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11: 26.2 Basic Definitions
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►For the actual partitions () for see Table 26.4.1.
►The integers whose sum is are referred to as the parts in the partition.
The example has six parts, three of which equal 1.
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12: Bibliography F
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Sur certaines sommes des intégral-cosinus.
Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
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Tables of Elliptic Integrals of the First, Second, and Third Kind.
Technical report
Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.
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A Table of the Complete Elliptic Integral of the First Kind for Complex Values of the Modulus. Part I.
Technical report
Technical Report ARL 69-0172, Aerospace Research Laboratories, Office of Aerospace Research, Wright-Patterson Air Force Base, Ohio.
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On weighted polynomial approximation on the whole real axis.
Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.
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13: Bibliography M
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Rational approximations, software and test methods for sine and cosine integrals.
Numer. Algorithms 12 (3-4), pp. 259–272.
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Numerical Methods for Roots of Polynomials. Part I.
Studies in Computational Mathematics, Vol. 14, Elsevier, Amsterdam.
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Calculation of the modified Bessel functions of the second kind with complex argument.
Math. Comp. 20 (95), pp. 407–412.
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Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems.
Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.
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The -analogue of the Laguerre polynomials.
J. Math. Anal. Appl. 81 (1), pp. 20–47.
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14: 18.40 Methods of Computation
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►It is now necessary to take the limit of , and the imaginary part is the required Stieltjes–Perron inversion:
…The question is then: how is this possible given only , rather than itself? often converges to smooth results for off the real axis for at a distance greater than the pole spacing of the , this may then be followed by approximate numerical analytic continuation via fitting to lower order continued fractions (either Padé, see §3.11(iv), or pointwise continued fraction approximants, see Schlessinger (1968, Appendix)), to and evaluating these on the real axis in regions of higher pole density that those of the approximating function.
Results of low ( to decimal digits) precision for are easily obtained for to .
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15: 26.9 Integer Partitions: Restricted Number and Part Size
§26.9 Integer Partitions: Restricted Number and Part Size
… ► denotes the number of partitions of into at most parts. … … ►Conjugation establishes a one-to-one correspondence between partitions of into at most parts and partitions of into parts with largest part less than or equal to . … ►equivalently, partitions into at most parts either have exactly parts, in which case we can subtract one from each part, or they have strictly fewer than parts. …16: 25.5 Integral Representations
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25.5.2
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25.5.4
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25.5.5
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►In (25.5.15)–(25.5.19), , is the digamma function, and is Euler’s constant (§5.2).
(25.5.16) is also valid for , .
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17: 6.16 Mathematical Applications
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►By integration by parts
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►If we assume Riemann’s hypothesis that all nonreal zeros of have real part of (§25.10(i)), then
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18: 5.22 Tables
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►Abramowitz and Stegun (1964, Chapter 6) tabulates , , , and for to 10D; and for to 10D; , , , , , , , and for to 8–11S; for to 20S.
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►Zhang and Jin (1996, pp. 70, 71, and 73) tabulates the real and imaginary parts of , , and for , to 8S.
19: Bibliography B
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Pionic atoms.
Annual Review of Nuclear and Particle Science 20, pp. 467–508.
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A program for computing the Riemann zeta function for complex argument.
Comput. Phys. Comm. 20 (3), pp. 441–445.
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Coulomb functions (negative energies).
Comput. Phys. Comm. 20 (3), pp. 447–458.
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Some solutions of the problem of forced convection.
Philos. Mag. Series 7 20, pp. 322–343.
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Bessel functions and modular relations of higher type and hyperbolic differential equations.
Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.
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