Lagrange inversion theorem

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11: 22.18 Mathematical Applications

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Lemniscate

… ►Inversely: … ►See Akhiezer (1990, Chapter 8) and McKean and Moll (1999, Chapter 2) for discussions of the inverse mapping. … ►

§22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem

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12: Bibliography B

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Abelian Functions: Abel’s Theorem and the Allied Theory of Theta Functions. Cambridge University Press, Cambridge.

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Notes:

Reprint of the 1897 original, with a foreword by Igor Krichever.

External Links:

ISBN 0-521-49877-5, MathReview (John B. Little), ZentralBlatt

Referenced by:

§20.9(ii).

Encodings:

BibTeX

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Barycentric Lagrange interpolation. SIAM Rev. 46 (3), pp. 501–517.

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External Links:

ISSN 0036-1445, Document, Link, MathReview (Mao Dong Ye), ZentralBlatt

Referenced by:

§3.3(i).

Encodings:

BibTeX

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Waves and Thom’s theorem. Advances in Physics 25 (1), pp. 1–26.

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External Links:

Document

Referenced by:

§36.14(i).

Encodings:

BibTeX

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Rational Chebyshev approximations for the inverse of the error function. Math. Comp. 30 (136), pp. 827–830.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§7.17(iii), §7.17(iii).

Encodings:

BibTeX

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13: 35.2 Laplace Transform

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Inversion Formula

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Convolution Theorem

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14: Bibliography G

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On mean convergence of extended Lagrange interpolation. J. Comput. Appl. Math. 43 (1-2), pp. 19–35.

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Notes:

Orthogonal polynomials and numerical methods

External Links:

ISSN 0377-0427, Document, Link, MathReview (Mircea Ivan), ZentralBlatt

Referenced by:

Erratum (V1.0.28) for Table 18.3.1.

Encodings:

BibTeX

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A theorem on the numerators of the Bernoulli numbers. Amer. Math. Monthly 97 (2), pp. 136–138.

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External Links:

ISSN 0002-9890, FullText, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.10(iv), §24.16(iii).

Encodings:

BibTeX

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Multilateral summation theorems for ordinary and basic hypergeometric series in $\mathrm{U}(n)$ . SIAM J. Math. Anal. 18 (6), pp. 1576–1596.

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External Links:

ISSN 0036-1410, Document, MathReview, ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

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15: 4.27 Sums

§4.27 Sums

►For sums of trigonometric and inverse trigonometric functions see Gradshteyn and Ryzhik (2000, Chapter 1), Hansen (1975, §§14–42), Oberhettinger (1973), and Prudnikov et al. (1986a, Chapter 5).

16: 19.26 Addition Theorems

§19.26 Addition Theorems

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19.26.11 $$R_C(x+\lambda ,y+\lambda )+R_C(x+\mu ,y+\mu )=R_C(x,y),$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\lambda$: positive, $x$: positive, $y$: positive and $\mu >0$: parameter

Referenced by:

§19.26(i)

Permalink:

http://dlmf.nist.gov/19.26.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §19.26(i), §19.26 and Ch.19

… ►The equations inverse to $z+\lambda =(\sqrt{z}+\sqrt{x})(\sqrt{z}+\sqrt{y})$ and the two other equations obtained by permuting $x,y,z$ (see (19.26.19)) are … ►

19.26.25 $$R_C(x,y)=2R_C(x+\lambda ,y+\lambda ),$$ $\lambda =y+2\sqrt{x}\sqrt{y}$.

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\lambda$: positive, $x$: positive and $y$: positive

Referenced by:

§19.26(iii)

Permalink:

http://dlmf.nist.gov/19.26.E25

Encodings:

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See also:

Annotations for §19.26(iii), §19.26 and Ch.19

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19.26.27 $$R_C(x^2,x^2-\theta )=2R_C(s^2,s^2-\theta ),$$ $s=x+\sqrt{x^2-\theta }$, $\theta \ne x^2$ or $s^2$.

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $x$: positive

Permalink:

http://dlmf.nist.gov/19.26.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §19.26(iii), §19.26 and Ch.19

17: 3.11 Approximation Techniques

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3.11.6 $$T_n\left(x\right)=\cos \left(n\arccos x\right),$$ $-1\le x\le 1$.

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $\cos z$: cosine function and $\arccos z$: arccosine function

Referenced by:

§3.11(ii)

Permalink:

http://dlmf.nist.gov/3.11.E6

Encodings:

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See also:

Annotations for §3.11(ii), §3.11 and Ch.3

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Laplace Transform Inversion

►Numerical inversion of the Laplace transform (§1.14(iii)) … ►If $J=n+1$, then $p_n(x)$ is the Lagrange interpolation polynomial for the set $x_1,x_2,\mathrm{\dots },x_J$ (§3.3(i)). … ►For many applications a spline function is a more adaptable approximating tool than the Lagrange interpolation polynomial involving a comparable number of parameters; see §3.3(i), where a single polynomial is used for interpolating $f(x)$ on the complete interval $[a,b]$. …

18: Bibliography L

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Démonstration d’un Théoréme d’Arithmétique. Nouveau Mém. Acad. Roy. Sci. Berlin, pp. 123–133 (French).

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Notes:

Reprinted in Oeuvres, Vol. 3, pp. 189–201, Gauthier-Villars, Paris, 1867

External Links:

FullText

Referenced by:

§27.13(iii).

Encodings:

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19: 27.15 Chinese Remainder Theorem

§27.15 Chinese Remainder Theorem

►The Chinese remainder theorem states that a system of congruences $x\equiv a_1(mod{m}_1),\mathrm{\dots },x\equiv a_k(mod{m}_k)$, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod $m$), where $m$ is the product of the moduli. ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod $m_1$), (mod $m_2$), (mod $m_3$), and (mod $m_4$), respectively. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $(modm)$, which is correct to 20 digits. …

20: 4.47 Approximations

§4.47 Approximations

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§4.47(i) Chebyshev-Series Expansions

►Clenshaw (1962) and Luke (1975, Chapter 3) give 20D coefficients for $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\cot$, $\arcsin$, $\arctan$, $\mathrm{arcsinh}$. … ►Hart et al. (1968) give $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\cot$, $\arcsin$, $\arccos$, $\arctan$, $\sinh$, $\cosh$, $\tanh$, $\mathrm{arcsinh}$, $\mathrm{arccosh}$. … ►Luke (1975, Chapter 3) supplies real and complex approximations for $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\arctan$, $\mathrm{arcsinh}$. …