L’Hôpital rule for derivatives

♦ 21—30 of 348 matching pages ♦

(0.002 seconds)

21—30 of 348 matching pages

21: Bibliography Y

… ►

G. D. Yakovleva (1969) Tables of Airy Functions and Their Derivatives. Izdat. Nauka, Moscow (Russian).

ⓘ

Referenced by:: 4th item.
Encodings:: BibTeX

… ►

H. A. Yamani and W. P. Reinhardt (1975)

L

-squared discretizations of the continuum: Radial kinetic energy and the Coulomb Hamiltonian. Phys. Rev. A 11 (4), pp. 1144–1156.

ⓘ

Referenced by:: §18.39(iv), §18.39(iv), §18.39(iv), §18.40(ii).
Encodings:: BibTeX

… ►

A. Yu. Yeremin, I. E. Kaporin, and M. K. Kerimov (1988) Computation of the derivatives of the Riemann zeta-function in the complex domain. USSR Comput. Math. and Math. Phys. 28 (4), pp. 115–124.

ⓘ

Notes:: Includes Fortran programs.
External Links:: Document, ZentralBlatt
Referenced by:: §25.18(i), §25.21(iii).
Encodings:: BibTeX

…

22: 34.5 Basic Properties: $\mathit{6j}$ Symbol

… ►

34.5.11

{(2j_{1}+1)\left((J_{3}+J_{2}-J_{1})(L_{3}+L_{2}-J_{1})-2(J_{3}L_{3}+J_{2}L_{2% }-J_{1}L_{1})\right)\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}}\\ =j_{1}E(j_{1}+1)\begin{Bmatrix}j_{1}+1&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}+(j_{1}+1)E(j_{1})\begin{Bmatrix}j_{1}-1&j_{2}&j% _{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix},

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $l,l_{r}$ : non-negative integers or non-negative integers plus one half., $J_{r}$ , $L_{r}$ and $E(j)$
Permalink:: http://dlmf.nist.gov/34.5.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(iii), §34.5 and Ch.34

… ►

L_{r}=l_{r}(l_{r}+1),

… ►Equations (34.5.15) and (34.5.16) are the sum rules. They constitute addition theorems for the

\mathit{6j}

symbol. …

23: 14.30 Spherical and Spheroidal Harmonics

… ►

14.30.11

\mathrm{L}^{2}Y_{{l},{m}}=\hbar^{2}l(l+1)Y_{{l},{m}},

l=0,1,2,\dots

ⓘ

Symbols:: $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, $m$ : integer, $l$ : nonnegative integer and $\mathrm{L}$ : angular momentum operator
Referenced by:: §18.39(ii)
Permalink:: http://dlmf.nist.gov/14.30.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §14.30(iv), §14.30 and Ch.14

… ►Here, in spherical coordinates,

\mathrm{L}^{2}

is the squared angular momentum operator: ►

14.30.12

\mathrm{L}^{2}=-\hbar^{2}\left(\frac{1}{\sin\theta}\frac{\partial}{\partial% \theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{{\sin}^% {2}\theta}\frac{{\partial}^{2}}{{\partial\phi}^{2}}\right),

ⓘ

Defines:: $\mathrm{L}$ : angular momentum operator (locally)
Symbols:: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ and $\sin\NVar{z}$ : sine function
Referenced by:: §18.39(ii)
Permalink:: http://dlmf.nist.gov/14.30.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §14.30(iv), §14.30 and Ch.14

►and

\mathrm{L}_{z}

is the

z

component of the angular momentum operator ►

14.30.13

\mathrm{L}_{z}=-\mathrm{i}\hbar\frac{\partial}{\partial\phi};

ⓘ

Defines:: $\mathrm{L}_{z}$ : $z$ component of the angular momentum operator (locally)
Symbols:: $\mathrm{i}$ : imaginary unit, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ and $z$ : complex variable
Referenced by:: §14.30(iv), Erratum (V1.1.0) for Additions
Permalink:: http://dlmf.nist.gov/14.30.E13
Encodings:: pMML, png, TeX
Addition (effective with 1.1.0):: This equation was added.
See also:: Annotations for §14.30(iv), §14.30 and Ch.14

…

24: Bibliography

… ►

G. Allasia and R. Besenghi (1987a) Numerical computation of Tricomi’s psi function by the trapezoidal rule. Computing 39 (3), pp. 271–279.

ⓘ

External Links:: ISSN 0010-485X, Document, MathReview, ZentralBlatt
Referenced by:: §13.29(iii).
Encodings:: BibTeX

►

G. Allasia and R. Besenghi (1991) Numerical evaluation of the Kummer function with complex argument by the trapezoidal rule. Rend. Sem. Mat. Univ. Politec. Torino 49 (3), pp. 315–327.

ⓘ

External Links:: ISSN 0373-1243, MathReview, ZentralBlatt
Referenced by:: §13.29(iii).
Encodings:: BibTeX

►

G. Allasia and R. Besenghi (1987b) Numerical calculation of incomplete gamma functions by the trapezoidal rule. Numer. Math. 50 (4), pp. 419–428.

ⓘ

External Links:: ISSN 0029-599X, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §8.25(ii).
Encodings:: BibTeX

… ►

A. Apelblat (1989) Derivatives and integrals with respect to the order of the Struve functions

\mathbf{H}_{\nu}(x)

and

\mathbf{L}_{\nu}(x)

. J. Math. Anal. Appl. 137 (1), pp. 17–36.

ⓘ

External Links:: ISSN 0022-247X, Document, MathReview (A. McD. Mercer), ZentralBlatt
Referenced by:: §11.4(vi), §11.7(iv).
Encodings:: BibTeX

… ►

T. M. Apostol (1985b) Note on the trivial zeros of Dirichlet

L

-functions. Proc. Amer. Math. Soc. 94 (1), pp. 29–30.

ⓘ

External Links:: ISSN 0002-9939, FullText, MathReview (Jerzy Kaczorowski), ZentralBlatt
Referenced by:: (25.15.7), (25.15.8), §25.15(ii), §25.15.
Encodings:: BibTeX

…

25: 18.34 Bessel Polynomials

… ►For the confluent hypergeometric function

{{}_{1}F_{1}}

and the generalized hypergeometric function

{{}_{2}F_{0}}

, the Laguerre polynomial

L^{(\alpha)}_{n}

and the Whittaker function

W_{\kappa,\mu}

see §16.2(ii), §16.2(iv), (18.5.12), and (13.14.3), respectively. ►

18.34.1

y_{n}\left(x;a\right)={{}_{2}F_{0}}\left({-n,n+a-1\atop-};-\frac{x}{2}\right)=% {\left(n+a-1\right)_{n}}\left(\frac{x}{2}\right)^{n}{{}_{1}F_{1}}\left({-n% \atop-2n-a+2};\frac{2}{x}\right)=n!\left(-\tfrac{1}{2}x\right)^{n}L^{(1-a-2n)}% _{n}\left(2x^{-1}\right)=\left(\tfrac{1}{2}x\right)^{1-\frac{1}{2}a}{\mathrm{e% }}^{1/x}W_{1-\frac{1}{2}a,\frac{1}{2}(a-1)+n}\left(2x^{-1}\right).

ⓘ

Defines:: $y_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Bessel polynomial
Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ : $=M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ notation for the Kummer confluent hypergeometric function, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer and $x$ : real variable
Proved:: Grosswald (1978, p.38, Theorem 1); Combine (18.5.12) and items (i), (v), (vii) of the mentioned theorem for $b=2$ (proved)
Referenced by:: (18.34.2), (18.34.7_1), §18.34(i), §18.34(i), Erratum (V1.2.0) for Equation (18.34.1)
Permalink:: http://dlmf.nist.gov/18.34.E1
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was updated to include the definition of Bessel polynomials in terms of Laguerre polynomials and the Whittaker confluent hypergeometric function.
See also:: Annotations for §18.34(i), §18.34 and Ch.18

… ►where primes denote derivatives with respect to

x

. … ►

18.34.7_1

\phi_{n}(x;\lambda)={\mathrm{e}}^{-\lambda\,{\mathrm{e}}^{-x}}\left(2\lambda\,% {\mathrm{e}}^{-x}\right)^{\lambda-\frac{1}{2}}\ifrac{y_{n}\left(\lambda^{-1}{% \mathrm{e}}^{x};2-2\lambda\right)}{n!}=(-1)^{n}{\mathrm{e}}^{-\lambda\,{% \mathrm{e}}^{-x}}\left(2\lambda\,{\mathrm{e}}^{-x}\right)^{\lambda-n-\frac{1}{% 2}}L^{(2\lambda-2n-1)}_{n}\left(2\lambda\,{\mathrm{e}}^{-x}\right)=(2\lambda)^% {-\frac{1}{2}}\,{\mathrm{e}}^{x/2}\,W_{\lambda,n+\frac{1}{2}-\lambda}\left(2% \lambda{\mathrm{e}}^{-x}\right)/n!\,,

n=0,1,\dots,N=\left\lceil\lambda-\frac{3}{2}\right\rceil

\lambda>\frac{1}{2}

ⓘ

Symbols:: $y_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Bessel polynomial, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\left\lceil\NVar{x}\right\rceil$ : ceiling of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $N$ : positive integer, $n$ : nonnegative integer and $x$ : real variable
Proof sketch:: The second and third equality follow from (18.34.1).
Referenced by:: (18.34.7_2), (18.34.7_3), §18.34(iii), (18.39.17), (18.39.18), §18.39(i), Erratum (V1.2.0) Section 18.34
Permalink:: http://dlmf.nist.gov/18.34.E7_1
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.34(iii), §18.34 and Ch.18

… ►

18.34.7_2

\left(\frac{{\mathrm{d}}^{2}}{{\mathrm{d}x}^{2}}-\lambda^{2}\left({\mathrm{e}}% ^{-2x}-2{\mathrm{e}}^{-x}\right)-\left(\lambda-\left(n+\tfrac{1}{2}\right)% \right)^{2}\right)\phi_{n}(x;\lambda)=0.

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $n$ : nonnegative integer and $x$ : real variable
Proof sketch:: This follows by straightforward computation from (18.34.7) and (18.34.7_1).
Referenced by:: (18.39.17), (18.39.18)
Permalink:: http://dlmf.nist.gov/18.34.E7_2
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.34(iii), §18.34 and Ch.18

…

26: 23.10 Addition Theorems and Other Identities

… ►

23.10.10

\sigma\left(2z\right)=-\wp'\left(z\right){\sigma}^{4}\left(z\right).

ⓘ

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\mathbb{L}$ : lattice and $z$ : complex
A&S Ref:: 18.4.8
Referenced by:: §23.10(ii)
Permalink:: http://dlmf.nist.gov/23.10.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §23.10(ii), §23.10 and Ch.23

… ►

23.10.17

\wp\left(cz|c\mathbb{L}\right)=c^{-2}\wp\left(z|\mathbb{L}\right),

ⓘ

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, $z$ : complex and $c$ : constant
A&S Ref:: 18.2.2
Permalink:: http://dlmf.nist.gov/23.10.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §23.10(iv), §23.10 and Ch.23

►

23.10.18

\zeta\left(cz|c\mathbb{L}\right)=c^{-1}\zeta\left(z|\mathbb{L}\right),

ⓘ

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, $z$ : complex and $c$ : constant
A&S Ref:: 18.2.3
Permalink:: http://dlmf.nist.gov/23.10.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §23.10(iv), §23.10 and Ch.23

►

23.10.19

\sigma\left(cz|c\mathbb{L}\right)=c\sigma\left(z|\mathbb{L}\right).

ⓘ

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, $z$ : complex and $c$ : constant
A&S Ref:: 18.2.4
Permalink:: http://dlmf.nist.gov/23.10.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §23.10(iv), §23.10 and Ch.23

►Also, when

\mathbb{L}

is replaced by

c\mathbb{L}

the lattice invariants

g_{2}

and

g_{3}

are divided by

c^{4}

and

c^{6}

, respectively. …

27: 18.27 $q$ -Hahn Class

… ►In the

q

-Hahn class OP’s the role of the operator

\ifrac{\mathrm{d}}{\mathrm{d}x}

in the Jacobi, Laguerre, and Hermite cases is played by the

q

-derivative

\mathcal{D}_{q}

, as defined in (17.2.41). … ►

18.27.14_6

\lim_{q\to 1}p_{n}\left((1-q)x;q^{\alpha},0;q\right)=\frac{n!}{{\left(\alpha+1% \right)_{n}}}L^{(\alpha)}_{n}\left(x\right).

ⓘ

Symbols:: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b};\NVar{q}\right)$ : little $q$ -Jacobi polynomial, $q$ : real variable, $n$ : nonnegative integer and $x$ : real variable
Source:: Koekoek et al. (2010, (14.12.12), (14.20.12))
Referenced by:: §18.27(iv), §18.27(iv), Erratum (V1.2.0) Section 18.27
Permalink:: http://dlmf.nist.gov/18.27.E14_6
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.27(iv), §18.27(iv), §18.27 and Ch.18

… ►

18.27.16

\int_{0}^{\infty}L^{(\alpha)}_{n}\left(x;q\right)L^{(\alpha)}_{m}\left(x;q% \right)\frac{x^{\alpha}}{\left(-x;q\right)_{\infty}}\,\mathrm{d}x=\frac{\left(% q^{\alpha+1};q\right)_{n}}{\left(q;q\right)_{n}q^{n}}h_{0}^{(1)}\delta_{n,m},

\alpha>-1

ⓘ

Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x};\NVar{q}\right)$ : $q$ -Laguerre polynomial, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable and $h_{n}$
Permalink:: http://dlmf.nist.gov/18.27.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §18.27(v), §18.27 and Ch.18

… ►

18.27.17

\sum_{y=-\infty}^{\infty}L^{(\alpha)}_{n}\left(cq^{y};q\right)L^{(\alpha)}_{m}% \left(cq^{y};q\right)\frac{q^{y(\alpha+1)}}{\left(-cq^{y};q\right)_{\infty}}=% \frac{\left(q^{\alpha+1};q\right)_{n}}{\left(q;q\right)_{n}q^{n}}h_{0}^{(2)}% \delta_{n,m},

\alpha>-1

c>0

ⓘ

Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x};\NVar{q}\right)$ : $q$ -Laguerre polynomial, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $y$ : real variable, $q$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $h_{n}$
Permalink:: http://dlmf.nist.gov/18.27.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §18.27(v), §18.27 and Ch.18

… ►

18.27.17_3

\lim_{q\to 1}L^{(\alpha)}_{n}\left((1-q)x;q\right)=L^{(\alpha)}_{n}\left(x% \right).

ⓘ

Symbols:: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x};\NVar{q}\right)$ : $q$ -Laguerre polynomial, $q$ : real variable, $n$ : nonnegative integer and $x$ : real variable
Source:: Koekoek et al. (2010, (14.21.18))
Referenced by:: §18.27(v), Erratum (V1.2.0) Section 18.27
Permalink:: http://dlmf.nist.gov/18.27.E17_3
Encodings:: pMML, png, TeX
Modification (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.27(v), §18.27(v), §18.27 and Ch.18

…

28: 23.14 Integrals

… ►

23.14.1

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

ⓘ

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

►

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,

ⓘ

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

►

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.

ⓘ

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

…

29: 23.2 Definitions and Periodic Properties

… ►The generators of a given lattice

\mathbb{L}

are not unique. …where

a, b, c, d

are integers, then

2\chi_{1}

2\chi_{3}

are generators of

\mathbb{L}

iff … ►When

z\notin\mathbb{L}

the functions are related by … ►When it is important to display the lattice with the functions they are denoted by

\wp\left(z|\mathbb{L}\right)

\zeta\left(z|\mathbb{L}\right)

, and

\sigma\left(z|\mathbb{L}\right)

, respectively. … ►If

2\omega_{1}

2\omega_{3}

is any pair of generators of

\mathbb{L}