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1: 2.11 Remainder Terms; Stokes Phenomenon
Secondly, the asymptotic series represents an infinite class of functions, and the remainder depends on which member we have in mind. … In the transition through θ = π , erfc ( 1 2 ρ c ( θ ) ) changes very rapidly, but smoothly, from one form to the other; compare the graph of its modulus in Figure 2.11.1 in the case ρ = 100 . … uniformly with respect to ph z in each case. …
2: 23.23 Tables
2 in Abramowitz and Stegun (1964) gives values of ( z ) , ( z ) , and ζ ( z ) to 7 or 8D in the rectangular and rhombic cases, normalized so that ω 1 = 1 and ω 3 = i a (rectangular case), or ω 1 = 1 and ω 3 = 1 2 + i a (rhombic case), for a = 1. …05, and in the case of ( z ) the user may deduce values for complex z by application of the addition theorem (23.10.1). …
3: 2.6 Distributional Methods
2.6.6 S ( x ) 2 π 3 s = 0 ( - 1 ) s ( - 1 3 s ) x - s - ( 1 / 3 ) , x .
Similarly, in the case α = 1 , we define …In either case, we define the distribution associated with f n ( t ) by …
4: 18.31 Bernstein–Szegő Polynomials
In consequence, p n ( cos θ ) can be given explicitly in terms of ρ ( cos θ ) and sines and cosines, provided that < 2 n in the first case, < 2 n + 2 in the second case, and < 2 n + 1 in the third case. …
5: 12.18 Methods of Computation
Because PCFs are special cases of confluent hypergeometric functions, the methods of computation described in §13.29 are applicable to PCFs. …
6: 17.16 Mathematical Applications
Many special cases of q -series arise in the theory of partitions, a topic treated in §§27.14(i) and 26.9. …
7: 16.18 Special Cases
§16.18 Special Cases
The F 1 1 and F 1 2 functions introduced in Chapters 13 and 15, as well as the more general F q p functions introduced in the present chapter, are all special cases of the Meijer G -function. …As a corollary, special cases of the F 1 1 and F 1 2 functions, including Airy functions, Bessel functions, parabolic cylinder functions, Ferrers functions, associated Legendre functions, and many orthogonal polynomials, are all special cases of the Meijer G -function. …
8: 12.1 Special Notation
Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. … Whittaker’s notation D ν ( z ) is useful when ν is a nonnegative integer (Hermite polynomial case).
9: Possible Errors in DLMF
There are also cases where browser bugs or poor fonts can be misleading; you can verify MathML display by comparing the to the images or TeXfound under Encodings in the Info boxes (see About MathML). …
10: 16.25 Methods of Computation
In these cases integration, or recurrence, in either a forward or a backward direction is unstable. …