: 10.3 Graphics

… ►In the graphics shown in this subsection, height corresponds to the absolute value of the function and color to the phase. … ►

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Figure 10.3.10: $H_0^{(1)}\left(x+\mathrm{i}y\right)$, $-10\le x\le 5$, $-2.8\le y\le 4$. Principal value. … Magnify 3D Help

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Figure 10.3.12: $H_1^{(1)}\left(x+\mathrm{i}y\right)$, $-10\le x\le 5$, $-2.8\le y\le 4$. Principal value. … Magnify 3D Help

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Figure 10.3.14: $H_5^{(1)}\left(x+\mathrm{i}y\right)$, $-20\le x\le 10$, $-4\le y\le 4$. Principal value. … Magnify 3D Help

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Figure 10.3.15: $J_{5.5}\left(x+\mathrm{i}y\right)$, $-10\le x\le 10$, $-4\le y\le 4$. Principal value. … Magnify 3D Help

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7: 8.3 Graphics

… ►In the graphics shown in this subsection, height corresponds to the absolute value of the function and color to the phase. … ►

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Figure 8.3.8: $\mathrm{\Gamma }(0.25,x+\mathrm{i}y)$, $-3\le x\le 3$, $-3\le y\le 3$. Principal value. … Magnify 3D Help

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Figure 8.3.9: $\gamma (0.25,x+\mathrm{i}y)$, $-3\le x\le 3$, $-3\le y\le 3$. Principal value. … Magnify 3D Help

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Figure 8.3.14: $\mathrm{\Gamma }(2.5,x+\mathrm{i}y)$, $-2.2\le x\le 3$, $-3\le y\le 3$. Principal value. … Magnify 3D Help

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Figure 8.3.15: $\gamma (2.5,x+\mathrm{i}y)$, $-2.2\le x\le 3$, $-3\le y\le 3$. Principal value. … Magnify 3D Help

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8: 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). …

9: 4.3 Graphics

… ►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 $z=\ln w$ (principal value). …

10: 10.25 Definitions

… ►In particular, the principal branch of $I_{\nu }\left(z\right)$ is defined in a similar way: it corresponds to the principal value of ${(\frac{1}{2}z)}^{\nu }$, is analytic in $ℂ\setminus (-\mathrm{\infty },0]$, and two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$. … ►The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in $ℂ\setminus (-\mathrm{\infty },0]$, and two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$. …