via connection formulas
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§2.11(ii) Connection Formulas…
… ►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 ; 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 28–30.
… ►Subsequently and 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 and can be computed in a stable manner by integrating forwards, that is, from the origin toward infinity. …
On the asymptotic analysis of the Painlevé equations via the isomonodromy method.
Nonlinearity 7 (5), pp. 1291–1325.
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).
Connection formulae for the fourth Painlevé transcendent; Clarkson-McLeod solution.
J. Phys. A 31 (17), pp. 4073–4113.
… ►For other values of the , series solutions in powers of (possibly involving also ) can be constructed via a limiting process; compare §2.7(i) in the case of second-order differential equations. … ►We have the connection formula … ►Analytical continuation formulas for near are given in Bühring (1987b) for the case , and in Bühring (1992) for the general case. …
… ►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). …
Connection formulae for the first Painlevé transcendent in the complex domain.
Lett. Math. Phys. 27 (4), pp. 243–252.
Series expansions for the third incomplete elliptic integral via partial fraction decompositions.
J. Comput. Appl. Math. 207 (2), pp. 331–337.
On formulas involving both the Bernoulli and Fibonacci numbers.
Scripta Math. 23, pp. 27–35.
Connecting Jacobi elliptic functions with different modulus parameters.
Pramana 63 (5), pp. 921–936.
Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I.
Inverse Problems 20 (4), pp. 1165–1206.
Three-Point Formula… ►
Four-Point Formula… ►
Five-Point Formula… ►
Six-Point Formula… ►
… ►For conformal mappings via modular functions see Apostol (1990, §2.7). … ►It follows from the addition formula (23.10.1) that the points , , have zero sum iff , so that addition of points on the curve corresponds to addition of parameters on the torus ; see McKean and Moll (1999, §§2.11, 2.14). … ►If , then intersects the plane in a curve that is connected if ; if , 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). … ►