constant term
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21—30 of 140 matching pages
21: 11.6 Asymptotic Expansions
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§11.6(i) Large , Fixed
… ►where is an arbitrary small positive constant. If the series on the right-hand side of (11.6.1) is truncated after terms, then the remainder term is . … ►For re-expansions of the remainder terms in (11.6.1) and (11.6.2), see Dingle (1973, p. 445). … ►where is Euler’s constant (§5.2(ii)). …22: 18.18 Sums
23: 14.23 Values on the Cut
24: 32.10 Special Function Solutions
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has solutions expressible in terms of Airy functions (§9.2) iff
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then has solutions expressible in terms of Bessel functions (§10.2) iff
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has solutions expressible in terms of parabolic cylinder functions (§12.2) iff either
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then has solutions expressible in terms of Whittaker functions (§13.14(i)), iff
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has solutions expressible in terms of hypergeometric functions (§15.2(i)) iff
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25: 31.10 Integral Equations and Representations
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►Fuchs–Frobenius solutions are represented in terms of Heun functions by (31.10.1) with , , and with kernel chosen from
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26: 8.20 Asymptotic Expansions of
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8.20.2
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8.20.3
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again denoting an arbitrary small positive constant.
Where the sectors of validity of (8.20.2) and (8.20.3) overlap the contribution of the first term on the right-hand side of (8.20.3) is exponentially small compared to the other contribution; compare §2.11(ii).
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27: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
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10.46.1
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►The Laplace transform of can be expressed in terms of the Mittag-Leffler function:
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10.46.3
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28: 25.8 Sums
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25.8.3
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29: 8.22 Mathematical Applications
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8.22.1
►plays a fundamental role in re-expansions of remainder terms in asymptotic expansions, including exponentially-improved expansions and a smooth interpretation of the Stokes phenomenon.
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►The function , with and , has an intimate connection with the Riemann zeta function (§25.2(i)) on the critical line .
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8.22.2
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