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1: 36.15 Methods of Computation
§36.15(iii) Integration along Deformed Contour
Direct numerical evaluation can be carried out along a contour that runs along the segment of the real t -axis containing all real critical points of Φ and is deformed outside this range so as to reach infinity along the asymptotic valleys of exp ( i Φ ) . …
§36.15(iv) Integration along Finite Contour
This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of Φ , with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. …
2: 36.5 Stokes Sets
where j denotes a real critical point (36.4.1) or (36.4.2), and μ denotes a critical point with complex t or s , t , connected with j by a steepest-descent path (that is, a path where Φ = constant ) in complex t or ( s , t ) space. In the following subsections, only Stokes sets involving at least one real saddle are included unless stated otherwise. … The second sheet corresponds to x > 0 and it intersects the bifurcation set (§36.4) smoothly along the line generated by X = X 1 = 6.95643 , | Y | = | Y 1 | = 6.81337 . … the intersection lines with the bifurcation set are generated by | X | = X 2 = 0.45148 , Y = Y 2 = 0.59693 . … The distribution of real and complex critical points in Figures 36.5.5 and 36.5.6 follows from consistency with Figure 36.5.1 and the fact that there are four real saddles in the inner regions. …
3: Foreword
The online version, the NIST Digital Library of Mathematical Functions (DLMF), presents the same technical information along with extensions and innovative interactive features consistent with the new medium. … November 20, 2009 …
4: 10.3 Graphics
§10.3(i) Real Order and Variable
§10.3(ii) Real Order, Complex Variable
See accompanying text
Figure 10.3.10: H 0 ( 1 ) ( x + i y ) , 10 x 5 , 2.8 y 4 . …There is a cut along the negative real axis. Magnify 3D Help
See accompanying text
Figure 10.3.12: H 1 ( 1 ) ( x + i y ) , 10 x 5 , 2.8 y 4 . …There is a cut along the negative real axis. Magnify 3D Help
§10.3(iii) Imaginary Order, Real Variable
5: 20 Theta Functions
Chapter 20 Theta Functions
6: 25.12 Polylogarithms
The principal branch has a cut along the interval [ 1 , ) and agrees with (25.12.1) when | z | 1 ; see also §4.2(i). … For real or complex s and z the polylogarithm Li s ( z ) is defined by … The series also converges when | z | = 1 , provided that s > 1 . … valid when s > 0 and | ph ( 1 z ) | < π , or s > 1 and z = 1 . … valid when s > 0 , a > 0 or s > 1 , a = 0 . …
7: 25.5 Integral Representations
§25.5 Integral Representations
In (25.5.15)–(25.5.19), 0 < s < 1 , ψ ( x ) is the digamma function, and γ is Euler’s constant (§5.2). (25.5.16) is also valid for 0 < s < 2 , s 1 . …
25.5.19 ζ ( m + s ) = ( 1 ) m 1 Γ ( s ) sin ( π s ) π Γ ( m + s ) 0 ψ ( m ) ( 1 + x ) x s d x , m = 1 , 2 , 3 , .
where the integration contour is a loop around the negative real axis; it starts at , encircles the origin once in the positive direction without enclosing any of the points z = ± 2 π i , ± 4 π i , …, and returns to . …
8: Bibliography
  • M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.
  • M. Abramowitz and P. Rabinowitz (1954) Evaluation of Coulomb wave functions along the transition line. Physical Rev. (2) 96, pp. 77–79.
  • A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.
  • S. V. Aksenov, M. A. Savageau, U. D. Jentschura, J. Becher, G. Soff, and P. J. Mohr (2003) Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.
  • D. E. Amos (1989) Repeated integrals and derivatives of K Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.
  • 9: 4.3 Graphics
    §4.3(i) Real Arguments
    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). …Lines parallel to the real axis in the z -plane map onto rays in the w -plane, and lines parallel to the imaginary axis in the z -plane map onto circles centered at the origin in the w -plane. In the labeling of corresponding points r is a real parameter that can lie anywhere in the interval ( 0 , ) . …
    See accompanying text
    Figure 4.3.3: ln ( x + i y ) (principal value). There is a branch cut along the negative real axis. Magnify 3D Help
    10: 14.22 Graphics
    See accompanying text
    Figure 14.22.1: P 1 / 2 0 ( x + i y ) , 5 x 5 , 5 y 5 . There is a cut along the real axis from to 1 . Magnify 3D Help
    See accompanying text
    Figure 14.22.2: P 1 / 2 1 / 2 ( x + i y ) , 5 x 5 , 5 y 5 . There is a cut along the real axis from to 1 . Magnify 3D Help
    See accompanying text
    Figure 14.22.3: P 1 / 2 1 ( x + i y ) , 5 x 5 , 5 y 5 . There is a cut along the real axis from to 1 . Magnify 3D Help
    See accompanying text
    Figure 14.22.4: 𝑸 0 0 ( x + i y ) , 5 x 5 , 5 y 5 . There is a cut along the real axis from 1 to 1 . Magnify 3D Help