DLMF: Untitled Document

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J. Van Deun and R. Cools (2008) Integrating products of Bessel functions with an additional exponential or rational factor. Comput. Phys. Comm. 178 (8), pp. 578–590.

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Notes:: Associated computer program available online
External Links:: ISSN 0010-4655, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §10.74(vii), §10.74(vii).
Encodings:: BibTeX

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B. Ph. van Milligen and A. López Fraguas (1994) Expansion of vacuum magnetic fields in toroidal harmonics. Comput. Phys. Comm. 81 (1-2), pp. 74–90.

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External Links:: Document
Referenced by:: §14.31(i).
Encodings:: BibTeX

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L. Vietoris (1983) Dritter Beweis der die unvollständige Gammafunktion betreffenden Lochsschen Ungleichungen. Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 192 (1-3), pp. 83–91 (German).

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External Links:: ISSN 0723-9319, MathReview (Peter Schroth), ZentralBlatt
Referenced by:: §8.10.
Encodings:: BibTeX

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H. Volkmer (2023) Asymptotic expansion of the generalized hypergeometric function

{}_{p}F_{q}(z)

as

z\to\infty

for

p<q

. Anal. Appl. (Singap.) 21 (2), pp. 535–545.

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External Links:: ISSN 0219-5305, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §16.11(i), §16.11, Erratum (V1.1.9) for Section 16.11(i).
Encodings:: BibTeX

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19.11.1

F\left(\theta,k\right)+F\left(\phi,k\right)=F\left(\psi,k\right),

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Symbols:: $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\phi$ : real or complex argument, $k$ : real or complex modulus, $\theta$ : amplitude and $\psi$ : angle
Referenced by:: §19.11(i)
Permalink:: http://dlmf.nist.gov/19.11.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.11(i), §19.11 and Ch.19

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19.11.7

F\left(\phi,k\right)=K\left(k\right)-F\left(\theta,k\right),

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Symbols:: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\theta$ : amplitude
Permalink:: http://dlmf.nist.gov/19.11.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §19.11(ii), §19.11 and Ch.19

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19.11.12

F\left(\psi,k\right)=2F\left(\theta,k\right),

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Symbols:: $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $k$ : real or complex modulus, $\theta$ : amplitude and $\psi$ : angle
Permalink:: http://dlmf.nist.gov/19.11.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §19.11(iii), §19.11 and Ch.19

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R. Barakat and E. Parshall (1996) Numerical evaluation of the zero-order Hankel transform using Filon quadrature philosophy. Appl. Math. Lett. 9 (5), pp. 21–26.

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External Links:: ISSN 0893-9659, Document, MathReview, ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

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K. Bay, W. Lay, and A. Akopyan (1997) Avoided crossings of the quartic oscillator. J. Phys. A 30 (9), pp. 3057–3067.

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External Links:: ISSN 0305-4470, Document, MathReview (Gabriel Alvarez), ZentralBlatt
Referenced by:: §31.17(ii).
Encodings:: BibTeX

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P. L. Butzer, M. Hauss, and M. Leclerc (1992) Bernoulli numbers and polynomials of arbitrary complex indices. Appl. Math. Lett. 5 (6), pp. 83–88.

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External Links:: ISSN 0893-9659, MathReview, ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

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P. L. Butzer and M. Hauss (1992) Riemann zeta function: Rapidly converging series and integral representations. Appl. Math. Lett. 5 (2), pp. 83–88.

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External Links:: ISSN 0893-9659, Document, MathReview (Jacek Fabrykowski), ZentralBlatt
Referenced by:: §25.18(i).
Encodings:: BibTeX

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J. G. Byatt-Smith (2000) The Borel transform and its use in the summation of asymptotic expansions. Stud. Appl. Math. 105 (2), pp. 83–113.

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External Links:: ISSN 0022-2526, Document, MathReview (A. D. Wood), ZentralBlatt
Referenced by:: §2.11(v).
Encodings:: BibTeX

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23.10.13

\sigma\left(nz\right)=A_{n}e^{-n(n-1)(\eta_{1}+\eta_{3})z}\prod_{j=0}^{n-1}% \prod_{\ell=0}^{n-1}\sigma\left(z+\frac{2j}{n}\omega_{1}+\frac{2\ell}{n}\omega% _{3}\right),

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Symbols:: $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\mathrm{e}$ : base of natural logarithm, $\mathbb{L}$ : lattice, $n$ : integer, $z$ : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $A_{n}$ : coefficients and $\eta_{j}$ : complex number
Referenced by:: §23.10(iii)
Permalink:: http://dlmf.nist.gov/23.10.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §23.10(iii), §23.10 and Ch.23

►where ►

23.10.14

A_{n}=n\prod_{j=0}^{n-1}\prod_{\begin{subarray}{c}\ell=0\\ \ell\neq j\end{subarray}}^{n-1}\frac{1}{\sigma\left((2j\omega_{1}+2\ell\omega_% {3})/n\right)}.

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Symbols:: $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\mathbb{L}$ : lattice, $n$ : integer, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $A_{n}$ : coefficients
Referenced by:: §23.10(iii)
Permalink:: http://dlmf.nist.gov/23.10.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §23.10(iii), §23.10 and Ch.23

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23.10.15

A_{n}=\left(\frac{\pi^{2}G^{2}}{\omega_{1}}\right)^{n^{2}-1}\frac{q^{n(n-1)/2}% }{i^{n-1}}\exp\left(-\frac{(n-1)\eta_{1}}{3\omega_{1}}\left((2n-1)(\omega_{1}^% {2}+\omega_{3}^{2})+3(n-1)\omega_{1}\omega_{3}\right)\right),

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $q$ : nome, $n$ : integer, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $A_{n}$ : coefficients, $G$ and $\eta_{j}$ : complex number
Referenced by:: §23.10(iii)
Permalink:: http://dlmf.nist.gov/23.10.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §23.10(iii), §23.10 and Ch.23

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M. Abramowitz and P. Rabinowitz (1954) Evaluation of Coulomb wave functions along the transition line. Physical Rev. (2) 96, pp. 77–79.

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External Links:: Document, MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §33.12(i).
Encodings:: BibTeX

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G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen (1990) Functional inequalities for complete elliptic integrals and their ratios. SIAM J. Math. Anal. 21 (2), pp. 536–549.

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External Links:: ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt
Referenced by:: §19.9(i).
Encodings:: BibTeX

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T. M. Apostol (1952) Theorems on generalized Dedekind sums. Pacific J. Math. 2 (1), pp. 1–9.

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External Links:: MathReview (N. G. de Bruijn), ZentralBlatt
Referenced by:: (25.11.41), (25.11.42), §25.11(xii).
Encodings:: BibTeX

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H. Appel (1968) Numerical Tables for Angular Correlation Computations in

\alpha

-,

\beta

- and

\gamma

-Spectroscopy:

3j

-,

6j

-,

9j

-Symbols, F- and

\Gamma

-Coefficients. Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology, Springer-Verlag.

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External Links:: ISBN 978-3-540-04218-1
Referenced by:: §34.14.
Encodings:: BibTeX

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F. M. Arscott (1959) A new treatment of the ellipsoidal wave equation. Proc. London Math. Soc. (3) 9, pp. 21–50.

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External Links:: ISSN 0024-6115, Document, MathReview (J. Meixner), ZentralBlatt
Referenced by:: §29.11.
Encodings:: BibTeX

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H. E. Salzer (1955) Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms. Math. Tables Aids Comput. 9 (52), pp. 164–177.

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External Links:: FullText, MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §3.5(viii).
Encodings:: BibTeX

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T. C. Scott, G. Fee, and J. Grotendorst (2013) Asymptotic series of generalized Lambert

W

function. ACM Commun. Comput. Algebra 47 (3), pp. 75–83.

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External Links:: FullText, ZentralBlatt
Referenced by:: §4.13, §4.13.
Encodings:: BibTeX

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T. Shiota (1986) Characterization of Jacobian varieties in terms of soliton equations. Invent. Math. 83 (2), pp. 333–382.

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External Links:: ISSN 0020-9910, Document, MathReview (Bert van Geemen), ZentralBlatt
Referenced by:: §21.9.
Encodings:: BibTeX

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K. Srinivasa Rao, V. Rajeswari, and C. B. Chiu (1989) A new Fortran program for the

9

-

j

angular momentum coefficient. Comput. Phys. Comm. 56 (2), pp. 231–248.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §34.13.
Encodings:: BibTeX

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S. K. Suslov (2003) An Introduction to Basic Fourier Series. Developments in Mathematics, Vol. 9, Kluwer Academic Publishers, Dordrecht.

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External Links:: ISBN 1-4020-1221-7, MathReview (P. D. F. Ion), ZentralBlatt
Referenced by:: §17.3(i).
Encodings:: BibTeX

…

… ►For more inequalities involving the exponential function see Mitrinović (1964, pp. 73–77), Mitrinović (1970, pp. 266–271), and Bullen (1998, pp. 81–83).

… ►

K. Kajiwara and Y. Ohta (1996) Determinant structure of the rational solutions for the Painlevé II equation. J. Math. Phys. 37 (9), pp. 4693–4704.

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External Links:: ISSN 0022-2488, Document, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.8(ii).
Encodings:: BibTeX

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M. Kaneko (1997) Poly-Bernoulli numbers. J. Théor. Nombres Bordeaux 9 (1), pp. 221–228.

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External Links:: ISSN 1246-7405, PostScript, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

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M. K. Kerimov (2008) Overview of some new results concerning the theory and applications of the Rayleigh special function. Comput. Math. Math. Phys. 48 (9), pp. 1454–1507.

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External Links:: ISSN 0965-5425, Document, FullText
Referenced by:: §10.21(xiii), §10.21(xiii).
Encodings:: BibTeX

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E. Kreyszig (1957) On the zeros of the Fresnel integrals. Canad. J. Math. 9, pp. 118–131.

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External Links:: MathReview (W. C. Rheinboldt), ZentralBlatt
Referenced by:: §7.13(iii).
Encodings:: BibTeX

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M. D. Kruskal (1974) The Korteweg-de Vries Equation and Related Evolution Equations. In Nonlinear Wave Motion (Proc. AMS-SIAM Summer Sem., Clarkson Coll. Tech., Potsdam, N.Y., 1972), A. C. Newell (Ed.), Lectures in Appl. Math., Vol. 15, pp. 61–83.

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External Links:: MathReview (G. L. Lamb, Jr.), ZentralBlatt
Referenced by:: §22.19(iii).
Encodings:: BibTeX

…

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P. I. Hadži (1973) The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).

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External Links:: MathReview (L. Berg)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

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P. I. Hadži (1975b) Integrals containing the Fresnel functions

S(x)

and

C(x)

. Bul. Akad. Štiince RSS Moldoven. 1975 (3), pp. 48–60, 93 (Russian).

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External Links:: MathReview (J. Kaczmarski)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

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B. Hayes (2009) The higher arithmetic. American Scientist 97, pp. 364–368.

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External Links:: FullText
Referenced by:: §3.1(iv).
Encodings:: BibTeX

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J. R. Herndon (1961b) Algorithm 56: Complete elliptic integral of the second kind. Comm. ACM 4 (4), pp. 180–181.

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Notes:: Includes Algol program, maximum accuracy 8D. For certification see same journal 9 (1966), p. 12
External Links:: ISSN 0001-0782, Document
Referenced by:: §19.39(ii).
Encodings:: BibTeX

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M. H. Hull and G. Breit (1959) Coulomb Wave Functions. In Handbuch der Physik, Bd. 41/1, S. Flügge (Ed.), pp. 408–465.

ⓘ

External Links:: MathReview (G. Källén)
Referenced by:: §33.2(iii), §33.23(vii), §33.23(vii), §33.24, §33.5(ii), §33.7, Ch.33.
Encodings:: BibTeX

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10.21.19

j_{\nu,m},y_{\nu,m}\sim a-\frac{\mu-1}{8a}-\frac{4(\mu-1)(7\mu-31)}{3(8a)^{3}}% -\frac{32(\mu-1)(83\mu^{2}-982\mu+3779)}{15(8a)^{5}}-\frac{64(\mu-1)(6949\mu^{% 3}-1\;53855\mu^{2}+15\;85743\mu-62\;77237)}{105(8a)^{7}}-\dotsb,

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $j_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $J_{\nu}\left(x\right)$ , $y_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $Y_{\nu}\left(x\right)$ , $m$ : integer and $\nu$ : complex parameter
A&S Ref:: 9.5.12
Referenced by:: §10.21(vi), §10.21(vi)
Permalink:: http://dlmf.nist.gov/10.21.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §10.21(vi), §10.21 and Ch.10

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10.21.20

j_{\nu,m}',y_{\nu,m}'\sim b-\frac{\mu+3}{8b}-\frac{4(7\mu^{2}+82\mu-9)}{3(8b)^% {3}}-\frac{32(83\mu^{3}+2075\mu^{2}-3039\mu+3537)}{15(8b)^{5}}-\frac{64(6949% \mu^{4}+2\;96492\mu^{3}-12\;48002\mu^{2}+74\;14380\mu-58\;53627)}{105(8b)^{7}}% -\dotsb,

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $j_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $J_{\nu}\left(x\right)$ , $y_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $Y_{\nu}\left(x\right)$ , $m$ : integer and $\nu$ : complex parameter
A&S Ref:: 9.5.13
Referenced by:: §10.21(vi), §10.21(vi)
Permalink:: http://dlmf.nist.gov/10.21.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §10.21(vi), §10.21 and Ch.10

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10.21.21

j^{\prime\prime}_{\nu,m}\sim c-\frac{\mu+7}{8c}-\frac{28\mu^{2}+424\mu+1724}{3% (8c)^{3}}-\dotsb,

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Symbols:: $\sim$ : Poincaré asymptotic expansion, $m$ : integer, $\nu$ : complex parameter and $c$ : zero of cylinder function
Permalink:: http://dlmf.nist.gov/10.21.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §10.21(vi), §10.21 and Ch.10

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\beta^{\prime}_{2}=\tfrac{9}{350}{\alpha^{\prime}}^{2}+\tfrac{1}{100}{\alpha^{% \prime}}^{-1},

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J_{\nu}'\left(j_{\nu,1}\right)\sim-1.11310\;28\nu^{-\frac{2}{3}}\div(1+1.48460% \;6\nu^{-\frac{2}{3}}+0.43294\nu^{-\frac{4}{3}}-0.1943\nu^{-2}+0.019\nu^{-% \frac{8}{3}}+\cdots),

…

%E5%A4%A7%E9%83%BD%E4%BC%9A%E6%97%B6%E6%97%B6%E5%BD%A9%E7%BD%91%E5%9D%80%E3%80%90%E6%9D%8F%E5%BD%A9%E4%BD%93%E8%82%B2qee9.com%E3%80%919fdS

11—20 of 534 matching pages

11: Bibliography V

12: 19.11 Addition Theorems

13: Bibliography B

14: 23.10 Addition Theorems and Other Identities

15: Bibliography

16: Bibliography S

17: 4.5 Inequalities

18: Bibliography K

19: Bibliography H

20: 10.21 Zeros