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31: 19.25 Relations to Other Functions

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19.25.7 $$E(\varphi ,k)=2R_G(c-1,c-k^2,c)-(c-1)R_F(c-1,c-k^2,c)-\sqrt{c-1}\sqrt{c-k^2}/\sqrt{c},$$

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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, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.25(i), §19.25(i), §19.25(iii), Figure 19.3.4, Figure 19.3.4, Figure 19.3.4, §19.36(ii), Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.25.E7

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.25(i), §19.25 and Ch.19

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19.25.10 $$E(\varphi ,k)=k_{}^{\prime }{}_{}{}^2{R}_F(c-1,c-k^2,c)+\frac{1}{3}{k}^{2}k_{}^{\prime }{}_{}{}^2{R}_D(c-1,c,c-k^2)+k^2\sqrt{c-1}/\left(\sqrt{c}\sqrt{c-k^2}\right),$$ $c>k^2$,

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\varphi$: real or complex argument, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Referenced by:

§19.25(i), §19.30(i), §19.36(i), §19.6(iv), Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.25.E10

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.25(i), §19.25 and Ch.19

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19.25.35 $$z+2\omega =\pm R_F(\mathrm{\wp }\left(z\right)-e_1,\mathrm{\wp }\left(z\right)-e_2,\mathrm{\wp }\left(z\right)-e_3),$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots and $z$: complex number

Proof sketch:

Use (23.6.36) with $z=\mathrm{\wp }\left(\omega \right)$ as prescribed in the text that follows (23.6.36), substitute $u=t+\mathrm{\wp }\left(\omega \right)$ and compare with (19.16.1). The undetermined $\pm$ is a consequence of the multivaluedness of the square-roots in (19.16.4).

Referenced by:

(19.25.38), (19.25.40), §19.25(v), §19.25(vi), §19.25(vi), §20.9(i), Erratum (V1.1.0) for Subsection 19.25(vi)

Permalink:

http://dlmf.nist.gov/19.25.E35

Encodings:

pMML, png, TeX

Correction (effective with 1.1.0):

This equation has been corrected, namely the left-hand side which was originally $z$, has been replaced by $z+2\omega$ and the right-hand side has been multiplied by $\pm 1$.

See also:

Annotations for §19.25(vi), §19.25 and Ch.19

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19.25.36 $$\mathrm{\wp }\left(z\right)-e_j\in ℂ\setminus (-\mathrm{\infty },0],$$ $j=1,2,3.$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $ℂ$: complex plane, $\in$: element of, $(a,b]$: half-closed interval, $\setminus$: set subtraction, $j$: $(=1,2,3)$ and $z$: complex number

Referenced by:

(19.25.39)

Permalink:

http://dlmf.nist.gov/19.25.E36

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(vi), §19.25 and Ch.19

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19.25.38 $${\omega }_j=R_F(0,e_j-e_k,e_j-e_{\mathrm{\ell }}),$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $e_j$: Weierstrass lattice roots, $j$: $(=1,2,3)$ , $z$: complex number, $k$: $(=1,2,3)$ , $\mathrm{\ell }$: $(=1,2,3)$ and ${\omega }_j$: nonzero complex number

Proof sketch:

Take $z=-\omega$ in (19.25.35).

Referenced by:

§19.25(vi), §19.25(vi)

Permalink:

http://dlmf.nist.gov/19.25.E38

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(vi), §19.25 and Ch.19

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32: Bibliography K

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The computation of the sums of negative even powers of roots of Bessel functions. Doklady Akad. Nauk SSSR (N.S.) 77, pp. 561–564.

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§10.21(xiii).

Encodings:

BibTeX

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On the calculation of the multiple complex roots of the derivatives of cylindrical Bessel functions. Zh. Vychisl. Mat. i Mat. Fiz. 27 (11), pp. 1628–1639, 1758.

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 27(6), pp. 18–25

External Links:

ISSN 0044-4669, Document, MathReview (Hans-Jürgen Albrand), ZentralBlatt

Referenced by:

§10.21(ix), §10.74(vi), 5th item.

Encodings:

BibTeX

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Askey-Wilson Polynomials for Root Systems of Type $BC$ . In Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemp. Math., Vol. 138, pp. 189–204.

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External Links:

FullText, MathReview (Eric M. Opdam), ZentralBlatt

Referenced by:

§18.37(iii).

Encodings:

BibTeX

…

33: 1.3 Determinants, Linear Operators, and Spectral Expansions

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1.3.15

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Symbols:

$det$: determinant, $k$: integer, $n$: nonnegative integer and ${\omega }_j$: roots of unity

Permalink:

http://dlmf.nist.gov/1.3.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §1.3(ii), §1.3(ii), §1.3 and Ch.1

►where ${\omega }_1,{\omega }_2,\mathrm{\dots },{\omega }_n$ are the $n$th roots of unity (1.11.21). …

34: 19.21 Connection Formulas

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19.21.7 $$(x-y)R_D(y,z,x)+(z-y)R_D(x,y,z)=3R_F(x,y,z)-3y^{1/2}{x}^{-1/2}{z}^{-1/2},$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables and $R_F(x,y,z)$: symmetric elliptic integral of first kind

Referenced by:

§19.21(ii), §19.25(i), Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.21.E7

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.21(ii), §19.21 and Ch.19

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19.21.8 $$R_D(y,z,x)+R_D(z,x,y)+R_D(x,y,z)=3x^{-1/2}{y}^{-1/2}{z}^{-1/2},$$

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Symbols:

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

Referenced by:

§19.21(ii), §19.33(iii), Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.21.E8

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.21(ii), §19.21 and Ch.19

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19.21.10 $$2R_G(x,y,z)=z{R}_F(x,y,z)-\frac{1}{3}(x-z)(y-z)R_D(x,y,z)+x^{1/2}{y}^{1/2}{z}^{-1/2},$$ $z\ne 0$.

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind and $R_G(x,y,z)$: symmetric elliptic integral of second kind

Referenced by:

§19.20(ii), §19.21(i), §19.21(ii), §19.21(ii), §19.36(i), §19.36(ii), Erratum (V1.1.3) for Chapter 19, Erratum (V1.1.7) for Equation (19.21.10)

Permalink:

http://dlmf.nist.gov/19.21.E10

Encodings:

pMML, png, TeX

Correction (effective with 1.1.7):

An incorrect factor of 3 was removed in the last term on the right-hand side.

Suggested 2022-06-27 by Abdulhafeez Abdulsalam

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.21(ii), §19.21 and Ch.19

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35: 28.6 Expansions for Small $q$

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28.6.20 $$\underset{n\to \mathrm{\infty }}{lim\; inf}\frac{{\rho }_n^{(j)}}{n^2}\ge k{k}^{\prime }{(K\left(k\right))}^2=\mathrm{2.04183\hspace{0.33em}4}\mathrm{\dots },$$

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $lim\; inf$: least limit point, $n$: integer, $j$: integer, ${\rho }_n^{(j)}$: radii of convergence and $k$: root

Referenced by:

§28.6(i), §28.6(ii)

Permalink:

http://dlmf.nist.gov/28.6.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(i), §28.6 and Ch.28

►where $k$ is the unique root of the equation $2E\left(k\right)=K\left(k\right)$ in the interval $(0,1)$, and $k^{\prime }=\sqrt{1-k^2}$. …

36: 31.2 Differential Equations

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31.2.10 $$w(\xi )={\left(\mathrm{\wp }\left(\xi \right)-e_3\right)}^{(1-2\gamma )/4}{\left(\mathrm{\wp }\left(\xi \right)-e_2\right)}^{(1-2\delta )/4}{\left(\mathrm{\wp }\left(\xi \right)-e_1\right)}^{(1-2ϵ)/4}W(\xi ),$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $𝕃$: lattice and $W(\xi )$: solution

Permalink:

http://dlmf.nist.gov/31.2.E10

Encodings:

pMML, png, TeX

See also:

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

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37: 4.16 Elementary Properties

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Table 4.16.3: Trigonometric functions: interrelations. All square roots have their principal values when the functions are real, nonnegative, and finite.

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	$\sin \theta =a$	$\cos \theta =a$	$\tan \theta =a$	$\csc \theta =a$	$\sec \theta =a$	$\cot \theta =a$
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38: 23.9 Laurent and Other Power Series

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23.9.6 $$\mathrm{\wp }\left({\omega }_j+t\right)=e_j+(3{e_j}^2-5c_2)t^2+(10c_2{e}_j+21c_3)t^4+(7c_2{e_j}^2+21c_3{e}_j+5c_2^2)t^6+O\left(t^8\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $𝕃$: lattice, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $c_n$ and $t$: variable

Referenced by:

§23.9

Permalink:

http://dlmf.nist.gov/23.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §23.9 and Ch.23

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39: 28.7 Analytic Continuation of Eigenvalues

… ►The normal values are simple roots of the corresponding equations (28.2.21) and (28.2.22). …

40: 2.8 Differential Equations with a Parameter

… ►Let $c=-0.36604\mathrm{\dots }$ be the real root of the equation … ►

2.8.20 $$\mathrm{envAi}\left(x\right)=\mathrm{envBi}\left(x\right)={\left({\mathrm{Ai}}^2\left(x\right)+{\mathrm{Bi}}^2\left(x\right)\right)}^{1/2},$$ ,

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{envAi}\left(x\right)$: envelope of Airy function $\mathrm{Ai}\left(x\right)$ , $\mathrm{envBi}\left(x\right)$: envelope of Airy function $\mathrm{Bi}\left(x\right)$ and $c$: root

Permalink:

http://dlmf.nist.gov/2.8.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(iii), §2.8 and Ch.2

… ►Let $x=X_{\nu }$ be the smallest positive root of the equation … ►

2.8.34 $$\mathrm{env}{J}_{\nu }(x)=\mathrm{env}{Y}_{\nu }(x)={\left(J_{\nu }^2(x)+Y_{\nu }^2(x)\right)}^{1/2},$$ .

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\mathrm{env}{J}_{\nu }\left(x\right)$: envelope of Bessel function $J_{\nu }\left(x\right)$ , $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$ , $\nu$: real nonnegative constant and $X_{\nu }$: smallest positive root

Permalink:

http://dlmf.nist.gov/2.8.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(iv), §2.8 and Ch.2

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