.%E9%98%BF%E6%A0%B9%E5%BB%B7%E4%B8%96%E7%95%8C%E6%9D%AF%E9%A2%84%E9%80%89%E8%B5%9B_%E3%80%8E%E7%BD%91%E5%9D%80%3A68707.vip%E3%80%8F%E4%B8%96%E7%95%8C%E6%9D%AF%E5%A1%9E%E5%B0%94%E7%BB%B4%E4%BA%9A%E5%AF%B9%E5%93%A5_b5p6v3_2022%E5%B9%B411%E6%9C%8828%E6%97%A521%E6%97%B621%E5%88%8647%E7%A7%92_6smwawga4

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21: Bibliography C

… ►

B. C. Carlson (1977a) Elliptic integrals of the first kind. SIAM J. Math. Anal. 8 (2), pp. 231–242.

ⓘ

External Links:: Document, MathReview (H. E. Fettis), ZentralBlatt
Referenced by:: §19.29(iii).
Encodings:: BibTeX

… ►

T. M. Cherry (1948) Expansions in terms of parabolic cylinder functions. Proc. Edinburgh Math. Soc. (2) 8, pp. 50–65.

ⓘ

External Links:: MathReview (I. I. Hirschman, Jr.), ZentralBlatt
Referenced by:: §12.16.
Encodings:: BibTeX

… ►

C. W. Clark (1979) Coulomb phase shift. American Journal of Physics 47 (8), pp. 683–684.

ⓘ

External Links:: Document
Referenced by:: §5.20.
Encodings:: BibTeX

… ►

C. W. Clenshaw, F. W. J. Olver, and P. R. Turner (1989) Level-Index Arithmetic: An Introductory Survey. In Numerical Analysis and Parallel Processing (Lancaster, 1987), P. R. Turner (Ed.), Lecture Notes in Math., Vol. 1397, pp. 95–168.

ⓘ

External Links:: MathReview Entry, ZentralBlatt
Referenced by:: §3.1(iv).
Encodings:: BibTeX

… ►

W. J. Cody and K. E. Hillstrom (1967) Chebyshev approximations for the natural logarithm of the gamma function. Math. Comp. 21 (98), pp. 198–203.

ⓘ

External Links:: FullText, ZentralBlatt
Referenced by:: §5.23(i).
Encodings:: BibTeX

…

22: 9.2 Differential Equation

… ►

9.2.2

w=\operatorname{Ai}\left(z\right),\;\operatorname{Bi}\left(z\right),\;% \operatorname{Ai}\left(ze^{\mp 2\pi\mathrm{i}/3}\right).

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable and $w$ : ODE solution
Source:: Olver (1997b, pp. 392–393, 413–414)
Permalink:: http://dlmf.nist.gov/9.2.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §9.2(i), §9.2 and Ch.9

… ►

9.2.3

\operatorname{Ai}\left(0\right)=\frac{1}{3^{2/3}\Gamma\left(\tfrac{2}{3}\right% )}=0.35502\;80538\ldots,

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function and $\Gamma\left(\NVar{z}\right)$ : gamma function
Source:: Olver (1997b, (1.03), p. 392)
A&S Ref:: 10.4.4 (with more digits)
Permalink:: http://dlmf.nist.gov/9.2.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §9.2(ii), §9.2 and Ch.9

►

9.2.4

\operatorname{Ai}'\left(0\right)=-\frac{1}{3^{1/3}\Gamma\left(\tfrac{1}{3}% \right)}=-0.25881\;94037\ldots,

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function and $\Gamma\left(\NVar{z}\right)$ : gamma function
Source:: Olver (1997b, (1.03), p. 392)
A&S Ref:: 10.4.5 (with more digits)
Permalink:: http://dlmf.nist.gov/9.2.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §9.2(ii), §9.2 and Ch.9

►

9.2.5

\operatorname{Bi}\left(0\right)=\frac{1}{3^{1/6}\Gamma\left(\tfrac{2}{3}\right% )}=0.61492\;66274\ldots,

ⓘ

Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function and $\Gamma\left(\NVar{z}\right)$ : gamma function
Source:: Olver (1997b, (1.11), p. 393)
A&S Ref:: 10.4.4 (in different form)
Referenced by:: (9.12.15)
Permalink:: http://dlmf.nist.gov/9.2.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §9.2(ii), §9.2 and Ch.9

►

9.2.6

\operatorname{Bi}'\left(0\right)=\frac{3^{1/6}}{\Gamma\left(\tfrac{1}{3}\right% )}=0.44828\;83573\ldots.

ⓘ

Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function and $\Gamma\left(\NVar{z}\right)$ : gamma function
Source:: Olver (1997b, (1.11), p. 393)
A&S Ref:: 10.4.5 (in different form)
Referenced by:: (9.12.15)
Permalink:: http://dlmf.nist.gov/9.2.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §9.2(ii), §9.2 and Ch.9

…

23: 23.10 Addition Theorems and Other Identities

… ►(23.10.8) continues to hold when

e_{1}

e_{2}

e_{3}

are permuted cyclically. … ►For

n=2,3,\dots

, …where … ►

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

… ►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. …

24: 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

…

25: 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}

, and

\omega_{2}

is defined by (23.2.1), then …

26: 23.9 Laurent and Other Power Series

… ►

23.9.1

c_{n}=(2n-1)\sum_{w\in\mathbb{L}\setminus\{0\}}w^{-2n},

n=2,3,4,\dots

ⓘ

Symbols:: $\in$ : element of, $\setminus$ : set subtraction, $\mathbb{L}$ : lattice, $n$ : integer and $c_{n}$
Referenced by:: §20.6
Permalink:: http://dlmf.nist.gov/23.9.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §23.9 and Ch.23

… ►

23.9.2

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

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

ⓘ

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

… ►Explicit coefficients

c_{n}

in terms of

c_{2}

and

c_{3}

are given up to

c_{19}

in Abramowitz and Stegun (1964, p. 636). ►For

j=1,2,3

, and with

e_{j}

as in §23.3(i), … ►For

a_{m,n}

with

m=0,1,\dots,12

and

n=0,1,\dots,8

, see Abramowitz and Stegun (1964, p. 637).

27: 23.7 Quarter Periods

… ►

23.7.1

\wp\left(\tfrac{1}{2}\omega_{1}\right)=e_{1}+\sqrt{(e_{1}-e_{3})(e_{1}-e_{2})}% =e_{1}+\omega_{1}^{-2}(K\left(k\right))^{2}k^{\prime},

ⓘ

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 18.3.18
Permalink:: http://dlmf.nist.gov/23.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §23.7 and Ch.23

►

23.7.2

\wp\left(\tfrac{1}{2}\omega_{2}\right)=e_{2}-i\sqrt{(e_{1}-e_{2})(e_{2}-e_{3})% }=e_{2}-i\omega_{1}^{-2}(K\left(k\right))^{2}kk^{\prime},

ⓘ

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, $\mathbb{L}$ : lattice, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 18.3.18
Permalink:: http://dlmf.nist.gov/23.7.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §23.7 and Ch.23

►

23.7.3

\wp\left(\tfrac{1}{2}\omega_{3}\right)=e_{3}-\sqrt{(e_{1}-e_{3})(e_{2}-e_{3})}% =e_{3}-\omega_{1}^{-2}(K\left(k\right))^{2}k,

ⓘ

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $k$ : modulus
A&S Ref:: 18.3.18
Permalink:: http://dlmf.nist.gov/23.7.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §23.7 and Ch.23

…

28: 23.3 Differential Equations

… ►and are denoted by

e_{1},e_{2},e_{3}

. … ►Let

{g_{2}}^{3}\neq 27{g_{3}}^{2}

, or equivalently

\Delta

be nonzero, or

e_{1},e_{2},e_{3}

be distinct. Given

g_{2}

and

g_{3}

there is a unique lattice

\mathbb{L}

such that (23.3.1) and (23.3.2) are satisfied. … ►Conversely,

g_{2}

g_{3}

, and the set

\{e_{1},e_{2},e_{3}\}

are determined uniquely by the lattice

\mathbb{L}

independently of the choice of generators. However, given any pair of generators

2\omega_{1}

2\omega_{3}

\mathbb{L}

, and with

\omega_{2}

defined by (23.2.1), we can identify the

e_{j}

individually, via …

29: 23.1 Special Notation

… ►The main functions treated in this chapter are the Weierstrass

\wp

-function

\wp\left(z\right)=\wp\left(z|\mathbb{L}\right)=\wp\left(z;g_{2},g_{3}\right)

; the Weierstrass zeta function

\zeta\left(z\right)=\zeta\left(z|\mathbb{L}\right)=\zeta\left(z;g_{2},g_{3}\right)

; the Weierstrass sigma function

\sigma\left(z\right)=\sigma\left(z|\mathbb{L}\right)=\sigma\left(z;g_{2},g_{3}\right)

; the elliptic modular function

\lambda\left(\tau\right)

; Klein’s complete invariant

J\left(\tau\right)

; Dedekind’s eta function

\eta\left(\tau\right)

. … ►Whittaker and Watson (1927) requires only

\Im\left(\omega_{3}/\omega_{1}\right)\neq 0

, instead of

\Im\left(\omega_{3}/\omega_{1}\right)>0

. Abramowitz and Stegun (1964, Chapter 18) considers only rectangular and rhombic lattices (§23.5);

\omega_{1}

\omega_{3}

are replaced by

\omega

\omega^{\prime}

for the former and by

\omega_{2}

\omega^{\prime}

for the latter. Silverman and Tate (1992) and Koblitz (1993) replace

2\omega_{1}

and

2\omega_{3}

\omega_{1}

and

\omega_{3}

, respectively. Walker (1996) normalizes

2\omega_{1}=1

2\omega_{3}=\tau

, and uses homogeneity (§23.10(iv)). …

30: 23.6 Relations to Other Functions

… ►In this subsection

2\omega_{1}

2\omega_{3}

are any pair of generators of the lattice

\mathbb{L}

, and the lattice roots

e_{1}

e_{2}

e_{3}

are given by (23.3.9). … ►Again, in Equations (23.6.16)–(23.6.26),

2\omega_{1},2\omega_{3}

are any pair of generators of the lattice

\mathbb{L}

and

e_{1},e_{2},e_{3}

are given by (23.3.9). … ►Also,

\mathbb{L}_{\mspace{1.0mu}1}

\mathbb{L}_{\mspace{1.0mu}2}

\mathbb{L}_{\mspace{1.0mu}3}

are the lattices with generators

(4K,2\mathrm{i}{K^{\prime}})

(2K-2\mathrm{i}{K^{\prime}},2K+2\mathrm{i}{K^{\prime}})

(2K,4\mathrm{i}{K^{\prime}})

, respectively. … ►Let

z

be on the perimeter of the rectangle with vertices

0,2\omega_{1},2\omega_{1}+2\omega_{3},2\omega_{3}

. … ►Let

z

be a point of

\mathbb{C}

different from

e_{1},e_{2},e_{3}

, and define

w

by …