Jacobi%E2%80%99s

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21: 18.14 Inequalities

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Jacobi

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Jacobi

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Jacobi

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Szegő–Szász Inequality

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Jacobi

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22: 20.8 Watson’s Expansions

§20.8 Watson’s Expansions

►

20.8.1

\frac{\theta_{2}\left(0,q\right)\theta_{3}\left(z,q\right)\theta_{4}\left(z,q% \right)}{\theta_{2}\left(z,q\right)}=2\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}q% ^{n^{2}}e^{i2nz}}{q^{-n}e^{-iz}+q^{n}e^{iz}}.

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $n$ : integer, $z$ : complex and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.8.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §20.8 and Ch.20

…

23: 22.2 Definitions

… ►

Glaisher’s Notation

… ►Let

\mathrm{p}

\mathrm{q}

\mathrm{r}

be any three of the letters

\mathrm{s}

\mathrm{c}

\mathrm{d}

\mathrm{n}

. …

\operatorname{ss}\left(z,k\right)=1

. ►The six functions containing the letter

\mathrm{s}

in their two-letter name are odd in

z

; the other six are even in

z

. ►In terms of Neville’s theta functions (§20.1) …

24: 18.6 Symmetry, Special Values, and Limits to Monomials

… ►For Jacobi, ultraspherical, Chebyshev, Legendre, and Hermite polynomials, see Table 18.6.1. … ►

Table 18.6.1: Classical OP’s: symmetry and special values.

► ►►

$p_{n}(x)$	$p_{n}(-x)$	$p_{n}(1)$	$p_{2n}(0)$	$p_{2n+1}^{\prime}(0)$
…

► ►

§18.6(ii) Limits to Monomials

►

18.6.2

\lim_{\alpha\to\infty}\frac{P^{(\alpha,\beta)}_{n}\left(x\right)}{P^{(\alpha,% \beta)}_{n}\left(1\right)}=\left(\frac{1+x}{2}\right)^{n},

ⓘ

Symbols:: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.6(ii)
Permalink:: http://dlmf.nist.gov/18.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §18.6(ii), §18.6 and Ch.18

►

18.6.3

\lim_{\beta\to\infty}\frac{P^{(\alpha,\beta)}_{n}\left(x\right)}{P^{(\alpha,% \beta)}_{n}\left(-1\right)}=\left(\frac{1-x}{2}\right)^{n},

ⓘ

Symbols:: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.6(ii)
Permalink:: http://dlmf.nist.gov/18.6.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §18.6(ii), §18.6 and Ch.18

…

25: 20.15 Tables

… ►This reference gives

\theta_{j}\left(x,q\right)

j=1,2,3,4

, and their logarithmic

x

-derivatives to 4D for

x/\pi=0(.1)1

\alpha=0(9^{\circ})90^{\circ}

, where

\alpha

is the modular angle given by ►

20.15.1

\sin\alpha={\theta_{2}}^{2}\left(0,q\right)/{\theta_{3}}^{2}\left(0,q\right)=k.

ⓘ

Defines:: $\alpha$ : modular angle (locally)
Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\sin\NVar{z}$ : sine function and $q$ : nome
Referenced by:: §20.15, §20.15
Permalink:: http://dlmf.nist.gov/20.15.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §20.15 and Ch.20

►Spenceley and Spenceley (1947) tabulates

\theta_{1}\left(x,q\right)/\theta_{2}\left(0,q\right)

\theta_{2}\left(x,q\right)/\theta_{2}\left(0,q\right)

\theta_{3}\left(x,q\right)/\theta_{4}\left(0,q\right)

\theta_{4}\left(x,q\right)/\theta_{4}\left(0,q\right)

to 12D for

u=0(1^{\circ})90^{\circ}

\alpha=0(1^{\circ})89^{\circ}

, where

u=2x/(\pi{\theta_{3}}^{2}\left(0,q\right))

and

\alpha

is defined by (20.15.1), together with the corresponding values of

\theta_{2}\left(0,q\right)

and

\theta_{4}\left(0,q\right)

. ►Lawden (1989, pp. 270–279) tabulates

\theta_{j}\left(x,q\right)

j=1,2,3,4

, to 5D for

x=0(1^{\circ})90^{\circ}

q=0.1(.1)0.9

, and also

q

to 5D for

k^{2}=0(.01)1

. ►Tables of Neville’s theta functions

\theta_{s}\left(x,q\right)

\theta_{c}\left(x,q\right)

\theta_{d}\left(x,q\right)

\theta_{n}\left(x,q\right)

(see §20.1) and their logarithmic

x

-derivatives are given in Abramowitz and Stegun (1964, pp. 582–585) to 9D for

\varepsilon,\alpha=0(5^{\circ})90^{\circ}

, where (in radian measure)

\varepsilon=x/{\theta_{3}}^{2}\left(0,q\right)=\pi x/(2K\left(k\right))

, and

\alpha

is defined by (20.15.1). …

26: 22.15 Inverse Functions

… ►are denoted respectively by … ►

22.15.5

-K\leq\operatorname{arcsn}\left(x,k\right)\leq K,

ⓘ

Symbols:: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\operatorname{arcsn}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $x$ : real and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.15.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §22.15(i), §22.15 and Ch.22

►

22.15.6

0\leq\operatorname{arccn}\left(x,k\right)\leq 2K,

ⓘ

Symbols:: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\operatorname{arccn}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $x$ : real and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.15.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §22.15(i), §22.15 and Ch.22

… ►Equations (22.15.1) and (22.15.4), for

\operatorname{arcsn}\left(x,k\right)

, are equivalent to (22.15.12) and also to … ►

§22.15(ii) Representations as Elliptic Integrals

…

27: 20.1 Special Notation

… ►When

\tau

is fixed the notation is often abbreviated in the literature as

\theta_{j}(z)

, or even as simply

\theta_{j}

, it being then understood that the argument is the primary variable. … ►Primes on the

\theta

symbols indicate derivatives with respect to the argument of the

\theta

function. … ►Jacobi’s original notation:

\Theta(z|\tau)

\Theta_{1}(z|\tau)

\mathrm{H}(z|\tau)

\mathrm{H}_{1}(z|\tau)

, respectively, for

\theta_{4}\left(u\middle|\tau\right)

\theta_{3}\left(u\middle|\tau\right)

\theta_{1}\left(u\middle|\tau\right)

\theta_{2}\left(u\middle|\tau\right)

, where

u=z/{\theta_{3}}^{2}\left(0\middle|\tau\right)

. … ►Neville’s notation:

\theta_{s}(z|\tau)

\theta_{c}(z|\tau)

\theta_{d}(z|\tau)

\theta_{n}(z|\tau)

, respectively, for

{\theta_{3}}^{2}\left(0\middle|\tau\right)\ifrac{\theta_{1}\left(u\middle|\tau% \right)}{\theta_{1}'\left(0\middle|\tau\right)}

\ifrac{\theta_{2}\left(u\middle|\tau\right)}{\theta_{2}\left(0\middle|\tau% \right)}

\ifrac{\theta_{3}\left(u\middle|\tau\right)}{\theta_{3}\left(0\middle|\tau% \right)}

\ifrac{\theta_{4}\left(u\middle|\tau\right)}{\theta_{4}\left(0\middle|\tau% \right)}

, where again

u=z/{\theta_{3}}^{2}\left(0\middle|\tau\right)

. … ►McKean and Moll’s notation:

\vartheta_{j}(z|\tau)=\theta_{j}\left(\pi z\middle|\tau\right)

j=1,2,3,4

. …

28: 22.5 Special Values

… ►For example, at

z=K+iK^{\prime}

\operatorname{sn}\left(z,k\right)=1/k

\ifrac{\mathrm{d}\operatorname{sn}\left(z,k\right)}{\mathrm{d}z}=0

. … ►Table 22.5.2 gives

\operatorname{sn}\left(z,k\right)

\operatorname{cn}\left(z,k\right)

\operatorname{dn}\left(z,k\right)

for other special values of

z

. For example,

\operatorname{sn}\left(\frac{1}{2}K,k\right)=(1+k^{\prime})^{-1/2}

. … ►

Table 22.5.3: Limiting forms of Jacobian elliptic functions as

k\to 0

► ►►►

$\operatorname{sn}\left(z,k\right)$ $\;\to\;$	$\sin z$	$\operatorname{cd}\left(z,k\right)$ $\;\to\;$	$\cos z$	$\operatorname{dc}\left(z,k\right)$ $\;\to\;$	$\sec z$	$\operatorname{ns}\left(z,k\right)$ $\;\to\;$	$\csc z$
…
$\operatorname{dn}\left(z,k\right)$ $\;\to\;$	$1$	$\operatorname{nd}\left(z,k\right)$ $\;\to\;$	$1$	$\operatorname{sc}\left(z,k\right)$ $\;\to\;$	$\tan z$	$\operatorname{cs}\left(z,k\right)$ $\;\to\;$	$\cot z$

► …

29: 20.2 Definitions and Periodic Properties

… ►

§20.2(i) Fourier Series

… ►Corresponding expansions for

\theta_{j}'\left(z\middle|\tau\right)

j=1,2,3,4

, can be found by differentiating (20.2.1)–(20.2.4) with respect to

z

. … ►For fixed

\tau

, each

\theta_{j}\left(z\middle|\tau\right)

is an entire function of

z

with period

2\pi

;

\theta_{1}\left(z\middle|\tau\right)

is odd in

z

and the others are even. For fixed

z

, each of

\ifrac{\theta_{1}\left(z\middle|\tau\right)}{\sin z}

\ifrac{\theta_{2}\left(z\middle|\tau\right)}{\cos z}

\theta_{3}\left(z\middle|\tau\right)

, and

\theta_{4}\left(z\middle|\tau\right)

is an analytic function of

\tau

for

\Im\tau>0

, with a natural boundary

\Im\tau=0

, and correspondingly, an analytic function of

q

for

\left|q\right|<1

with a natural boundary

\left|q\right|=1

. … ►For

m,n\in\mathbb{Z}

, the

z

-zeros of

\theta_{j}\left(z\middle|\tau\right)

j=1,2,3,4

, are

(m+n\tau)\pi

(m+\tfrac{1}{2}+n\tau)\pi

(m+\tfrac{1}{2}+(n+\tfrac{1}{2})\tau)\pi

(m+(n+\tfrac{1}{2})\tau)\pi

respectively.

30: 18.9 Recurrence Relations and Derivatives

… ►For

p_{n}(x)=P^{(\alpha,\beta)}_{n}\left(x\right)

, … ►For

p_{n}(x)=P^{(\alpha,\beta)}_{n}\left(x\right)

, … ►

Jacobi

… ►

Jacobi

… ►Further

n

-th derivative formulas relating two different Jacobi polynomials can be obtained from §15.5(i) by substitution of (18.5.7). …