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11—20 of 620 matching pages
11: 33.16 Connection Formulas
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§33.16(i) and in Terms of and
… ►where is given by (33.2.5) or (33.2.6). ►§33.16(ii) and in Terms of and when
… ►and again define by (33.14.11) or (33.14.12). … ►and again define by (33.14.11) or (33.14.12). …12: Bibliography C
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Generalized hypergeometric functions and the evaluation of scalar one-loop integrals in Feynman diagrams.
J. Comput. Appl. Math. 115 (1-2), pp. 93–99.
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Elliptic integrals of the first kind.
SIAM J. Math. Anal. 8 (2), pp. 231–242.
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Coulomb phase shift.
American Journal of Physics 47 (8), pp. 683–684.
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Level-Index Arithmetic: An Introductory Survey.
In Numerical Analysis and Parallel Processing (Lancaster, 1987), P. R. Turner (Ed.),
Lecture Notes in Math., Vol. 1397, pp. 95–168.
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Inverse Acoustic and Electromagnetic Scattering Theory.
2nd edition, Applied Mathematical Sciences, Vol. 93, Springer-Verlag, Berlin.
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13: Bibliography S
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Time propagation of partial differential equations using the short iterative Lanczos method and finite-element discrete variable representation.
Adv. Quantum Chem. 72, pp. 95–127.
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Hypergeometric Functions and Their Applications.
Texts in Applied Mathematics, Vol. 8, Springer-Verlag, New York.
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Coulomb functions analytic in the energy.
Comput. Phys. Comm. 25 (1), pp. 87–95.
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Représentation asymptotique des fonctions de Mathieu et des fonctions d’onde sphéroidales.
Trans. Amer. Math. Soc. 66 (1), pp. 93–134 (French).
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Liouville-Green-Olver approximations for complex difference equations.
J. Approx. Theory 96 (2), pp. 301–322.
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14: 27.12 Asymptotic Formulas: Primes
15: 1.14 Integral Transforms
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►In this subsection we let .
►If is absolutely integrable on , then is continuous, as , and
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►If and are absolutely integrable on , then so is , and its Fourier transform is , where is the Fourier transform of .
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►If and are continuous and absolutely integrable on , and for all , then for all .
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►In this subsection we let , , , and .
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16: 33.17 Recurrence Relations and Derivatives
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17: 26.10 Integer Partitions: Other Restrictions
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denotes the number of partitions of into distinct parts.
denotes the number of partitions of into at most distinct parts.
denotes the number of partitions of into parts with difference at least .
…If more than one restriction applies, then the restrictions are separated by commas, for example, .
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►Note that , with strict inequality for .
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18: 6.7 Integral Representations
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6.7.1
, ,
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6.7.3
, ,
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6.7.4
, .
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►Many integrals with exponentials and rational functions, for example, integrals of the type , where is an arbitrary rational function, can be represented in finite form in terms of the function and elementary functions; see Lebedev (1965, p. 42).
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►For collections of integral representations see Bierens de Haan (1939, pp. 56–59, 72–73, 82–84, 121, 133–136, 155, 179–181, 223, 225–227,
230, 259–260, 374, 377, 397–398, 408, 416, 424, 431, 438–439, 442–444, 488,
496–500, 567–571, 585, 602, 638, 675–677), Corrington (1961), Erdélyi et al. (1954a, vol. 1, pp. 267–270), Geller and Ng (1969), Nielsen (1906b), Oberhettinger (1974, pp. 244–246), Oberhettinger and Badii (1973, pp. 364–371), and Watrasiewicz (1967).
19: 10.32 Integral Representations
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10.32.19
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20: Bibliography G
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Basic Hypergeometric Series.
Encyclopedia of Mathematics and its Applications, Vol. 35, Cambridge University Press, Cambridge.
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Basic Hypergeometric Series.
Second edition, Encyclopedia of Mathematics and its Applications, Vol. 96, Cambridge University Press, Cambridge.
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Algorithm 236: Bessel functions of the first kind.
Comm. ACM 7 (8), pp. 479–480.
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Algorithm 259: Legendre functions for arguments larger than one.
Comm. ACM 8 (8), pp. 488–492.
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Computational aspects of three-term recurrence relations.
SIAM Rev. 9 (1), pp. 24–82.
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