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21: 9.18 Tables
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Miller (1946) tabulates , for , for ; , for ; , for ; , , , (respectively , , , ) for . 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 , , , for to 8S and for to 9D.
Sherry (1959) tabulates , , , , ; 20S.
§9.18(vii) Generalized Airy Functions
…22: 3.8 Nonlinear Equations
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►No explicit general formulas exist when .
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►Consider and .
We have and .
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►Starting this iteration in the neighborhood of one of the four zeros , sequences are generated that converge to these zeros.
…In general the Julia set of an analytic function is a fractal, that is, a set that is self-similar.
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23: 9.13 Generalized Airy Functions
§9.13 Generalized Airy Functions
►§9.13(i) Generalizations from the Differential Equation
… ► ►§9.13(ii) Generalizations from Integral Representations
… ►24: 16.23 Mathematical Applications
§16.23 Mathematical Applications
… ►These equations are frequently solvable in terms of generalized hypergeometric functions, and the monodromy of generalized hypergeometric functions plays an important role in describing properties of the solutions. … ►§16.23(ii) Random Graphs
… ►§16.23(iv) Combinatorics and Number Theory
…25: Bibliography K
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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library.
ACM Trans. Math. Software 20 (4), pp. 447–459.
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Methods of computing the Riemann zeta-function and some generalizations of it.
USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I.
Inverse Problems 20 (4), pp. 1165–1206.
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The Askey scheme as a four-manifold with corners.
Ramanujan J. 20 (3), pp. 409–439.
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Some special cases of the generalized hypergeometric function
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J. Comput. Appl. Math. 78 (1), pp. 79–95.
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26: 5.11 Asymptotic Expansions
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►Wrench (1968) gives exact values of up to .
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►In the case
the factor is replaced with 4.
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►In terms of generalized Bernoulli polynomials (§24.16(i)), we have for ,
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5.11.17
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►For the error term in (5.11.19) in the case
and , see Olver (1995).
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27: 36.5 Stokes Sets
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►They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4).
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►The first sheet corresponds to and is generated as a solution of Equations (36.5.6)–(36.5.9).
The second sheet corresponds to and it intersects the bifurcation set (§36.4) smoothly along the line generated by , .
For the second sheet is generated by a second solution of (36.5.6)–(36.5.9), and for it is generated by the roots of the polynomial equation
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►the intersection lines with the bifurcation set are generated by , .
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28: 8 Incomplete Gamma and Related
Functions
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29: 28 Mathieu Functions and Hill’s Equation
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