# values on the cut

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##### 1: 14.23 Values on the Cut

###### §14.23 Values on the Cut

…##### 2: 4.37 Inverse Hyperbolic Functions

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►The

*principal values*(or*principal branches*) of the inverse $\mathrm{sinh}$, $\mathrm{cosh}$, and $\mathrm{tanh}$ 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

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►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 $\mathrm{ln}z$ is two-valued on the cut, and discontinuous across the cut. … ►This is an analytic function of $z$ on $\u2102\setminus (-\mathrm{\infty},0]$, 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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##### 5: 4.23 Inverse Trigonometric Functions

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►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

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►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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##### 7: 8.3 Graphics

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►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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##### 8: 4.15 Graphics

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►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 $w=\mathrm{sin}z$ and $z=\mathrm{arcsin}w$ (principal value).
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##### 9: 4.3 Graphics

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►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).
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##### 10: 10.25 Definitions

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►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 $\u2102\setminus (-\mathrm{\infty},0]$, 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 $\u2102\setminus (-\mathrm{\infty},0]$, and two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi $. …