Jacobi inversion formula

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1: 20.9 Relations to Other Functions

… ►The relations (20.9.1) and (20.9.2) between $k$ and $\tau$ (or $q$) are solutions of Jacobi’s inversion problem; see Baker (1995) and Whittaker and Watson (1927, pp. 480–485). …

2: 20.11 Generalizations and Analogs

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3: 22.21 Tables

§22.21 Tables

►Spenceley and Spenceley (1947) tabulates $\mathrm{sn}(Kx,k)$, $\mathrm{cn}(Kx,k)$, $\mathrm{dn}(Kx,k)$, $\mathrm{am}(Kx,k)$, $ℰ(Kx,k)$ for $\arcsin k=1^{\circ }(1^{\circ }){89}^{\circ }$ and $x=0\left(\frac{1}{90}\right)1$ to 12D, or 12 decimals of a radian in the case of $\mathrm{am}(Kx,k)$. … ►Lawden (1989, pp. 280–284 and 293–297) tabulates $\mathrm{sn}(x,k)$, $\mathrm{cn}(x,k)$, $\mathrm{dn}(x,k)$, $ℰ(x,k)$, $\mathrm{Z}\left(x|k\right)$ to 5D for $k=0.1(.1)0.9$, $x=0(.1)X$, where $X$ ranges from 1. … … ►Tables of theta functions (§20.15) can also be used to compute the twelve Jacobian elliptic functions by application of the quotient formulas given in §22.2.

4: 18.18 Sums

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Jacobi

… ►For formulas for Jacobi and Laguerre polynomials analogous to (18.18.8) and (18.18.9), see (Koornwinder, 1975b, 1977). … ►

§18.18(iv) Connection and Inversion Formulas

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Jacobi

… ►For the Poisson kernel of Jacobi polynomials (the Bailey formula) see Bailey (1938). …

5: 22.20 Methods of Computation

… ►and the inverse sine has its principal value (§4.23(ii)). …This formula for $\mathrm{dn}$ becomes unstable near $x=K$. … ►To compute $\mathrm{sn}$, $\mathrm{cn}$, $\mathrm{dn}$ to 10D when $x=0.8$, $k=0.65$. … ►

§22.20(v) Inverse Functions

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6: 18.3 Definitions

§18.3 Definitions

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As given by a Rodrigues formula (18.5.5).

… ►However, most of these formulas can be obtained by specialization of formulas for Jacobi polynomials, via (18.7.4)–(18.7.6). … ►

Jacobi on Other Intervals

… ►For $\nu \in ℝ$ and $N>-\frac{1}{2}$ a finite system of Jacobi polynomials $P_n^{(-N-1+\mathrm{i}\nu ,-N-1-\mathrm{i}\nu )}\left(\mathrm{i}x\right)$ (called pseudo Jacobi polynomials or Routh–Romanovski polynomials) is orthogonal on $(-\mathrm{\infty },\mathrm{\infty })$ with $w(x)={\left(1+x^2\right)}^{-N-1}{\mathrm{e}}^{2\nu \arctan x}$. …

7: Errata

… ►We have also incorporated material on continuous $q$-Jacobi polynomials, and several new limit transitions. … ►

The title of the paragraph which was previously “Gasper’s $q$-Analog of Clausen’s Formula” has been changed to “Gasper’s $q$-Analog of Clausen’s Formula (16.12.2)”.

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The wording was changed to make the integration variable more apparent.

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Additional keywords are being added to formulas (an ongoing project); these are visible in the associated ‘info boxes’ linked to the icons to the right of each formula, and provide better search capabilities.

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Special cases of normalization of Jacobi polynomials for which the general formula is undefined have been stated explicitly in Table 18.3.1.

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8: Bibliography E

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The numerical inversion of two classes of Kontorovich-Lebedev transform by direct quadrature. J. Comput. Appl. Math. 61 (1), pp. 43–72.

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External Links:

ISSN 0377-0427, Document, MathReview (J. P. Singhal), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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Uniform asymptotic expansions of the Jacobi polynomials and an associated function. Math. Comp. 25 (114), pp. 309–315.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§18.15(i).

Encodings:

BibTeX

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The Euler-Maclaurin formula revisited. J. Austral. Math. Soc. Ser. B 40 (E), pp. E27–E76 (electronic).

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External Links:

ISSN 0334-2700, FullText, MathReview (J. Németh), ZentralBlatt

Referenced by:

§2.10(i).

Encodings:

BibTeX

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A formula including Legendre’s $E{K}^{\prime }+K{E}^{\prime }-K{K}^{\prime }=\frac{1}{2}\pi$ . Messenger of Math. 33, pp. 31–32.

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Notes:

Errata on p. 45 of the same volume.

External Links:

ZentralBlatt

Referenced by:

§15.16.

Encodings:

BibTeX

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9: 15.9 Relations to Other Functions

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Jacobi

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§15.9(ii) Jacobi Function

… ►The Jacobi transform is defined as …with inverse … …

10: 18.17 Integrals

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Jacobi

… ►For formulas for Jacobi and Laguerre polynomials analogous to (18.17.5) and (18.17.6), see Koornwinder (1974, 1977). … ►For similar formulas for ultraspherical polynomials see Durand (1975), and for Jacobi and Laguerre polynomials see Durand (1978). … ►

Jacobi

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Jacobi

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