Jacobi original notation

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2: 18.27 $q$ -Hahn Class

… ►

§18.27(iii) Big $q$ -Jacobi Polynomials

… ►Alternative definitions and notations are … ►

From Big $q$ -Jacobi to Jacobi

… ►

From Big $q$ -Jacobi to Little $q$ -Jacobi

… ►

From Little $q$ -Jacobi to Jacobi

…

A sentence was added after (1.2.1) to refer to (1.2.6) as the definition of the binomial coefficient $\genfrac{(}{)}{0.0pt}{}{z}{k}$ when $z$ is complex. As a notational clarification, wherever $n$ appeared originally in (1.2.6)–(1.2.9), it was replaced by $z$ .

… ►

Table 22.4.3

Originally a minus sign was missing in the entries for $\operatorname{cd}u$ and $\operatorname{dc}u$ in the second column (headed $z+K+iK^{\prime}$ ). The correct entries are $-k^{-1}\operatorname{ns}z$ and $-k\operatorname{sn}z$ . Note: These entries appear online but not in the published print edition. More specifically, Table 22.4.3 in the published print edition is restricted to the three Jacobian elliptic functions $\operatorname{sn},\operatorname{cn},\operatorname{dn}$ , whereas Table 22.4.3 covers all 12 Jacobian elliptic functions.

	$u$
	$z+K$	$z+K+\mathrm{i}K^{\prime}$	$z+\mathrm{i}K^{\prime}$	$z+2K$	$z+2K+2\mathrm{i}K^{\prime}$	$z+2\mathrm{i}K^{\prime}$
$\vdots$
$\operatorname{cd}u$	$-\operatorname{sn}z$	$-k^{-1}\operatorname{ns}z$	$k^{-1}\operatorname{dc}z$	$-\operatorname{cd}z$	$-\operatorname{cd}z$	$\operatorname{cd}z$
$\vdots$
$\operatorname{dc}u$	$-\operatorname{ns}z$	$-k\operatorname{sn}z$	$k\operatorname{cd}z$	$-\operatorname{dc}z$	$-\operatorname{dc}z$	$\operatorname{dc}z$
$\vdots$

Reported 2014-02-28 by Svante Janson.

… ►

Equation (22.6.7)

22.6.7

\operatorname{dn}\left(2z,k\right)=\frac{{\operatorname{dn}}^{2}\left(z,k% \right)-k^{2}{\operatorname{sn}}^{2}\left(z,k\right){\operatorname{cn}}^{2}% \left(z,k\right)}{1-k^{2}{\operatorname{sn}}^{4}\left(z,k\right)}=\frac{{% \operatorname{dn}}^{4}\left(z,k\right)+k^{2}{k^{\prime}}^{2}{\operatorname{sn}% }^{4}\left(z,k\right)}{1-k^{2}{\operatorname{sn}}^{4}\left(z,k\right)}

Originally the term $k^{2}{\operatorname{sn}}^{2}\left(z,k\right){\operatorname{cn}}^{2}\left(z,k\right)$ was given incorrectly as $k^{2}{\operatorname{sn}}^{2}\left(z,k\right){\operatorname{dn}}^{2}\left(z,k\right)$ .

Reported 2014-02-28 by Svante Janson.

… ►

Table 22.5.4

Originally the limiting form for $\operatorname{sc}\left(z,k\right)$ in the last line of this table was incorrect ( $\cosh z$ , instead of $\sinh z$ ).

$\operatorname{sn}\left(z,k\right)$ $\;\to\;$	$\tanh z$	$\operatorname{cd}\left(z,k\right)$ $\;\to\;$	$1$	$\operatorname{dc}\left(z,k\right)$ $\;\to\;$	$1$	$\operatorname{ns}\left(z,k\right)$ $\;\to\;$	$\coth z$
$\operatorname{cn}\left(z,k\right)$ $\;\to\;$	$\operatorname{sech}z$	$\operatorname{sd}\left(z,k\right)$ $\;\to\;$	$\sinh z$	$\operatorname{nc}\left(z,k\right)$ $\;\to\;$	$\cosh z$	$\operatorname{ds}\left(z,k\right)$ $\;\to\;$	$\operatorname{csch}z$
$\operatorname{dn}\left(z,k\right)$ $\;\to\;$	$\operatorname{sech}z$	$\operatorname{nd}\left(z,k\right)$ $\;\to\;$	$\cosh z$	$\operatorname{sc}\left(z,k\right)$ $\;\to\;$	$\sinh z$	$\operatorname{cs}\left(z,k\right)$ $\;\to\;$	$\operatorname{csch}z$

Reported 2010-11-23.

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Equation (22.16.14)

22.16.14

\mathcal{E}\left(x,k\right)=\int_{0}^{\operatorname{sn}\left(x,k\right)}\sqrt{% \frac{1-k^{2}t^{2}}{1-t^{2}}}\,\mathrm{d}t

Originally this equation appeared with the upper limit of integration as $x$ , rather than $\operatorname{sn}\left(x,k\right)$ .

Reported 2010-07-08 by Charles Karney.

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4: 22.9 Cyclic Identities

… ►

§22.9(i) Notation

►The following notation is a generalization of that of Khare and Sukhatme (2002). … ►

22.9.1

s_{m,p}^{(2)}=\operatorname{sn}\left(z+2p^{-1}(m-1)K\left(k\right),k\right),

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Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $z$ : complex, $k$ : modulus, $m$ : positive integer, $p$ : positive integer and $s_{m,p}^{(r)}$ : cyclic quantity
Permalink:: http://dlmf.nist.gov/22.9.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §22.9(i), §22.9 and Ch.22

… ►The argument

z

is suppressed in the above notation, as all cyclic identities are independent of

z

. … ►

22.9.7

\kappa=\operatorname{dn}\left(2K\left(k\right)/3,k\right),

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Defines:: $\kappa$ : change of variable (locally)
Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind and $k$ : modulus
Referenced by:: §22.9(iii), §22.9(iv)
Permalink:: http://dlmf.nist.gov/22.9.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §22.9(ii), §22.9(ii), §22.9 and Ch.22

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5: 22.4 Periods, Poles, and Zeros

… ►

§22.4(ii) Graphical Interpretation via Glaisher’s Notation

… ►Using the p,q notation of (22.2.10), Figure 22.4.2 serves as a mnemonic for the poles, zeros, periods, and half-periods of the 12 Jacobian elliptic functions as follows. …Then: (a) In any lattice unit cell

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

has a simple zero at

z=\mbox{p}

and a simple pole at

z=\mbox{q}

. (b) The difference between p and the nearest q is a half-period of

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

. … ►For example,

\operatorname{sn}\left(z+K,k\right)=\operatorname{cd}\left(z,k\right)

. …

6: Bibliography J

… ►

C. G. J. Jacobi (1829) Fundamenta Nova Theoriae Functionum Ellipticarum. Regiomonti, Sumptibus fratrum Bornträger.

ⓘ

Notes:: Reprinted in Werke, Vol. I, pp. 49–239, Herausgegeben von C. W. Borchardt, Berlin, Verlag von G. Reimer, 1881.
External Links:: Werke TOC
Referenced by:: §27.13(iv).
Encodings:: BibTeX

… ►

E. Jahnke and F. Emde (1945) Tables of Functions with Formulae and Curves. 4th edition, Dover Publications, New York.

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Notes:: Contains corrections and enlargements of earlier editions. Originally published by B. G. Teubner, Leipzig, in 1933. In German and English.
External Links:: MathReview, ZentralBlatt
Referenced by:: 2nd item, §20.15, §5.1, Notations P, E. Jahnke, F. Emde, and F. Lösch (1966).
Encodings:: BibTeX

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J. K. M. Jansen (1977) Simple-periodic and Non-periodic Lamé Functions. Mathematical Centre Tracts, No. 72, Mathematisch Centrum, Amsterdam.

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External Links:: ISBN 90-6196-130-0, MathReview (V. P. Madan)
Referenced by:: §29.1, §29.1, §29.18(i), §29.19(i), §29.20(i), §29.20(i), §29.22(ii), §29.3(v), §29.6(i), §29.6(ii), §29.6(iii), §29.6(iv), Ch.29, Notations L, Notations L, Notations L, Notations L.
Encodings:: BibTeX

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H. Jeffreys and B. S. Jeffreys (1956) Methods of Mathematical Physics. 3rd edition, Cambridge University Press, Cambridge.

ⓘ

Notes:: Reprinted in 1999.
External Links:: MathReview, ZentralBlatt
Referenced by:: §10.1, §12.17, Ch.14, Ch.6, §7.18(iv), Ch.9, Notations H, Notations H, Notations K.
Encodings:: BibTeX

… ►

C. Jordan (1939) Calculus of Finite Differences. Hungarian Agent Eggenberger Book-Shop, Budapest.

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Notes:: Third edition published in 1965 by Chelsea Publishing Co., New York, and American Mathematical Society, AMS Chelsea Book Series, Providence, RI
External Links:: MathReview (W. E. Milne), ZentralBlatt
Referenced by:: §26.1, §26.1, §26.8(vii), §5.16, Notations S.
Encodings:: BibTeX

…

7: Bibliography

… ►

M. Abramowitz and I. A. Stegun (Eds.) (1964) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..

ⓘ

Notes:: Corrections appeared in later printings up to the 10th Printing, December, 1972. Reproductions by other publishers, in whole or in part, have been available since 1965.
External Links:: FullText, MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §10.1, §10.21(ix), §10.47(i), §10.68(iv), §10.70, 4th item, 2nd item, 4th item, 5th item, 1st item, 1st item, 4th item, 1st item, 1st item, Ch.10, §11.10(vi), 1st item, 1st item, Ch.11, 1st item, Ch.12, 3rd item, Ch.13, 1st item, Ch.14, Ch.15, §18.14(i), §18.3, §18.41(i), §18.41(ii), §18.5(iv), §18.8, §19.1, §19.14(i), §19.14(ii), §19.25(v), §19.36(iii), §19.37(ii), §19.37(ii), §19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.38, Ch.19, §20.15, Ch.20, §22.1, §22.15(ii), Ch.22, §23.1, §23.20(i), §23.23, §23.23, §23.9, §23.9, §23.9, Ch.23, §24.2(iv), §24.20, Ch.24, 1st item, §26.1, §26.1, §26.2, §26.21, §26.3(i), §26.4(i), §26.8(i), §27.2(ii), §27.21, §28.1, Ch.28, §30.1, Ch.30, 1st item, Ch.33, §4.44, §4.46, §5.11(i), §5.22(ii), §5.22(iii), §5.7(i), Ch.5, 1st item, 1st item, 1st item, Ch.6, 1st item, 2nd item, 1st item, Ch.7, (8.17.24), §8.22(ii), 1st item, Ch.8, 1st item, 1st item, 1st item, §9.8(i), Ch.9, Profile Frank W. J. Olver, About the Project, Preface, Possible Errors in DLMF, Notations E, Notations E, Notations F, Notations F, Notations H, Notations H, Notations J, Notations K, Notations P, Notations P, Notations S, Notations Y, H. E. Fettis and J. C. Caslin (1973).
Encodings:: BibTeX

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V. I. Arnol^′d (1997) Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, Vol. 60, Springer-Verlag, New York.

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Notes:: Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein, Corrected reprint of the second (1989) edition
External Links:: ISBN 0-387-96890-3, MathReview
Referenced by:: §21.5(i).
Encodings:: BibTeX

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F. M. Arscott (1964b) Periodic Differential Equations. An Introduction to Mathieu, Lamé, and Allied Functions. International Series of Monographs in Pure and Applied Mathematics, Vol. 66, Pergamon Press, The Macmillan Co., New York.

ⓘ

External Links:: MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §28.1, §28.1, §28.10(i), §28.11, §28.12(ii), §28.17, §28.2(ii), §28.2(iii), §28.2(iv), §28.28(i), §28.28(ii), §28.29(i), §28.32(i), §28.32(ii), §28.4(i), §28.5(i), Ch.28, §29.1, §29.11, §29.12(i), §29.12(ii), §29.12(iii), §29.14, §29.15(ii), §29.15(ii), §29.17(i), §29.18(iii), Ch.29, §30.1, §30.10, §30.11(i), §30.11(vi), §30.2(ii), §30.4(iii), §30.6, §30.8(i), §30.8(ii), Ch.30, §31.4, §31.5, §31.9(ii), Notations F, Notations G, Notations M, Notations N.
Encodings:: BibTeX

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R. Askey and J. Fitch (1969) Integral representations for Jacobi polynomials and some applications. J. Math. Anal. Appl. 26 (2), pp. 411–437.

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External Links:: Document, MathReview (D. L. Colton), ZentralBlatt
Referenced by:: §18.17(iv).
Encodings:: BibTeX

… ►

R. Askey and B. Razban (1972) An integral for Jacobi polynomials. Simon Stevin 46, pp. 165–169.

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External Links:: MathReview (H. M. Srivastava), ZentralBlatt
Referenced by:: §18.17(viii).
Encodings:: BibTeX

…

8: 19.25 Relations to Other Functions

… ►For the notation see §§22.2, 22.15, and 22.16(i). … ►If

{\operatorname{cs}}^{2}\left(u,k\right)\geq 0

, then ►

19.25.30

\operatorname{am}\left(u,k\right)=R_{C}\left({\operatorname{cs}}^{2}\left(u,k% \right),{\operatorname{ns}}^{2}\left(u,k\right)\right),

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\operatorname{am}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s amplitude function, $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function and $k$ : real or complex modulus
Referenced by:: §19.25(v)
Permalink:: http://dlmf.nist.gov/19.25.E30
Encodings:: pMML, png, TeX
See also:: Annotations for §19.25(v), §19.25 and Ch.19

… ►where we assume

0\leq x^{2}\leq 1

x=\operatorname{sn}

\operatorname{cn}

, or

\operatorname{cd}

;

x^{2}\geq 1

x=\operatorname{ns}

\operatorname{nc}

, or

\operatorname{dc}

;

x

real if

x=\operatorname{cs}

\operatorname{sc}

;

k^{\prime}\leq x\leq 1

x=\operatorname{dn}

;

1\leq x\leq 1/k^{\prime}

x=\operatorname{nd}

;

x^{2}\geq{k^{\prime}}^{2}

x=\operatorname{ds}

;

0\leq x^{2}\leq 1/{k^{\prime}}^{2}

x=\operatorname{sd}

. … ►For the notation see §23.2 and §23.3. …

9: 18.9 Recurrence Relations and Derivatives

… ►The notation of §18.2(iv) will be used. … ►For

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

, … ►For

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

, … ►

Jacobi

… ►

Jacobi

…

10: Bibliography F

… ►

J. Faraut and A. Korányi (1994) Analysis on Symmetric Cones. Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford-New York.

ⓘ

External Links:: ISBN 0-19-853477-9, MathReview (Hong Ming Ding), ZentralBlatt
Referenced by:: §35.1, §35.4(i), §35.7(i), §35.7(iii), §35.8(i), §35.8(iv), §35.8(v), §35.8(v), Ch.35, Notations J, Notations K.
Encodings:: BibTeX

… ►

V. Fock (1945) Diffraction of radio waves around the earth’s surface. Acad. Sci. USSR. J. Phys. 9, pp. 255–266.

ⓘ

External Links:: MathReview, ZentralBlatt
Referenced by:: §9.1, Notations U, Notations V.
Encodings:: BibTeX

… ►

W. B. Ford (1960) Studies on Divergent Series and Summability & The Asymptotic Developments of Functions Defined by Maclaurin Series. Chelsea Publishing Co., New York.

ⓘ

Notes:: Reprint with corrections of two books published originally in 1916 and 1936.
External Links:: MathReview
Referenced by:: §2.10(iii).
Encodings:: BibTeX

… ►

T. Fort (1948) Finite Differences and Difference Equations in the Real Domain. Clarendon Press, Oxford.

ⓘ

External Links:: MathReview (W. J. Trjitzinsky), ZentralBlatt
Referenced by:: §26.1, §26.1, Notations S, Notations S.
Encodings:: BibTeX

… ►

C. L. Frenzen and R. Wong (1985b) A uniform asymptotic expansion of the Jacobi polynomials with error bounds. Canad. J. Math. 37 (5), pp. 979–1007.

ⓘ

External Links:: ISSN 0008-414X, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.15(i), §18.16(ii).
Encodings:: BibTeX

…

Jacobi original notation

1—10 of 11 matching pages

1: 20.1 Special Notation

2: 18.27 $q$ -Hahn Class

§18.27(iii) Big $q$ -Jacobi Polynomials

From Big $q$ -Jacobi to Jacobi

From Big $q$ -Jacobi to Little $q$ -Jacobi

From Little $q$ -Jacobi to Jacobi

3: Errata

4: 22.9 Cyclic Identities

§22.9(i) Notation

5: 22.4 Periods, Poles, and Zeros

§22.4(ii) Graphical Interpretation via Glaisher’s Notation

6: Bibliography J

7: Bibliography

8: 19.25 Relations to Other Functions

9: 18.9 Recurrence Relations and Derivatives

Jacobi

Jacobi

10: Bibliography F

Jacobi original notation

1—10 of 11 matching pages

1: 20.1 Special Notation

2: 18.27 q -Hahn Class

§18.27(iii) Big q -Jacobi Polynomials

From Big q -Jacobi to Jacobi

From Big q -Jacobi to Little q -Jacobi

From Little q -Jacobi to Jacobi

3: Errata

4: 22.9 Cyclic Identities

§22.9(i) Notation

5: 22.4 Periods, Poles, and Zeros

§22.4(ii) Graphical Interpretation via Glaisher’s Notation

6: Bibliography J

7: Bibliography

8: 19.25 Relations to Other Functions

9: 18.9 Recurrence Relations and Derivatives

Jacobi

Jacobi

10: Bibliography F

2: 18.27 $q$ -Hahn Class

§18.27(iii) Big $q$ -Jacobi Polynomials

From Big $q$ -Jacobi to Jacobi

From Big $q$ -Jacobi to Little $q$ -Jacobi

From Little $q$ -Jacobi to Jacobi