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1: 34.6 Definition: Symbol
§34.6 Definition: Symbol
βΊThe symbol may be defined either in terms of symbols or equivalently in terms of symbols: βΊ
34.6.1
βΊ
34.6.2
βΊThe symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments.
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2: 19.2 Definitions
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βΊThe integral for is well defined if , and the Cauchy principal value (§1.4(v)) of is taken if vanishes at an interior point of the integration path.
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βΊ
§19.2(iv) A Related Function:
… βΊFormulas involving that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using . … βΊWhen and are positive, is an inverse circular function if and an inverse hyperbolic function (or logarithm) if : …For the special cases of and see (19.6.15). …3: Bibliography H
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βΊ
Integrals containing the Fresnel functions and
.
Bul. Akad. Ε tiince RSS Moldoven. 1975 (3), pp. 48–60, 93 (Russian).
βΊ
Expansions for the probability function in series of ΔebyΕ‘ev polynomials and Bessel functions.
Bul. Akad. Ε tiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).
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βΊ
Algorithm 56: Complete elliptic integral of the second kind.
Comm. ACM 4 (4), pp. 180–181.
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βΊ
Coulomb Wave Functions.
In Handbuch der Physik, Bd. 41/1, S. Flügge (Ed.),
pp. 408–465.
βΊ
Variable precision exponential function.
ACM Trans. Math. Software 12 (2), pp. 79–91.
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4: Bibliography D
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βΊ
Formes canoniques des équations confluentes de l’équation de Heun.
Ann. Soc. Sci. Bruxelles Sér. I 92 (1-2), pp. 53–78.
βΊ
Sur les équations confluentes de l’équation de Heun.
Ann. Soc. Sci. Bruxelles Sér. I 92 (3), pp. 151–189.
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βΊ
Handbuch der Laplace-Transformation. Bd. II. Anwendungen der Laplace-Transformation. 1. Abteilung.
Birkhäuser Verlag, Basel und Stuttgart (German).
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βΊ
Theta functions and non-linear equations.
Uspekhi Mat. Nauk 36 (2(218)), pp. 11–80 (Russian).
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βΊ
Product formulas and Nicholson-type integrals for Jacobi functions. I. Summary of results.
SIAM J. Math. Anal. 9 (1), pp. 76–86.
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5: 23.21 Physical Applications
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βΊThe Weierstrass function plays a similar role for cubic potentials in canonical form .
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βΊFor applications to soliton solutions of the Korteweg–de Vries (KdV) equation see McKean and Moll (1999, p. 91), Deconinck and Segur (2000), and Walker (1996, §8.1).
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βΊwhere are the corresponding Cartesian coordinates and , , are constants.
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βΊ
23.21.3
βΊAnother form is obtained by identifying , , as lattice roots (§23.3(i)), and setting
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6: 18.13 Continued Fractions
7: 24.2 Definitions and Generating Functions
8: Bibliography R
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βΊ
Congruence properties of partitions.
Math. Z. 9 (1-2), pp. 147–153.
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βΊ
Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow.
Studies in Appl. Math. 53, pp. 91–110.
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βΊ
Erratum to:Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging.
Computing in Science and Engineering 23 (4), pp. 91.
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βΊ
Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function.
Mathematics 9 (16).
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βΊ
Another proof of the triple sum formula for Wigner -symbols.
J. Math. Phys. 40 (12), pp. 6689–6691.
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9: Bibliography N
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βΊ
Generalization of Binet’s Gamma function formulas.
Integral Transforms Spec. Funct. 24 (8), pp. 597–606.
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βΊ
On the large argument asymptotics of the Lommel function via Stieltjes transforms.
Asymptot. Anal. 91 (3-4), pp. 265–281.
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βΊ
On the calculation of elliptic integrals of the second and third kinds.
Zastos. Mat. 11, pp. 91–94.
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βΊ
The asymptotic behavior of the general real solution of the third Painlevé equation.
Dokl. Akad. Nauk SSSR 283 (5), pp. 1161–1165 (Russian).
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βΊ
High-frequency scattering by an impenetrable sphere.
Ann. Physics 34 (1), pp. 23–95.
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10: Bibliography
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βΊ
Some determinants of Bernoulli, Euler and related numbers.
Portugal. Math. 18, pp. 91–99.
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βΊ
On basic hypergeometric series, mock theta functions, and partitions. II.
Quart. J. Math. Oxford Ser. (2) 17, pp. 132–143.
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βΊ
Umbral calculus, Bailey chains, and pentagonal number theorems.
J. Combin. Theory Ser. A 91 (1-2), pp. 464–475.
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βΊ
Theorems on generalized Dedekind sums.
Pacific J. Math. 2 (1), pp. 1–9.
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βΊ
A new treatment of the ellipsoidal wave equation.
Proc. London Math. Soc. (3) 9, pp. 21–50.
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