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11: 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). Elsewhere the generalized hypergeometric function is a multivalued function that is analytic except for possible branch points at z = 0 , 1 , and . Unless indicated otherwise it is assumed that in the DLMF generalized hypergeometric functions assume their principal values. … In general the series (16.2.1) diverges for all nonzero values of z . … (However, except where indicated otherwise in the DLMF we assume that when p > q + 1 at least one of the a k is a nonpositive integer.) …
12: 5.9 Integral Representations
(The fractional powers have their principal values.) … where the contour begins at - , circles the origin once in the positive direction, and returns to - . t - z has its principal value where t crosses the positive real axis, and is continuous. … where | ph z | < π / 2 and the inverse tangent has its principal value. …
5.9.18 γ = - 0 e - t ln t d t = 0 ( 1 1 + t - e - t ) d t t = 0 1 ( 1 - e - t ) d t t - 1 e - t d t t = 0 ( e - t 1 - e - t - e - t t ) d t .
13: 15.6 Integral Representations
In (15.6.2) the point 1 / z lies outside the integration contour, t b - 1 and ( t - 1 ) c - b - 1 assume their principal values where the contour cuts the interval ( 1 , ) , and ( 1 - z t ) a = 1 at t = 0 . …
14: 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 . … It has a branch point at z = 0 for all ν . 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 . … Except where indicated otherwise it is assumed throughout the DLMF that the symbols I ν ( z ) and K ν ( z ) denote the principal values of these functions. …
15: 3.11 Approximation Techniques
A sufficient condition for p n ( x ) to be the minimax polynomial is that | ϵ n ( x ) | attains its maximum at n + 2 distinct points in [ a , b ] and ϵ n ( x ) changes sign at these consecutive maxima. … There exists a unique solution of this minimax problem and there are at least k + + 2 values x j , a x 0 < x 1 < < x k + + 1 b , such that m j = m , where … is called a Padé approximant at zero of f if … A general procedure is to approximate F by a rational function R (vanishing at infinity) and then approximate f by r = - 1 R . … Given n + 1 distinct points x k in the real interval [ a , b ] , with ( a = ) x 0 < x 1 < < x n - 1 < x n ( = b ), on each subinterval [ x k , x k + 1 ] , k = 0 , 1 , , n - 1 , a low-degree polynomial is defined with coefficients determined by, for example, values f k and f k of a function f and its derivative at the nodes x k and x k + 1 . …
16: 10.2 Definitions
This differential equation has a regular singularity at z = 0 with indices ± ν , and an irregular singularity at z = of rank 1 ; compare §§2.7(i) and 2.7(ii). … 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 ] . … Each solution has a branch point at z = 0 for all ν . 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 ] . …
17: 6.16 Mathematical Applications
Hence, if x is fixed and n , then S n ( x ) 1 4 π , 0 , or - 1 4 π according as 0 < x < π , x = 0 , or - π < x < 0 ; compare (6.2.14). … The first maximum of 1 2 Si ( x ) for positive x occurs at x = π and equals ( 1.1789 ) × 1 4 π ; compare Figure 6.3.2. Hence if x = π / ( 2 n ) and n , then the limiting value of S n ( x ) overshoots 1 4 π by approximately 18%. Similarly if x = π / n , then the limiting value of S n ( x ) undershoots 1 4 π by approximately 10%, and so on. …
See accompanying text
Figure 6.16.2: The logarithmic integral li ( x ) , together with vertical bars indicating the value of π ( x ) for x = 10 , 20 , , 1000 . Magnify
18: 25.12 Polylogarithms
Other notations and names for Li 2 ( z ) include S 2 ( z ) (Kölbig et al. (1970)), Spence function Sp ( z ) (’t Hooft and Veltman (1979)), and L 2 ( z ) (Maximon (2003)). In the complex plane Li 2 ( z ) has a branch point at z = 1 . The principal branch has a cut along the interval [ 1 , ) and agrees with (25.12.1) when | z | 1 ; see also §4.2(i). …
See accompanying text
Figure 25.12.2: Absolute value of the dilogarithm function | Li 2 ( x + i y ) | , - 20 x 20 , - 20 y 20 . Principal value. … Magnify 3D Help
For other values of z , Li s ( z ) is defined by analytic continuation. …
19: 3.7 Ordinary Differential Equations
§3.7(ii) Taylor-Series Method: Initial-Value Problems
If the solution w ( z ) that we are seeking grows in magnitude at least as fast as all other solutions of (3.7.1) as we pass along 𝒫 from a to b , then w ( z ) and w ( z ) may be computed in a stable manner for z = z 0 , z 1 , , z P by successive application of (3.7.5) for j = 0 , 1 , , P - 1 , beginning with initial values w ( a ) and w ( a ) . … Similarly, if w ( z ) is decaying at least as fast as all other solutions along 𝒫 , then we may reverse the labeling of the z j along 𝒫 and begin with initial values w ( b ) and w ( b ) .
§3.7(iii) Taylor-Series Method: Boundary-Value Problems
The latter is especially useful if the endpoint b of 𝒫 is at , or if the differential equation is inhomogeneous. …
20: 4.15 Graphics
See accompanying text
Figure 4.15.4: arctan x and arccot 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). …Lines parallel to the real axis in the z -plane map onto ellipses in the w -plane with foci at w = ± 1 , and lines parallel to the imaginary axis in the z -plane map onto rectangular hyperbolas confocal with the ellipses. In the labeling of corresponding points r is a real parameter that can lie anywhere in the interval ( 0 , ) . … In the graphics shown in this subsection height corresponds to the absolute value of the function and color to the phase. …