# 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 $\operatorname{arccsch}z$, $\operatorname{arcsech}z$, and $\operatorname{arccoth}z$, 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 $\mathbb{C}\setminus(-\infty,0]$, and is two-valued and discontinuous on the cut shown in Figure 4.2.1, unless $a\in\mathbb{Z}$. …
##### 4: 6.3 Graphics Figure 6.3.3: | E 1 ⁡ ( x + i ⁢ y ) | , - 4 ≤ x ≤ 4 , - 4 ≤ y ≤ 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 $0 then $1/(x+i0)=(1/x)-i0$. …
##### 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. … Figure 10.3.10: H 0 ( 1 ) ⁡ ( x + i ⁢ y ) , - 10 ≤ x ≤ 5 , - 2.8 ≤ y ≤ 4 . Principal value. … Magnify 3D Help Figure 10.3.12: H 1 ( 1 ) ⁡ ( x + i ⁢ y ) , - 10 ≤ x ≤ 5 , - 2.8 ≤ y ≤ 4 . Principal value. … Magnify 3D Help Figure 10.3.14: H 5 ( 1 ) ⁡ ( x + i ⁢ y ) , - 20 ≤ x ≤ 10 , - 4 ≤ y ≤ 4 . Principal value. … Magnify 3D Help Figure 10.3.15: J 5.5 ⁡ ( x + i ⁢ y ) , - 10 ≤ x ≤ 10 , - 4 ≤ y ≤ 4 . Principal value. … Magnify 3D Help
##### 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. … Figure 8.3.8: Γ ⁡ ( 0.25 , x + i ⁢ y ) , - 3 ≤ x ≤ 3 , - 3 ≤ y ≤ 3 . Principal value. … Magnify 3D Help Figure 8.3.9: γ ⁡ ( 0.25 , x + i ⁢ y ) , - 3 ≤ x ≤ 3 , - 3 ≤ y ≤ 3 . Principal value. … Magnify 3D Help Figure 8.3.14: Γ ⁡ ( 2.5 , x + i ⁢ y ) , - 2.2 ≤ x ≤ 3 , - 3 ≤ y ≤ 3 . Principal value. … Magnify 3D Help Figure 8.3.15: γ ⁡ ( 2.5 , x + i ⁢ y ) , - 2.2 ≤ x ≤ 3 , - 3 ≤ y ≤ 3 . Principal value. … Magnify 3D Help
##### 8: 4.15 Graphics
Figure 4.15.7 illustrates the conformal mapping of the strip $-\tfrac{1}{2}\pi<\Re z<\tfrac{1}{2}\pi$ onto the whole $w$-plane cut along the real axis from $-\infty$ to $-1$ and $1$ to $\infty$, where $w=\sin z$ and $z=\operatorname{arcsin}w$ (principal value). …
##### 9: 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 $(\tfrac{1}{2}z)^{\nu}$, is analytic in $\mathbb{C}\setminus(-\infty,0]$, and two-valued and discontinuous on the cut $\operatorname{ph}z=\pm\pi$. … The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in $\mathbb{C}\setminus(-\infty,0]$, and two-valued and discontinuous on the cut $\operatorname{ph}z=\pm\pi$. …
##### 10: 4.3 Graphics
Figure 4.3.2 illustrates the conformal mapping of the strip $-\pi<\Im z<\pi$ onto the whole $w$-plane cut along the negative real axis, where $w=e^{z}$ and $z=\ln w$ (principal value). …