doubly-periodic%20forms
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21: 10.73 Physical Applications
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►and on separation of variables we obtain solutions of the form
, from which a solution satisfying prescribed boundary conditions may be constructed.
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►on assuming a time dependence of the form
.
…See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25).
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►With the spherical harmonic defined as in §14.30(i), the solutions are of the form
with , , , or , depending on the boundary conditions.
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22: 8.26 Tables
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Khamis (1965) tabulates for , to 10D.
Abramowitz and Stegun (1964, pp. 245–248) tabulates for , to 7D; also for , to 6S.
Pagurova (1961) tabulates for , to 4-9S; for , to 7D; for , to 7S or 7D.
Zhang and Jin (1996, Table 19.1) tabulates for , to 7D or 8S.
23: 23 Weierstrass Elliptic and Modular
Functions
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24: 26.6 Other Lattice Path Numbers
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is the number of paths from to that are composed of directed line segments of the form
, , or .
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is the number of lattice paths from to that stay on or above the line and are composed of directed line segments of the form
, , or .
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is the number of lattice paths from to that stay on or above the line , are composed of directed line segments of the form
or , and for which there are exactly occurrences at which a segment of the form
is followed by a segment of the form
.
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is the number of paths from to that stay on or above the diagonal and are composed of directed line segments of the form
, , or .
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25: 3.4 Differentiation
26: 23.9 Laurent and Other Power Series
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►Also, Abramowitz and Stegun (1964, (18.5.25)) supplies the first 22 terms in the reverted form of (23.9.2) as .
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27: 33.18 Limiting Forms for Large
§33.18 Limiting Forms for Large
…28: Bibliography
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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On the degrees of irreducible factors of higher order Bernoulli polynomials.
Acta Arith. 62 (4), pp. 329–342.
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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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Critical points of smooth functions, and their normal forms.
Uspehi Mat. Nauk 30 (5(185)), pp. 3–65 (Russian).
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Commentary on the Paper “Beiträge zur Theorie der Toeplitzschen Form”.
In Gábor Szegő, Collected Papers. Vol. 1,
Contemporary Mathematicians, pp. 303–305.
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29: 26.12 Plane Partitions
30: 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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A vortex filament moving without change of form.
J. Fluid Mech. 112, pp. 397–409.
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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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