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1: 32.8 Rational Solutions
P II P VI  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. … More generally, … In general, P IV  has rational solutions iff either … In the general case assume δ 0 , so that as in §32.2(ii) we may set δ = 1 2 . … In the general case, P VI  has rational solutions if …
2: 28.35 Tables
  • National Bureau of Standards (1967) includes the eigenvalues a n ( q ) , b n ( q ) for n = 0 ( 1 ) 3 with q = 0 ( .2 ) 20 ( .5 ) 37 ( 1 ) 100 , and n = 4 ( 1 ) 15 with q = 0 ( 2 ) 100 ; Fourier coefficients for ce n ( x , q ) and se n ( x , q ) for n = 0 ( 1 ) 15 , n = 1 ( 1 ) 15 , respectively, and various values of q in the interval [ 0 , 100 ] ; joining factors g e , n ( q ) , f e , n ( q ) for n = 0 ( 1 ) 15 with q = 0 ( .5  to  10 ) 100 (but in a different notation). Also, eigenvalues for large values of q . Precision is generally 8D.

  • 3: 9.18 Tables
  • Miller (1946) tabulates Ai ( x ) , Ai ( x ) for x = 20 ( .01 ) 2 ; log 10 Ai ( x ) , Ai ( x ) / Ai ( x ) for x = 0 ( .1 ) 25 ( 1 ) 75 ; Bi ( x ) , Bi ( x ) for x = 10 ( .1 ) 2.5 ; log 10 Bi ( x ) , Bi ( x ) / Bi ( x ) for x = 0 ( .1 ) 10 ; M ( x ) , N ( x ) , θ ( x ) , ϕ ( x ) (respectively F ( x ) , G ( x ) , χ ( x ) , ψ ( x ) ) for x = 80 ( 1 ) 30 ( .1 ) 0 . Precision is generally 8D; slightly less for some of the auxiliary functions. Extracts from these tables are included in Abramowitz and Stegun (1964, Chapter 10), together with some auxiliary functions for large arguments.

  • Zhang and Jin (1996, p. 337) tabulates Ai ( x ) , Ai ( x ) , Bi ( x ) , Bi ( x ) for x = 0 ( 1 ) 20 to 8S and for x = 20 ( 1 ) 0 to 9D.

  • Miller (1946) tabulates a k , Ai ( a k ) , a k , Ai ( a k ) , k = 1 ( 1 ) 50 ; b k , Bi ( b k ) , b k , Bi ( b k ) , k = 1 ( 1 ) 20 . Precision is 8D. Entries for k = 1 ( 1 ) 20 are reproduced in Abramowitz and Stegun (1964, Chapter 10).

  • Sherry (1959) tabulates a k , Ai ( a k ) , a k , Ai ( a k ) , k = 1 ( 1 ) 50 ; 20S.

  • §9.18(vii) Generalized Airy Functions
    4: 3.8 Nonlinear Equations
    From this graph we estimate an initial value x 0 = 4.65 . … No explicit general formulas exist when n 5 . … Table 3.8.3 gives the successive values of s and t . … Consider x = 20 and j = 19 . We have p ( 20 ) = 19 ! and a 19 = 1 + 2 + + 20 = 210 . …
    5: 5.11 Asymptotic Expansions
    Wrench (1968) gives exact values of g k up to g 20 . Spira (1971) corrects errors in Wrench’s results and also supplies exact and 45D values of g k for k = 21 , 22 , , 30 . … uniformly for bounded real values of x . … In terms of generalized Bernoulli polynomials B n ( ) ( x ) 24.16(i)), we have for k = 0 , 1 , ,
    5.11.17 G k ( a , b ) = ( a b k ) B k ( a b + 1 ) ( a ) ,
    6: Bibliography K
  • R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.
  • M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
  • A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.
  • U. J. Knottnerus (1960) Approximation Formulae for Generalized Hypergeometric Functions for Large Values of the Parameters. J. B. Wolters, Groningen.
  • T. H. Koornwinder (2009) The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.
  • 7: Bibliography N
  • D. Naylor (1989) On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
  • National Bureau of Standards (1967) Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors. 2nd edition, National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..
  • W. J. Nellis and B. C. Carlson (1966) Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.
  • T. D. Newton (1952) Coulomb Functions for Large Values of the Parameter η . Technical report Atomic Energy of Canada Limited, Chalk River, Ontario.
  • E. W. Ng and M. Geller (1969) A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
  • 8: 36.5 Stokes Sets
    They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4). … The first sheet corresponds to x < 0 and is generated as a solution of Equations (36.5.6)–(36.5.9). 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 . For | Y | > Y 1 the second sheet is generated by a second solution of (36.5.6)–(36.5.9), and for | Y | < Y 1 it is generated by the roots of the polynomial equation … the intersection lines with the bifurcation set are generated by | X | = X 2 = 0.45148 , Y = Y 2 = 0.59693 . …
    9: Bibliography I
  • IMSL (commercial C, Fortran, and Java libraries) IMSL Nuerical Libraries..
  • K. Inkeri (1959) The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.
  • Inverse Symbolic Calculator (website) Centre for Experimental and Constructive Mathematics, Simon Fraser University, Canada.
  • M. E. H. Ismail (2000a) An electrostatics model for zeros of general orthogonal polynomials. Pacific J. Math. 193 (2), pp. 355–369.
  • 10: Bibliography L
  • A. Laforgia and M. E. Muldoon (1988) Monotonicity properties of zeros of generalized Airy functions. Z. Angew. Math. Phys. 39 (2), pp. 267–271.
  • P. W. Lawrence, R. M. Corless, and D. J. Jeffrey (2012) Algorithm 917: complex double-precision evaluation of the Wright ω function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.
  • E. W. Leaver (1986) Solutions to a generalized spheroidal wave equation: Teukolsky’s equations in general relativity, and the two-center problem in molecular quantum mechanics. J. Math. Phys. 27 (5), pp. 1238–1265.
  • W. R. Leeb (1979) Algorithm 537: Characteristic values of Mathieu’s differential equation. ACM Trans. Math. Software 5 (1), pp. 112–117.
  • D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.