DLMF: Untitled Document

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FDLIBM (free C library)

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Notes:: The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.
External Links:: GAMS (package FDLIBM)
Referenced by:: § â€£ Software Cross Index, § â€£ Software Cross Index.
Encodings:: BibTeX

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S. Fempl (1960) Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

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External Links:: MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §6.15.
Encodings:: BibTeX

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H. E. Fettis and J. C. Caslin (1964) Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

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Notes:: Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.
Referenced by:: §19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).
Encodings:: BibTeX

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A. Fletcher (1948) Guide to tables of elliptic functions. Math. Tables and Other Aids to Computation 3 (24), pp. 229–281.

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External Links:: FullText, MathReview (S. C. van Veen)
Referenced by:: §19.37(i).
Encodings:: BibTeX

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G. Freud (1969) On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

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External Links:: ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)
Referenced by:: §18.32.
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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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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S. Bochner (1952) Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.

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External Links:: MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §35.5(i).
Encodings:: BibTeX

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R. Bulirsch (1969b) Numerical calculation of elliptic integrals and elliptic functions. III. Numer. Math. 13 (4), pp. 305–315.

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Notes:: Algol 60 programs for the incomplete elliptic integral of the third kind, and for a general complete elliptic integral, are included.
External Links:: ISSN 0029-599X, Document, MathReview, ZentralBlatt
Referenced by:: §19.2(iii), §19.36(ii), §19.36(ii), §19.39(ii), §19.39(iii), R. Bulirsch (1965a).
Encodings:: BibTeX

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R. Bulirsch (1965b) Numerical calculation of elliptic integrals and elliptic functions. Numer. Math. 7 (1), pp. 78–90.

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Notes:: Algol 60 programs are included for real and complex elliptic integrals, and for Jacobian elliptic functions of real argument.
External Links:: ISSN 0029-599X, Document, MathReview (M. Feliciano), ZentralBlatt
Referenced by:: §19.1, §19.2(iii), §19.36(ii), §19.39(iii), §22.22(ii).
Encodings:: BibTeX

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§20.11 Generalizations and Analogs

… â–ºAs in §20.11(ii), the modulus

k

of elliptic integrals (§19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in

q

-series via (20.9.1). … … â–ºFor applications to rapidly convergent expansions for

\pi

see Chudnovsky and Chudnovsky (1988), and for applications in the construction of elliptic-hypergeometric series see Rosengren (2004). â–º

§20.11(iv) Theta Functions with Characteristics

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S. Lang (1987) Elliptic Functions. 2nd edition, Graduate Texts in Mathematics, Vol. 112, Springer-Verlag, New York.

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Notes:: With an appendix by J. Tate
External Links:: ISBN 0-387-96508-4, MathReview, ZentralBlatt
Referenced by:: §22.18(iv).
Encodings:: BibTeX

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P. W. Lawrence, R. M. Corless, and D. J. Jeffrey (2012) Algorithm 917: complex double-precision evaluation of the Wright

\omega

function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.

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External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §4.13, 2nd item, §4.48(iv).
Encodings:: BibTeX

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D. K. Lee (1990) Application of theta functions for numerical evaluation of complete elliptic integrals of the first and second kinds. Comput. Phys. Comm. 60 (3), pp. 319–327.

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External Links:: ISSN 0010-4655, Document, MathReview, ZentralBlatt
Referenced by:: §19.36(iii), §19.5.
Encodings:: BibTeX

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D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

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External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.11.
Encodings:: BibTeX

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Y. L. Luke (1968) Approximations for elliptic integrals. Math. Comp. 22 (103), pp. 627–634.

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External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §19.38.
Encodings:: BibTeX

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J. Igusa (1972) Theta Functions. Springer-Verlag, New York.

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Notes:: Die Grundlehren der mathematischen Wissenschaften, Band 194
External Links:: ISBN 3540056998, MathReview (H. Klingen), ZentralBlatt
Referenced by:: §21.5(ii), §21.8, Ch.21.
Encodings:: BibTeX

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Y. Ikebe, Y. Kikuchi, and I. Fujishiro (1991) Computing zeros and orders of Bessel functions. J. Comput. Appl. Math. 38 (1-3), pp. 169–184.

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External Links:: ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt
Referenced by:: §10.74(vi), §3.8(iii).
Encodings:: BibTeX

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E. L. Ince (1932) Tables of the elliptic cylinder functions. Proc. Roy. Soc. Edinburgh Sect. A 52, pp. 355–433.

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External Links:: ZentralBlatt
Referenced by:: 4th item, 2nd item.
Encodings:: BibTeX

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E. L. Ince (1940a) The periodic Lamé functions. Proc. Roy. Soc. Edinburgh 60, pp. 47–63.

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External Links:: MathReview (H. Bateman), ZentralBlatt
Referenced by:: §29.1, §29.15(ii), 1st item, §29.7(i).
Encodings:: BibTeX

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K. Inkeri (1959) The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

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External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.12(i).
Encodings:: BibTeX

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N. I. Akhiezer (1990) Elements of the Theory of Elliptic Functions. Translations of Mathematical Monographs, Vol. 79, American Mathematical Society, Providence, RI.

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Notes:: Translated from the second Russian edition by H. H. McFaden
External Links:: ISBN 0-8218-4532-2, MathReview (Glenn Stevens), ZentralBlatt
Referenced by:: §22.18(ii), §22.18(iv).
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. 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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G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen (1992a) Functional inequalities for hypergeometric functions and complete elliptic integrals. SIAM J. Math. Anal. 23 (2), pp. 512–524.

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External Links:: ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt
Referenced by:: §19.9(i).
Encodings:: BibTeX

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G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen (1992b) Hypergeometric Functions and Elliptic Integrals. In Current Topics in Analytic Function Theory, H. M. Srivastava and S. Owa (Eds.), pp. 48–85.

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External Links:: MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: §19.9(i).
Encodings:: BibTeX

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§23.14 Integrals

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23.14.1

\int\wp\left(z\right)\,\mathrm{d}z=-\zeta\left(z\right),

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Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathbb{L}$ : lattice and $z$ : complex
A&S Ref:: 18.6.1
Permalink:: http://dlmf.nist.gov/23.14.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §23.14 and Ch.23

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23.14.2

\int{\wp}^{2}\left(z\right)\,\mathrm{d}z=\frac{1}{6}\wp'\left(z\right)+\frac{1% }{12}g_{2}z,

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Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathbb{L}$ : lattice and $z$ : complex
A&S Ref:: 18.7.1
Permalink:: http://dlmf.nist.gov/23.14.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §23.14 and Ch.23

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23.14.3

\int{\wp}^{3}\left(z\right)\,\mathrm{d}z=\frac{1}{120}\wp'''\left(z\right)-% \frac{3}{20}g_{2}\zeta\left(z\right)+\frac{1}{10}g_{3}z.

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Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathbb{L}$ : lattice and $z$ : complex
A&S Ref:: 18.7.2
Permalink:: http://dlmf.nist.gov/23.14.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §23.14 and Ch.23

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A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:: Includes Fortran program claiming 12S accuracy
External Links:: ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt
Referenced by:: 3rd item, §10.77(ix).
Encodings:: BibTeX

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Ya. I. GranovskiÄ, I. M. Lutzenko, and A. S. Zhedanov (1992) Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:: ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

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A. G. Greenhill (1892) The Applications of Elliptic Functions. MacMillan, London.

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Notes:: Reprinted by Dover Publications Inc., New York, 1959.
External Links:: Link, ZentralBlatt
Referenced by:: §19.30(iii), Ch.19.
Encodings:: BibTeX

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A. G. Greenhill (1959) The Applications of Elliptic Functions. Dover Publications Inc., New York.

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Notes:: Reprint of Macmillan, London edition, 1892
External Links:: MathReview, ZentralBlatt
Referenced by:: §22.18(iii), §22.19(iv), §22.8(iii).
Encodings:: BibTeX

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A. J. Guttmann and T. Prellberg (1993) Staircase polygons, elliptic integrals, Heun functions, and lattice Green functions. Phys. Rev. E 47 (4), pp. R2233–R2236.

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External Links:: Document
Referenced by:: §19.35(ii).
Encodings:: BibTeX

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B. C. Carlson (1965) On computing elliptic integrals and functions. J. Math. and Phys. 44, pp. 36–51.

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External Links:: MathReview (R. Nicolovius), ZentralBlatt
Referenced by:: §19.36(ii), §19.36(ii).
Encodings:: BibTeX

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B. C. Carlson (2006b) Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric

R

-functions. Math. Comp. 75 (255), pp. 1309–1318.

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External Links:: ISSN 0025-5718, Document, MathReview, ZentralBlatt
Referenced by:: §19.25(v), §19.29(i), §22.14(ii).
Encodings:: BibTeX

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B. C. Carlson (2008) Power series for inverse Jacobian elliptic functions. Math. Comp. 77 (263), pp. 1615–1621.

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External Links:: ISSN 0025-5718, Document, MathReview (Shaun Cooper), ZentralBlatt
Referenced by:: §19.25(v), §22.15(ii).
Encodings:: BibTeX

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A. Cayley (1961) An Elementary Treatise on Elliptic Functions. Dover Publications, New York (English).

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Notes:: Second edition of Cayley (1895), unabridged and corrected
External Links:: ZentralBlatt
Referenced by:: §19.11(i), Erratum (V1.0.24) for Subsection 19.11(i).
Encodings:: BibTeX

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M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

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External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §13.29(v), §15.19(v).
Encodings:: BibTeX

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§23.9 Laurent and Other Power Series

… â–º

23.9.2

\wp\left(z\right)=\frac{1}{z^{2}}+\sum_{n=2}^{\infty}c_{n}z^{2n-2},

0<|z|<|z_{0}|

,

â“˜

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\mathbb{L}$ : lattice, $n$ : integer, $z$ : complex and $c_{n}$
A&S Ref:: 18.5.1
Referenced by:: §23.9, §23.9
Permalink:: http://dlmf.nist.gov/23.9.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §23.9 and Ch.23

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23.9.3

\zeta\left(z\right)=\frac{1}{z}-\sum_{n=2}^{\infty}\frac{c_{n}}{2n-1}z^{2n-1},

0<|z|<|z_{0}|

.

â“˜

Symbols:: $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\mathbb{L}$ : lattice, $n$ : integer, $z$ : complex and $c_{n}$
A&S Ref:: 18.5.5
Referenced by:: §23.9
Permalink:: http://dlmf.nist.gov/23.9.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §23.9 and Ch.23

… â–º

c_{2}=\frac{1}{20}g_{2},

… â–º

23.9.7

\sigma\left(z\right)=\sum_{m,n=0}^{\infty}a_{m,n}(10c_{2})^{m}(56c_{3})^{n}% \frac{z^{4m+6n+1}}{(4m+6n+1)!},

â“˜

Symbols:: $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $!$ : factorial (as in $n!$ ), $\mathbb{L}$ : lattice, $n$ : integer, $m$ : integer, $z$ : complex, $c_{n}$ and $a_{m,n}$ : coefficients
A&S Ref:: 18.5.6
Referenced by:: §23.9
Permalink:: http://dlmf.nist.gov/23.9.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §23.9 and Ch.23

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for%20elliptic%20functions

11—20 of 31 matching pages

11: Bibliography F

12: Bibliography B

13: 20.11 Generalizations and Analogs

§20.11 Generalizations and Analogs

§20.11(iv) Theta Functions with Characteristics

14: Bibliography L

15: Bibliography I

16: Bibliography

17: 23.14 Integrals

§23.14 Integrals

18: Bibliography G

19: Bibliography C

20: 23.9 Laurent and Other Power Series

§23.9 Laurent and Other Power Series