asymptotic formula
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31: Gergő Nemes
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►Nemes has research interests in asymptotic analysis, Écalle theory, exact WKB analysis, and special functions.
►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions.
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32: 6.18 Methods of Computation
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►For large and , expansions in inverse factorial series (§6.10(i)) or asymptotic expansions (§6.12) are available.
The attainable accuracy of the asymptotic expansions can be increased considerably by exponential improvement.
Also, other ranges of can be covered by use of the continuation formulas of §6.4.
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►For example, the Gauss–Laguerre formula (§3.5(v)) can be applied to (6.2.2); see Todd (1954) and Tseng and Lee (1998).
For an application of the Gauss–Legendre formula (§3.5(v)) see Tooper and Mark (1968).
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33: 5.11 Asymptotic Expansions
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►For explicit formulas for in terms of Stirling numbers see Nemes (2013a), and for asymptotic expansions of as see Boyd (1994) and Nemes (2015a).
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34: 5.21 Methods of Computation
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►An effective way of computing in the right half-plane is backward recurrence, beginning with a value generated from the asymptotic expansion (5.11.3).
…For the left half-plane we can continue the backward recurrence or make use of the reflection formula (5.5.3).
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35: 10.49 Explicit Formulas
§10.49 Explicit Formulas
►§10.49(i) Unmodified Functions
… ►§10.49(ii) Modified Functions
… ►§10.49(iii) Rayleigh’s Formulas
… ►§10.49(iv) Sums or Differences of Squares
…36: 35.9 Applications
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►These references all use results related to the integral formulas (35.4.7) and (35.5.8).
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►The asymptotic approximations of §35.7(iv) are applied in numerous statistical contexts in Butler and Wood (2002).
►In chemistry, Wei and Eichinger (1993) expresses the probability density functions of macromolecules in terms of generalized hypergeometric functions of matrix argument, and develop asymptotic approximations for these density functions.
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37: 2.3 Integrals of a Real Variable
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►Then
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►For the Fourier integral
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§2.3(ii) Watson’s Lemma
… ►Then … ►§2.3(vi) Asymptotics of Mellin Transforms
…38: 32.11 Asymptotic Approximations for Real Variables
§32.11 Asymptotic Approximations for Real Variables
… ►Connection formulas for and are given by … ►The connection formulas for are … ►The connection formulas relating (32.11.25) and (32.11.26) are … ►Connection formulas for and are given by …39: 8.11 Asymptotic Approximations and Expansions
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►This reference also contains explicit formulas for in terms of Stirling numbers and for the case an asymptotic expansion for as .
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40: 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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