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1: 14.23 Values on the Cut
§14.23 Values on the Cut
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 arccsch z , arcsech z , and 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 ( - , 0 ] , and is two-valued and discontinuous on the cut shown in Figure 4.2.1, unless a . …
4: 6.3 Graphics
See accompanying text
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 < x < then 1 / ( x + i 0 ) = ( 1 / x ) - 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. …
See accompanying text
Figure 10.3.10: H 0 ( 1 ) ( x + i y ) , - 10 x 5 , - 2.8 y 4 . Principal value. … Magnify 3D Help
See accompanying text
Figure 10.3.12: H 1 ( 1 ) ( x + i y ) , - 10 x 5 , - 2.8 y 4 . Principal value. … Magnify 3D Help
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Figure 10.3.14: H 5 ( 1 ) ( x + i y ) , - 20 x 10 , - 4 y 4 . Principal value. … Magnify 3D Help
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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. …
See accompanying text
Figure 8.3.8: Γ ( 0.25 , x + i y ) , - 3 x 3 , - 3 y 3 . Principal value. … Magnify 3D Help
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Figure 8.3.9: γ ( 0.25 , x + i y ) , - 3 x 3 , - 3 y 3 . Principal value. … Magnify 3D Help
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Figure 8.3.14: Γ ( 2.5 , x + i y ) , - 2.2 x 3 , - 3 y 3 . Principal value. … Magnify 3D Help
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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 - 1 2 π < z < 1 2 π onto the whole w -plane cut along the real axis from - to - 1 and 1 to , where w = sin z and z = arcsin w (principal value). …
9: 10.25 Definitions
In particular, the principal branch of I ν ( z ) is defined in a similar way: it corresponds to the principal value of ( 1 2 z ) ν , is analytic in ( - , 0 ] , and two-valued and discontinuous on the cut ph z = ± π . … The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in ( - , 0 ] , and two-valued and discontinuous on the cut ph z = ± π . …
10: 4.3 Graphics
Figure 4.3.2 illustrates the conformal mapping of the strip - π < z < π onto the whole w -plane cut along the negative real axis, where w = e z and z = ln w (principal value). …