principal%20values
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1: 25.12 Polylogarithms
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►Other notations and names for include (Kölbig et al. (1970)), Spence function (’t Hooft and Veltman (1979)), and (Maximon (2003)).
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►The principal branch has a cut along the interval and agrees with (25.12.1) when ; see also §4.2(i).
The remainder of the equations in this subsection apply to principal branches.
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►For other values of , is defined by analytic continuation.
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2: 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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3: 10.75 Tables
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Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of and , for , 9S.
4: 6.16 Mathematical Applications
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►Hence if and , then the limiting value of overshoots by approximately 18%.
Similarly if , then the limiting value of undershoots by approximately 10%, and so on.
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6.16.5
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5: 18.40 Methods of Computation
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►There are many ways to implement these first two steps, noting that the expressions for and of equation (18.2.30) are of little practical numerical value, see Gautschi (2004) and Golub and Meurant (2010).
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18.40.6
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►Results of low ( to decimal digits) precision for are easily obtained for to .
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6: 5.11 Asymptotic Expansions
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5.11.1
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5.11.2
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►Wrench (1968) gives exact values of up to .
Spira (1971) corrects errors in Wrench’s results and also supplies exact and 45D values of for .
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►uniformly for bounded real values of .
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7: 20.7 Identities
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►See Lawden (1989, pp. 19–20).
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►In the following equations , and all square roots assume their principal values.
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20.7.34
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8: 32.8 Rational Solutions
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– possess hierarchies of rational solutions for special values of the parameters which are generated from “seed solutions” using the Bäcklund transformations and often can be expressed in the form of determinants.
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32.8.3
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32.8.4
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32.8.5
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9: 19.36 Methods of Computation
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►All cases of , , , and are computed by essentially the same procedure (after transforming Cauchy principal values by means of (19.20.14) and (19.2.20)).
Complex values of the variables are allowed, with some restrictions in the case of that are sufficient but not always necessary.
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►Accurate values of for near 0 can be obtained from by (19.2.6) and (19.25.13).
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►This method loses significant figures in if and are nearly equal unless they are given exact values—as they can be for tables.
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►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20).
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