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1: 4.2 Definitions
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Figure 4.2.1: z -plane: Branch cut for ln z and z α . Magnify
Most texts extend the definition of the principal value to include the branch cut
2: 4.15 Graphics
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Figure 4.15.9: arcsin ( x + i y ) (principal value). There are branch cuts along the real axis from to 1 and 1 to . Magnify 3D Help
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Figure 4.15.11: arctan ( x + i y ) (principal value). There are branch cuts along the imaginary axis from i to i and i to i . Magnify 3D Help
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Figure 4.15.13: arccsc ( x + i y ) (principal value). There is a branch cut along the real axis from 1 to 1 . Magnify 3D Help
3: 19.3 Graphics
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Figure 19.3.7: K ( k ) as a function of complex k 2 for 2 ( k 2 ) 2 , 2 ( k 2 ) 2 . There is a branch cut where 1 < k 2 < . Magnify 3D Help
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Figure 19.3.8: E ( k ) as a function of complex k 2 for 2 ( k 2 ) 2 , 2 ( k 2 ) 2 . There is a branch cut where 1 < k 2 < . Magnify 3D Help
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Figure 19.3.9: ( K ( k ) ) as a function of complex k 2 for 2 ( k 2 ) 2 , 2 ( k 2 ) 2 . …On the branch cut ( k 2 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: ( K ( k ) ) as a function of complex k 2 for 2 ( k 2 ) 2 , 2 ( k 2 ) 2 . …On the upper edge of the branch cut ( k 2 1 ) it has the value K ( k ) if k 2 > 1 , and 1 4 π if k 2 = 1 . Magnify 3D Help
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Figure 19.3.12: ( E ( k ) ) as a function of complex k 2 for 2 ( k 2 ) 2 , 2 ( k 2 ) 2 . …On the upper edge of the branch cut ( k 2 > 1 ) it has the (negative) value K ( k ) E ( k ) , with limit 0 as k 2 1 + . Magnify 3D Help
4: 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. …
Figure 4.37.1: z -plane. Branch cuts for the inverse hyperbolic functions.
5: 4.3 Graphics
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Figure 4.3.3: ln ( x + i y ) (principal value). There is a branch cut along the negative real axis. Magnify 3D Help
6: 8.19 Generalized Exponential Integral
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Figure 8.19.2: E 1 2 ( x + i y ) , 4 x 4 , 4 y 4 . …There is a branch cut along the negative real axis. Magnify 3D Help
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Figure 8.19.3: E 1 ( x + i y ) , 4 x 4 , 4 y 4 . …There is a branch cut along the negative real axis. Magnify 3D Help
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Figure 8.19.4: E 3 2 ( x + i y ) , 3 x 3 , 3 y 3 . …There is a branch cut along the negative real axis. Magnify 3D Help
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Figure 8.19.5: E 2 ( x + i y ) , 3 x 3 , 3 y 3 . …There is a branch cut along the negative real axis. Magnify 3D Help
7: 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. …
Figure 4.23.1: z -plane. Branch cuts for the inverse trigonometric functions.
8: 4.13 Lambert W -Function
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Figure 4.13.2: The W ( z ) function on the first 5 Riemann sheets. … Magnify
where t 0 for W 0 , t 0 for W ± 1 on the relevant branch cuts, …
9: 15.2 Definitions and Analytical Properties
The branch obtained by introducing a cut from 1 to + on the real z -axis, that is, the branch in the sector | ph ( 1 z ) | π , is the principal branch (or principal value) of F ( a , b ; c ; z ) . … The difference between the principal branches on the two sides of the branch cut4.2(i)) is given by …
10: 3.10 Continued Fractions
For example, by converting the Maclaurin expansion of arctan z (4.24.3), we obtain a continued fraction with the same region of convergence ( | z | 1 , z ± i ), whereas the continued fraction (4.25.4) converges for all z except on the branch cuts from i to i and i to i . …