limiting%20form%20as%20a%20Bessel%20function
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11: Bibliography C
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Asymptotic behaviour of the zeros of the (generalized) Laguerre polynomial as the index and limiting formula relating Laguerre polynomials of large index and large argument to Hermite polynomials.
Lett. Nuovo Cimento (2) 23 (3), pp. 101–102.
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Sur la fonction qui détermine la totalité des nombres premiers inférieurs à une limite donnée.
Mem. Ac. Sc. St. Pétersbourg 6, pp. 141–157.
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Asymptotic estimates for generalized Stirling numbers.
Analysis (Munich) 20 (1), pp. 1–13.
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Validated computation of certain hypergeometric functions.
ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
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Coulomb effects in the Klein-Gordon equation for pions.
Phys. Rev. C 20 (2), pp. 696–704.
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12: 26.12 Plane Partitions
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►A plane partition, , of a positive integer , is a partition of in which the parts have been arranged in a 2-dimensional array that is weakly decreasing (nonincreasing) across rows and down columns.
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►The plane partition in Figure 26.12.1 is an example of a cyclically symmetric plane partition.
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►A plane partition is totally symmetric if it is both symmetric and cyclically symmetric.
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►The example of a strict shifted plane partition also satisfies the conditions of a descending plane partition.
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§26.12(iv) Limiting Form
…13: 35.5 Bessel Functions of Matrix Argument
§35.5 Bessel Functions of Matrix Argument
►§35.5(i) Definitions
… ►§35.5(ii) Properties
… ►§35.5(iii) Asymptotic Approximations
►For asymptotic approximations for Bessel functions of matrix argument, see Herz (1955) and Butler and Wood (2003).14: 8.26 Tables
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Khamis (1965) tabulates for , to 10D.
Pagurova (1963) tabulates and (with different notation) for , to 7D.
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.
15: 6.16 Mathematical Applications
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►These limits are not approached uniformly, however.
…Hence if and , then the limiting value of overshoots by approximately 18%.
Similarly if , then the limiting value of undershoots by approximately 10%, and so on.
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►It occurs with Fourier-series expansions of all piecewise continuous functions.
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16: 33.18 Limiting Forms for Large
§33.18 Limiting Forms for Large
…17: 23 Weierstrass Elliptic and Modular
Functions
Chapter 23 Weierstrass Elliptic and Modular Functions
…18: 18.40 Methods of Computation
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►Orthogonal polynomials can be computed from their explicit polynomial form by Horner’s scheme (§1.11(i)).
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►Given the power moments, , , can these be used to find a unique , a non-decreasing, real, function of , in the case that the moment problem is determined? Should a unique solution not exist the moment problem is then indeterminant.
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►It is now necessary to take the limit
of , and the imaginary part is the required Stieltjes–Perron inversion:
…Results of low ( to decimal digits) precision for are easily obtained for to .
Gautschi (2004, p. 119–120) has explored the
limit via the Wynn -algorithm, (3.9.11) to accelerate convergence, finding four to eight digits of precision in , depending smoothly on , for , for an example involving first numerator Legendre OP’s.
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19: Bibliography N
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On an integral transform involving a class of Mathieu functions.
SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
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Reduction and evaluation of elliptic integrals.
Math. Comp. 20 (94), pp. 223–231.
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Coulomb Functions for Large Values of the Parameter
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Technical report
Atomic Energy of Canada Limited, Chalk
River, Ontario.
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A table of integrals of the error functions.
J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
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Special Functions of Mathematical Physics: A Unified Introduction with Applications.
Birkhäuser Verlag, Basel.
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20: 27.2 Functions
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►Functions in this section derive their properties from the fundamental
theorem of arithmetic, which states that every integer can be represented uniquely as a product of prime powers,
…Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
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►An equivalent form states that the th prime (when the primes are listed in increasing order) is asymptotic to as :
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►Such a set is a reduced
residue system modulo .
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►where is a prime power with ; otherwise .
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