Jacobi zeta function

SearchAdvancedHelp

(0.007 seconds)

11—20 of 33 matching pages

11: 22.15 Inverse Functions

… ►

22.15.3 $$\mathrm{dn}(\zeta ,k)=x,$$ $k^{\prime }\le x\le 1$,

ⓘ

Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $x$: real, $k$: modulus, $k^{\prime }$: complementary modulus and $\zeta$: solution

Referenced by:

§22.15(i)

Permalink:

http://dlmf.nist.gov/22.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.15(i), §22.15 and Ch.22

… ►

22.15.10 $$\zeta =\pm \mathrm{arcdn}(x,k)+2mK,$$

ⓘ

Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{arcdn}(x,k)$: inverse Jacobian elliptic function, $x$: real, $k$: modulus, $\zeta$: solution and $m$: integer

Permalink:

http://dlmf.nist.gov/22.15.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §22.15(i), §22.15 and Ch.22

…

12: 23.6 Relations to Other Functions

… ►

23.6.13 $$\zeta \left(u\right)=\frac{{\eta }_1}{{\omega }_1}u+\frac{\pi }{2{\omega }_1}\frac{d}{dz}\ln {\theta }_1(z,q),$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\ln z$: principal branch of logarithm function, $𝕃$: lattice, $q$: nome, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, ${\eta }_j$: complex number and $u$: change of variable

Referenced by:

§23.22(i), §23.6(i)

Permalink:

http://dlmf.nist.gov/23.6.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §23.6(i), §23.6 and Ch.23

… ►

23.6.27 $$\zeta \left(z|{𝕃}_1\right)-\zeta \left(z+2K|{𝕃}_1\right)+\zeta \left(2K|{𝕃}_1\right)=\mathrm{ns}(z,k),$$

ⓘ

Symbols:

$\mathrm{ns}(z,k)$: Jacobian elliptic function, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $𝕃$: lattice, $z$: complex and $k$: modulus

Referenced by:

§23.6(ii), §23.6(ii)

Permalink:

http://dlmf.nist.gov/23.6.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §23.6(ii), §23.6 and Ch.23

►

23.6.28 $$\zeta \left(z|{𝕃}_2\right)-\zeta \left(z+2K|{𝕃}_2\right)+\zeta \left(2K|{𝕃}_2\right)=\mathrm{ds}(z,k),$$

ⓘ

Symbols:

$\mathrm{ds}(z,k)$: Jacobian elliptic function, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $𝕃$: lattice, $z$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/23.6.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §23.6(ii), §23.6 and Ch.23

►

23.6.29 $$\zeta \left(z|{𝕃}_3\right)-\zeta \left(z+2\mathrm{i}{K}^{\prime }|{𝕃}_3\right)-\zeta \left(2\mathrm{i}{K}^{\prime }|{𝕃}_3\right)=\mathrm{cs}(z,k).$$

ⓘ

Symbols:

$\mathrm{cs}(z,k)$: Jacobian elliptic function, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $𝕃$: lattice, $z$: complex and $k$: modulus

Referenced by:

§23.6(ii), §23.6(ii)

Permalink:

http://dlmf.nist.gov/23.6.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §23.6(ii), §23.6 and Ch.23

…

13: 31.2 Differential Equations

… ►

31.2.8 $$\frac{d^{2}w}{{d\zeta }^2}+\left((2\gamma -1)\frac{\mathrm{cn}\zeta \mathrm{dn}\zeta }{\mathrm{sn}\zeta }-(2\delta -1)\frac{\mathrm{sn}\zeta \mathrm{dn}\zeta }{\mathrm{cn}\zeta }-(2ϵ-1)k^2\frac{\mathrm{sn}\zeta \mathrm{cn}\zeta }{\mathrm{dn}\zeta }\right)\frac{dw}{d\zeta }+4k^2(\alpha \beta {\mathrm{sn}}^2\zeta -q)w=0.$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter, $\beta$: real or complex parameter, $\zeta$: change of variable and $k$: modulus

Referenced by:

§31.7(ii)

Permalink:

http://dlmf.nist.gov/31.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §31.2(iv), §31.2(iv), §31.2 and Ch.31

…

14: Bibliography E

… ►

Riemann’s Zeta Function. Academic Press, New York-London.

ⓘ

Notes:

Pure and Applied Mathematics, Vol. 58. Reprinted by Dover Publications Inc., Mineola, NY, 2001.

External Links:

MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

Ch.25.

Encodings:

BibTeX

… ►

An asymptotic expansion for the first derivative of the generalized Riemann zeta function. Math. Comp. 47 (175), pp. 347–350.

ⓘ

External Links:

ISSN 0025-5718, FullText, MathReview (S. Audinarayana Moorthy), ZentralBlatt

Referenced by:

(25.11.44), (25.11.45), §25.11(xii), §25.11(xii).

Encodings:

BibTeX

►

Ten Physical Applications of Spectral Zeta Functions. Lecture Notes in Physics. New Series m: Monographs, Vol. 35, Springer-Verlag, Berlin.

ⓘ

External Links:

ISBN 3-540-60230-5, MathReview (Alfred Actor), ZentralBlatt

Referenced by:

§25.17.

Encodings:

BibTeX

►

Uniform asymptotic expansions of the Jacobi polynomials and an associated function. Math. Comp. 25 (114), pp. 309–315.

ⓘ

External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§18.15(i).

Encodings:

BibTeX

… ►

A new series representation for $\zeta (3)$ . Amer. Math. Monthly 97 (3), pp. 219–220.

ⓘ

External Links:

ISSN 0002-9890, FullText, MathReview (Christian Radoux), ZentralBlatt

Referenced by:

(25.8.10).

Encodings:

BibTeX

…

15: 5.16 Sums

§5.16 Sums

… ►

5.16.2 $$\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k}{\psi }^{\prime }\left(k+1\right)=\zeta \left(3\right)=-\frac{1}{2}{\psi }^{\prime \prime }\left(1\right).$$

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\psi \left(z\right)$: psi (or digamma) function and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/5.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.16 and Ch.5

►For further sums involving the psi function see Hansen (1975, pp. 360–367). For sums of gamma functions see Andrews et al. (1999, Chapters 2 and 3) and §§15.2(i), 16.2. ►For related sums involving finite field analogs of the gamma and beta functions (Gauss and Jacobi sums) see Andrews et al. (1999, Chapter 1) and Terras (1999, pp. 90, 149).

16: 20.9 Relations to Other Functions

§20.9 Relations to Other Functions

►

§20.9(i) Elliptic Integrals

… ►

§20.9(ii) Elliptic Functions and Modular Functions

… ►The relations (20.9.1) and (20.9.2) between $k$ and $\tau$ (or $q$) are solutions of Jacobi’s inversion problem; see Baker (1995) and Whittaker and Watson (1927, pp. 480–485). … ►

§20.9(iii) Riemann Zeta Function

…

17: Bibliography C

… ►

A recurrence formula for $\zeta (2n)$ . Proc. Amer. Math. Soc. 12 (6), pp. 991–992.

ⓘ

External Links:

FullText, MathReview (R. D. James)

Referenced by:

§24.14(i).

Encodings:

BibTeX

… ►

The Riemann zeta-function and its derivatives. Proc. Roy. Soc. London Ser. A 450, pp. 477–499.

ⓘ

External Links:

ISSN 0962-8444, FullText, MathReview, ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

… ►

Chebyshev approximations for the Riemann zeta function. Math. Comp. 25 (115), pp. 537–547.

ⓘ

External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

… ►

On some series representations of the Hurwitz zeta function. J. Comput. Appl. Math. 216 (1), pp. 297–305.

ⓘ

External Links:

ISSN 0377-0427, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§25.11(iv), §25.11(iv).

Encodings:

BibTeX

►

An efficient algorithm for the Hurwitz zeta and related functions. J. Comput. Appl. Math. 225 (2), pp. 338–346.

ⓘ

External Links:

ISSN 0377-0427, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§25.18(i), §25.18(i).

Encodings:

BibTeX

…

18: 15.12 Asymptotic Approximations

… ►where … ►See also Dunster (1999) where the asymptotics of Jacobi polynomials is described; compare (15.9.1). … ►with the branch chosen to be continuous and $\beta >0$ when $\zeta >0$. Also, …where …

19: 25.5 Integral Representations

§25.5 Integral Representations

►

§25.5(i) In Terms of Elementary Functions

… ►

§25.5(ii) In Terms of Other Functions

… ►For ${\theta }_3$ see §20.2(i). … ►

§25.5(iii) Contour Integrals

…

20: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

§22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

… ►

22.12.2 $$2Kk\mathrm{sn}(2Kt,k)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\pi }{\sin \left(\pi (t-(n+\frac{1}{2})\tau )\right)}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left(\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^m}{t-m-(n+\frac{1}{2})\tau }\right),$$

ⓘ

Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\sin z$: sine function, $k$: modulus and $\tau$: ratio

Referenced by:

§22.12, §22.12

Permalink:

http://dlmf.nist.gov/22.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

… ►

22.12.8 $$2K\mathrm{dc}(2Kt,k)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\pi }{\sin \left(\pi (t+\frac{1}{2}-n\tau )\right)}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left(\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^m}{t+\frac{1}{2}-m-n\tau }\right),$$

ⓘ

Symbols:

$\mathrm{dc}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\sin z$: sine function, $k$: modulus and $\tau$: ratio

Permalink:

http://dlmf.nist.gov/22.12.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

… ►

22.12.11 $$2K\mathrm{ns}(2Kt,k)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\pi }{\sin \left(\pi (t-n\tau )\right)}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left(\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^m}{t-m-n\tau }\right),$$

ⓘ

Symbols:

$\mathrm{ns}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\sin z$: sine function, $k$: modulus and $\tau$: ratio

Permalink:

http://dlmf.nist.gov/22.12.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

►

22.12.12 $$2K\mathrm{ds}(2Kt,k)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^n\pi }{\sin \left(\pi (t-n\tau )\right)}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left(\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^{m+n}}{t-m-n\tau }\right),$$

ⓘ

Symbols:

$\mathrm{ds}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\sin z$: sine function, $k$: modulus and $\tau$: ratio

Permalink:

http://dlmf.nist.gov/22.12.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

…