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1: 2.11 Remainder Terms; Stokes Phenomenon
§2.11(ii) Connection Formulas
2: 31.18 Methods of Computation
Subsequently, the coefficients in the necessary connection formulas can be calculated numerically by matching the values of solutions and their derivatives at suitably chosen values of z ; see Laĭ (1994) and Lay et al. (1998). …The computation of the accessory parameter for the Heun functions is carried out via the continued-fraction equations (31.4.2) and (31.11.13) in the same way as for the Mathieu, Lamé, and spheroidal wave functions in Chapters 2830.
3: 11.13 Methods of Computation
Subsequently H ν ( z ) and L ν ( z ) are obtainable via (11.2.5) and (11.2.6). … Then from the limiting forms for small argument (§§11.2(i), 10.7(i), 10.30(i)), limiting forms for large argument (§§11.6(i), 10.7(ii), 10.30(ii)), and the connection formulas (11.2.5) and (11.2.6), it is seen that H ν ( x ) and L ν ( x ) can be computed in a stable manner by integrating forwards, that is, from the origin toward infinity. …
4: Bibliography I
  • A. R. Its, A. S. Fokas, and A. A. Kapaev (1994) On the asymptotic analysis of the Painlevé equations via the isomonodromy method. Nonlinearity 7 (5), pp. 1291–1325.
  • A. R. Its and A. A. Kapaev (1987) The method of isomonodromic deformations and relation formulas for the second Painlevé transcendent. Izv. Akad. Nauk SSSR Ser. Mat. 51 (4), pp. 878–892, 912 (Russian).
  • A. R. Its and A. A. Kapaev (1998) Connection formulae for the fourth Painlevé transcendent; Clarkson-McLeod solution. J. Phys. A 31 (17), pp. 4073–4113.
  • 5: 16.8 Differential Equations
    For other values of the b j , series solutions in powers of z (possibly involving also ln z ) can be constructed via a limiting process; compare §2.7(i) in the case of second-order differential equations. … We have the connection formulaAnalytical continuation formulas for F q q + 1 ( a ; b ; z ) near z = 1 are given in Bühring (1987b) for the case q = 2 , and in Bühring (1992) for the general case. …
    6: 9.13 Generalized Airy Functions
    Connection formulas for the solutions of (9.13.31) include … For further generalizations via integral representations see Chin and Hedstrom (1978), Janson et al. (1993, §10), and Kamimoto (1998). …
    7: Bibliography K
  • A. A. Kapaev and A. V. Kitaev (1993) Connection formulae for the first Painlevé transcendent in the complex domain. Lett. Math. Phys. 27 (4), pp. 243–252.
  • D. Karp, A. Savenkova, and S. M. Sitnik (2007) Series expansions for the third incomplete elliptic integral via partial fraction decompositions. J. Comput. Appl. Math. 207 (2), pp. 331–337.
  • R. P. Kelisky (1957) On formulas involving both the Bernoulli and Fibonacci numbers. Scripta Math. 23, pp. 27–35.
  • A. Khare and U. Sukhatme (2004) Connecting Jacobi elliptic functions with different modulus parameters. Pramana 63 (5), pp. 921–936.
  • 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.
  • 8: 3.3 Interpolation
    Three-Point Formula
    Four-Point Formula
    Five-Point Formula
    Six-Point Formula
    Seven-Point Formula
    9: 23.20 Mathematical Applications
    For conformal mappings via modular functions see Apostol (1990, §2.7). … It follows from the addition formula (23.10.1) that the points P j = P ( z j ) , j = 1 , 2 , 3 , have zero sum iff z 1 + z 2 + z 3 𝕃 , so that addition of points on the curve C corresponds to addition of parameters z j on the torus / 𝕃 ; see McKean and Moll (1999, §§2.11, 2.14). … If a , b , then C intersects the plane 2 in a curve that is connected if Δ 4 a 3 + 27 b 2 > 0 ; if Δ < 0 , then the intersection has two components, one of which is a closed loop. …The geometric nature of this construction is illustrated in McKean and Moll (1999, §2.14), Koblitz (1993, §§6, 7), and Silverman and Tate (1992, Chapter 1, §§3, 4): each of these references makes a connection with the addition theorem (23.10.1). …
    §23.20(iii) Factorization
    10: 14.15 Uniform Asymptotic Approximations
    Provided that μ - ν the corresponding expansions for P ν μ ( x ) and Q ν μ ( x ) can be obtained from the connection formulas (14.9.7), (14.9.9), and (14.9.10). … Approximations for P ν μ ( x ) and Q ν μ ( x ) can then be achieved via (14.9.7), (14.9.9), and (14.9.10). …