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21: 19.19 Taylor and Related Series

§19.19 Taylor and Related Series

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19.19.6 $$R_J(x,y,z,p)=R_{-\frac{3}{2}}(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2};x,y,z,p,p)$$

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function and $R_J(x,y,z,p)$: symmetric elliptic integral of third kind

Referenced by:

§19.19

Permalink:

http://dlmf.nist.gov/19.19.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.19 and Ch.19

… ►Special cases are given in (19.36.1) and (19.36.2).

22: 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

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22.12.1 $$\tau =\mathrm{i}{K}^{\prime }\left(k\right)/K\left(k\right),$$

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Symbols:

$K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $k$: modulus and $\tau$: ratio

Permalink:

http://dlmf.nist.gov/22.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

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

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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

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

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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

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22.12.13 $$2K\mathrm{cs}(2Kt,k)=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\frac{\pi }{\tan \left(\pi (t-n\tau )\right)}=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\left(\underset{M\to \mathrm{\infty }}{lim}\sum _{m=-M}^M\frac{1}{t-m-n\tau }\right).$$

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Symbols:

$\mathrm{cs}(z,k)$: 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, $\tan z$: tangent function, $k$: modulus and $\tau$: ratio

Referenced by:

§22.12

Permalink:

http://dlmf.nist.gov/22.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

23: 19.26 Addition Theorems

§19.26 Addition Theorems

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§19.26(ii) Case $x=0$

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§19.26(iii) Duplication Formulas

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19.26.20 $$R_D(x,y,z)=2R_D(x+\lambda ,y+\lambda ,z+\lambda )+\frac{3}{\sqrt{z}(z+\lambda )}.$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\lambda$: positive, $x$: positive, $y$: positive and $z$: positive

Referenced by:

§19.26(iii)

Permalink:

http://dlmf.nist.gov/19.26.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §19.26(iii), §19.26 and Ch.19

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19.26.21 $$2R_G(x,y,z)=4R_G(x+\lambda ,y+\lambda ,z+\lambda )-\lambda R_F(x,y,z)-\sqrt{x}-\sqrt{y}-\sqrt{z}.$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $R_G(x,y,z)$: symmetric elliptic integral of second kind, $\lambda$: positive, $x$: positive, $y$: positive and $z$: positive

Referenced by:

§19.36(i)

Permalink:

http://dlmf.nist.gov/19.26.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §19.26(iii), §19.26 and Ch.19

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24: 29.11 Lamé Wave Equation

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29.11.1 $$\frac{d^{2}w}{{dz}^2}+(h-\nu (\nu +1)k^2{\mathrm{sn}}^2(z,k)+k^2{\omega }^2{\mathrm{sn}}^4(z,k))w=0,$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $h$: real parameter, $k$: real parameter, $\nu$: real parameter, $\omega$: parameter and $w(z)$: solution

Referenced by:

§29.11, §29.11, §29.18(ii), §29.7(ii)

Permalink:

http://dlmf.nist.gov/29.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §29.11 and Ch.29

… ►In the case $\omega =0$, (29.11.1) reduces to Lamé’s equation (29.2.1). …

25: 22.18 Mathematical Applications

§22.18 Mathematical Applications

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§22.18(i) Lengths and Parametrization of Plane Curves

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§22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem

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26: 19.29 Reduction of General Elliptic Integrals

… ►All other cases are integrals of the second kind. …

27: 19.24 Inequalities

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§19.24(i) Complete Integrals

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§19.24(ii) Incomplete Integrals

… ►Special cases with $a=\pm \frac{1}{2}$ are (19.24.8) (because of (19.16.20), (19.16.23)), and …The same reference also gives upper and lower bounds for symmetric integrals in terms of their elementary degenerate cases. …

28: Errata

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A note was added after (22.20.5) to deal with cases when computation of $\mathrm{dn}(x,k)$ becomes numerically unstable near $x=K$.

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Two corrections have been made in this paragraph. First, the correct range of the initial displacement $a$ is . Previously it was $\sqrt{1/\beta }\le |a|\le \sqrt{2/\beta }$. Second, the correct period of the oscillations is $2K\left(k\right)/\sqrt{\eta }$. Previously it was given incorrectly as $4K\left(k\right)/\sqrt{\eta }$.

Reported 2014-05-02 by Svante Janson.

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29: 20.11 Generalizations and Analogs

… ►In the case $z=0$ identities for theta functions become identities in the complex variable $q$, with , that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). … ►As 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). However, in this case $q$ is no longer regarded as an independent complex variable within the unit circle, because $k$ is related to the variable $\tau =\tau (k)$ of the theta functions via (20.9.2). … ►For 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). …

30: 36.4 Bifurcation Sets

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Special Cases

… ►Elliptic umbilic bifurcation set (codimension three): for fixed $z$, the section of the bifurcation set is a three-cusped astroid … ►Elliptic umbilic cusp lines (ribs): … ►

§36.4(ii) Visualizations

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Figure 36.4.3: Bifurcation set of elliptic umbilic catastrophe. Magnify

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