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21: 28.8 Asymptotic Expansions for Large $q$

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28.8.1

\rselection{a_{m}\left(h^{2}\right)\\ b_{m+1}\left(h^{2}\right)}\sim-2h^{2}+2sh-\frac{1}{8}(s^{2}+1)-\frac{1}{2^{7}h% }(s^{3}+3s)-\frac{1}{2^{12}h^{2}}(5s^{4}+34s^{2}+9)-\frac{1}{2^{17}h^{3}}(33s^% {5}+410s^{3}+405s)-\frac{1}{2^{20}h^{4}}(63s^{6}+1260s^{4}+2943s^{2}+486)-% \frac{1}{2^{25}h^{5}}(527s^{7}+15617s^{5}+69001s^{3}+41607s)+\cdots.

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Symbols:: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $\sim$ : Poincaré asymptotic expansion, $m$ : integer and $h$ : parameter
A&S Ref:: 20.2.30 (in different form)
Referenced by:: §28.8(i), Erratum (V1.1.10) for Sections 28.6(i), 28.6(ii), 28.8(i), 28.8(ii)
Permalink:: http://dlmf.nist.gov/28.8.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §28.8(i), §28.8 and Ch.28

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28.8.2

b_{m+1}\left(h^{2}\right)-a_{m}\left(h^{2}\right)=\frac{2^{4m+5}}{m!}\left(% \frac{2}{\pi}\right)^{\ifrac{1}{2}}h^{m+(\ifrac{3}{2})}e^{-4h}\*{\left(1-\frac% {6m^{2}+14m+7}{32h}+O\left(\frac{1}{h^{2}}\right)\right)}.

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $m$ : integer and $h$ : parameter
A&S Ref:: 20.2.31
Permalink:: http://dlmf.nist.gov/28.8.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §28.8(i), §28.8 and Ch.28

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28.8.6

\widehat{C}_{m}\sim\left(\frac{\pi h}{2(m!)^{2}}\right)^{\ifrac{1}{4}}\left(1+% \frac{2m+1}{8h}+\dfrac{m^{4}+2m^{3}+263m^{2}+262m+108}{2048h^{2}}+\cdots\right% )^{-\ifrac{1}{2}},

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Defines:: $\widehat{C}_{m}$ : coefficient (locally)
Symbols:: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $m$ : integer and $h$ : parameter
A&S Ref:: 20.9.19 (in slightly different form)
Permalink:: http://dlmf.nist.gov/28.8.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §28.8(ii), §28.8 and Ch.28

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28.8.7

\widehat{S}_{m}\sim\left(\frac{\pi h}{2(m!)^{2}}\right)^{\ifrac{1}{4}}\left(1-% \frac{2m+1}{8h}+\dfrac{m^{4}+2m^{3}-121m^{2}-122m-84}{2048h^{2}}+\cdots\right)% ^{-\ifrac{1}{2}}.

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Defines:: $\widehat{S}_{m}$ : coefficient (locally)
Symbols:: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $m$ : integer and $h$ : parameter
A&S Ref:: 20.9.20 (in slightly different form)
Referenced by:: §28.8(ii), Erratum (V1.1.10) for Sections 28.6(i), 28.6(ii), 28.8(i), 28.8(ii)
Permalink:: http://dlmf.nist.gov/28.8.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §28.8(ii), §28.8 and Ch.28

… ►With additional restrictions on

z

, uniform asymptotic approximations for solutions of (28.2.1) and (28.20.1) are also obtained in terms of elementary functions by re-expansions of the Whittaker functions; compare §2.8(ii). …

22: 36 Integrals with Coalescing Saddles

…

23: Gergő Nemes

… ►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. …

24: Wolter Groenevelt

… ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …

25: Bibliography B

… ►

G. Backenstoss (1970) Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

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External Links:: Document
Referenced by:: §33.22(iv).
Encodings:: BibTeX

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A. Bañuelos and R. A. Depine (1980) A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.

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Notes:: Double-precision Fortran, maximum accuracy 10S.
External Links:: Document, Code, ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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M. N. Barber and B. W. Ninham (1970) Random and Restricted Walks: Theory and Applications. Gordon and Breach, New York.

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External Links:: ZentralBlatt
Referenced by:: §16.24(i).
Encodings:: BibTeX

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K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:: Double-precision Fortran code.
External Links:: Document, Code
Referenced by:: 1st item, 4th item.
Encodings:: BibTeX

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W. G. Bickley (1935) Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

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External Links:: Document, ZentralBlatt
Referenced by:: §10.73(iv).
Encodings:: BibTeX

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26: 33.24 Tables

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•

Abramowitz and Stegun (1964, Chapter 14) tabulates $F_{0}\left(\eta,\rho\right)$ , $G_{0}\left(\eta,\rho\right)$ , $F_{0}'\left(\eta,\rho\right)$ , and $G_{0}'\left(\eta,\rho\right)$ for $\eta=0.5(.5)20$ and $\rho=1(1)20$ , 5S; $C_{0}\left(\eta\right)$ for $\eta=0(.05)3$ , 6S.

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27: 19.36 Methods of Computation

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19.36.2

1-\tfrac{3}{14}E_{2}+\tfrac{1}{6}E_{3}+\tfrac{9}{88}E_{2}^{2}-\tfrac{3}{22}E_{% 4}-\tfrac{9}{52}E_{2}E_{3}+\tfrac{3}{26}E_{5}-\tfrac{1}{16}E_{2}^{3}+\tfrac{3}% {40}E_{3}^{2}+\tfrac{3}{20}E_{2}E_{4}+\tfrac{45}{272}E_{2}^{2}E_{3}-\tfrac{9}{% 68}(E_{3}E_{4}+E_{2}E_{5}).

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Symbols:: $E_{s}(\mathbf{z})$ : symmetric function
Referenced by:: §19.19
Permalink:: http://dlmf.nist.gov/19.36.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.36(i), §19.36 and Ch.19

… ►Complex values of the variables are allowed, with some restrictions in the case of

R_{J}

that are sufficient but not always necessary. … ►

19.36.8

R_{F}\left(t_{n}^{2},t_{n}^{2}+\theta c_{n}^{2},t_{n}^{2}+\theta a_{n}^{2}\right)

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $n$ : nonnegative integer
Referenced by:: §19.36(ii)
Permalink:: http://dlmf.nist.gov/19.36.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §19.36(ii), §19.36 and Ch.19

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19.36.13

2R_{G}\left(t_{0}^{2},t_{0}^{2}+\theta c_{0}^{2},t_{0}^{2}+\theta a_{0}^{2}% \right)=\left(t_{0}^{2}+\theta\sum_{m=0}^{\infty}2^{m-1}c_{m}^{2}\right)R_{C}% \left(T^{2}+\theta M^{2},T^{2}\right)+h_{0}+\sum_{m=1}^{\infty}2^{m}(h_{m}-h_{% m-1}).

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Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $m$ : nonnegative integer, $M$ : limit, $T$ : limit and $h_{n}$
Permalink:: http://dlmf.nist.gov/19.36.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §19.36(ii), §19.36(ii), §19.36 and Ch.19

… ►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). …

28: William P. Reinhardt

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20 Theta Functions

… ►In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.

29: 27.2 Functions

… ►Functions in this section derive their properties from the fundamental theorem of arithmetic, which states that every integer

n>1

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. …They tend to thin out among the large integers, but this thinning out is not completely regular. … ►the sum of the

k

th powers of the positive integers

m\leq n

that are relatively prime to

n

. … ►is the number of

k

-tuples of integers

\leq n

whose greatest common divisor is relatively prime to

n

. …

30: Bibliography

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M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

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External Links:: Document, MathReview (Mark Adler)
Referenced by:: §32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.
Encodings:: BibTeX

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A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

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External Links:: ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.12(iii).
Encodings:: BibTeX

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S. V. Aksenov, M. A. Savageau, U. D. Jentschura, J. Becher, G. Soff, and P. J. Mohr (2003) Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.

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Notes:: Includes algorithms for Lerch’s transcendent. C and Mathematica codes by Aksenov and Jentschura are available.
External Links:: Code, Document
Referenced by:: 1st item.
Encodings:: BibTeX

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D. E. Amos (1989) Repeated integrals and derivatives of

K

Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

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External Links:: ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt
Referenced by:: §10.43(iii).
Encodings:: BibTeX

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G. E. Andrews (1979) Plane partitions. III. The weak Macdonald conjecture. Invent. Math. 53 (3), pp. 193–225.

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External Links:: ISSN 0020-9910, Document, MathReview (Edward A. Bender), ZentralBlatt
Referenced by:: §26.12(i), §26.12(ii).
Encodings:: BibTeX

…