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principal branch (value)

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1: 4.1 Special Notation
k , m , n

integers.

2: 10.2 Definitions
§10.2(ii) Standard Solutions
The principal branch of J ν ( z ) corresponds to the principal value of ( 1 2 z ) ν 4.2(iv)) and is analytic in the z -plane cut along the interval ( - , 0 ] . … The principal branches correspond to principal values of the square roots in (10.2.5) and (10.2.6), again with a cut in the z -plane along the interval ( - , 0 ] . …
Table 10.2.1: Numerically satisfactory pairs of solutions of Bessel’s equation.
Pair Interval or Region
3: 4.2 Definitions
This is a multivalued function of z with branch point at z = 0 . The principal value, or principal branch, is defined by … Most texts extend the definition of the principal value to include the branch cut
4.2.18 ln 10 = 2.30258 50929 94045 68401 .
This result is also valid when z a has its principal value, provided that the branch of Ln w satisfies …
4: 16.2 Definition and Analytic Properties
The branch obtained by introducing a cut from 1 to + on the real axis, that is, the branch in the sector | ph ( 1 - z ) | π , is the principal branch (or principal value) of F q q + 1 ( a ; b ; z ) ; compare §4.2(i). …Unless indicated otherwise it is assumed that in the DLMF generalized hypergeometric functions assume their principal values. …
5: 14.21 Definitions and Basic Properties
When z is complex P ν ± μ ( z ) , Q ν μ ( z ) , and Q ν μ ( z ) are defined by (14.3.6)–(14.3.10) with x replaced by z : the principal branches are obtained by taking the principal values of all the multivalued functions appearing in these representations when z ( 1 , ) , and by continuity elsewhere in the z -plane with a cut along the interval ( - , 1 ] ; compare §4.2(i). … …
6: 10.25 Definitions
§10.25(ii) Standard Solutions
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 . …
7: 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 ) . … again with analytic continuation for other values of z , and with the principal branch defined in a similar way. …
8: 4.15 Graphics
See accompanying text
Figure 4.15.2: Arcsin x and Arccos x . Principal values are shown with thickened lines. Magnify
See accompanying text
Figure 4.15.4: arctan x and arccot x . Only principal values are shown. … Magnify
See accompanying text
Figure 4.15.6: arccsc x and arcsec x . Only principal values are shown. … Magnify
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). … In the graphics shown in this subsection height corresponds to the absolute value of the function and color to the phase. …
9: 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. …
10: 4.10 Integrals
4.10.7 0 x d t ln t = li ( x ) , x > 1 .