Gauss–Legendre formula

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1: 3.5 Quadrature

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Gauss–Legendre Formula

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Table 3.5.1: Nodes and weights for the 5-point Gauss–Legendre formula.

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$\pm x_k$		$w_k$
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Table 3.5.2: Nodes and weights for the 10-point Gauss–Legendre formula.

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$\pm x_k$	$w_k$
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Table 3.5.3: Nodes and weights for the 20-point Gauss–Legendre formula.

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$\pm x_k$	$w_k$
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Table 3.5.5: Nodes and weights for the 80-point Gauss–Legendre formula.

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$\pm x_k$	$w_k$
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2: 6.18 Methods of Computation

… ►For an application of the Gauss-Legendre formula (§3.5(v)) see Tooper and Mark (1968). …

3: 18.5 Explicit Representations

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§18.5(ii) Rodrigues Formulas

… ►Related formula: … ►and two similar formulas by symmetry; compare the second row in Table 18.6.1. … ►For corresponding formulas for Chebyshev, Legendre, and the Hermite ${\mathit{He}}_n$ polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11). … ►

Legendre

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4: Errata

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• Equations (4.45.8) and (4.45.9) have been replaced with equations that are better for numerically computing $\arctan x$.

• A new Subsection 13.29(v) Continued Fractions, has been added to cover computation of confluent hypergeometric functions by continued fractions.

• A new Subsection 14.5(vi) Addendum to §14.5(ii) $\mu \mathrm{=}\mathrm{0}$, $\nu \mathrm{=}\mathrm{2}$ , containing the values of Legendre and Ferrers functions for degree $\nu =2$ has been added.

• Subsection 14.18(iii) has been altered to identify Equations (14.18.6) and (14.18.7) as Christoffel’s Formulas.

• A new Subsection 15.19(v) Continued Fractions, has been added to cover computation of the Gauss hypergeometric functions by continued fractions.

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

• Bibliographic citations have been added in §§4.13, 5.6(i), 5.11(iii), 7.25(iii), 8.13(i), 10.37, 12.18, 14.11, 15.12(ii), 16.6, 18.16(ii), 18.16(iv), 18.24, 18.27(iv), 18.27(v),18.28(i), 24.13(i), 28.36(iii).

• Cross-references have been added in §§1.2(i), 10.19(iii), 10.23(ii), 17.2(iii), 18.15(iii), 19.2(iv), 19.16(i).

• Several small revisions have been made. For details see §§5.11(ii), 10.12, 10.19(ii), 18.9(i), 18.16(iv), 19.7(ii), 22.2, 32.11(v), 32.13(ii).

• Entries for the Sage computational system have been updated in the Software Index.

• The default document format for DLMF is now HTML5 which includes MathML providing better accessibility and display of mathematics.

• All interactive 3D graphics on the DLMF website have been recast using WebGL and X3DOM, improving portability and performance; WebGL it is now the default format.

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5: Frank W. J. Olver

… ►Olver joined NIST in 1961 after having been recruited by Milton Abramowitz to be the author of the Chapter “Bessel Functions of Integer Order” in the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, a publication which went on to become the most widely distributed and most highly cited publication in NIST’s history. … ►, Bessel functions, hypergeometric functions, Legendre functions). … ►In a review of that volume, Jet Wimp of Drexel University said that the papers “exemplify a redoubtable mathematical talent, the work of a man who has done more than almost anyone else in the 20th century to bestow on the discipline of applied mathematics the elegance and rigor that its earliest practitioners, such as Gauss and Laplace, would have wished for it. …

6: 15.9 Relations to Other Functions

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Legendre

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15.9.7 $$P_n\left(x\right)=F(\genfrac{}{}{0pt}{}{-n,n+1}{1};\frac{1-x}{2}).$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_{2}F_1(a,b;c;z)$ Gauss’ hypergeometric function, $P_n\left(x\right)$: Legendre polynomial, $x$: real variable and $n$: integer

Permalink:

http://dlmf.nist.gov/15.9.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

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§15.9(iv) Associated Legendre Functions; Ferrers Functions

►Any hypergeometric function for which a quadratic transformation exists can be expressed in terms of associated Legendre functions or Ferrers functions. … ►The following formulas apply with principal branches of the hypergeometric functions, associated Legendre functions, and fractional powers. …

7: 19.21 Connection Formulas

§19.21 Connection Formulas

… ►Legendre’s relation (19.7.1) can be written … ►The complete cases of $R_F$ and $R_G$ have connection formulas resulting from those for the Gauss hypergeometric function (Erdélyi et al. (1953a, §2.9)). … ►If and $y=z+1$, then as $p\to 0$ (19.21.6) reduces to Legendre’s relation (19.21.1). … ►Connection formulas for $R_{-a}(\mathbf{b};\mathbf{z})$ are given in Carlson (1977b, pp. 99, 101, and 123–124). …

8: Bibliography G

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Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.

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External Links:

ISSN 1023-6198, Document, FullText, MathReview (Thomas Britz), ZentralBlatt

Referenced by:

§17.7(iii).

Encodings:

BibTeX

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Werke. Band II. pp. 436–447 (German).

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Notes:

Reprinted by Georg Olms Verlag, Hildesheim, 1973.

External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§27.2(i).

Encodings:

BibTeX

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Gauss quadrature approximations to hypergeometric and confluent hypergeometric functions. J. Comput. Appl. Math. 139 (1), pp. 173–187.

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External Links:

ISSN 0377-0427, Document, MathReview (Hasan Taşeli), ZentralBlatt

Referenced by:

§13.29(iii), §15.19(iii).

Encodings:

BibTeX

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Calculation of Gauss quadrature rules. Math. Comp. 23 (106), pp. 221–230.

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Notes:

Loose microfiche suppl. A1–A10

External Links:

FullText, MathReview (F. Stenger), ZentralBlatt

Referenced by:

§3.5(vi).

Encodings:

BibTeX

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Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.

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External Links:

FullText, MathReview (B. M. Stewart), ZentralBlatt

Referenced by:

§24.15(iii), §24.6.

Encodings:

BibTeX

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9: 18.12 Generating Functions

… ►and a similar formula by symmetry; compare the second row in Table 18.6.1. For the hypergeometric function ${}_{2}F_1$ see §§15.1, 15.2(i). … ►

Legendre

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18.12.11 $$\frac{1}{\sqrt{1-2xz+z^2}}=\sum _{n=0}^{\mathrm{\infty }}{P}_n\left(x\right)z^n,$$ .

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.9.12

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.12 $${\mathrm{e}}^{xz}{J}_0\left(z\sqrt{1-x^2}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{P_n\left(x\right)}{n!}{z}^n.$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $P_n\left(x\right)$: Legendre polynomial, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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10: Bibliography Z

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Distribution of zeros of Gauss and Kummer hypergeometric functions. A semiclassical approach. Ann. Numer. Math. 2 (1-4), pp. 457–472.

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External Links:

ISSN 1021-2655, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§13.9(i), §15.13.

Encodings:

BibTeX

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Tablitsy funktsii Lezhandra $P_{-1/2+i\tau }^1(x)$ . Vyčisl. Centr Akad. Nauk SSSR, Moscow.

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Notes:

Translated title: Tables of the Legendre functions $P_{-1/2+i\tau }^1(x)$

External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

5th item.

Encodings:

BibTeX

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Tables of the Legendre functions $P_{-1/2+i\tau }(x)$. Part I. Translated by D. E. Brown. Mathematical Tables Series, Vol. 22, Pergamon Press, Oxford.

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Notes:

Translation of Žurina and Karmazina, Tablitsy funktsii Lezhandra $P_{\mathrm{-}\mathrm{1}\mathrm{/}\mathrm{2}\mathrm{+}i\mathit{}\tau }\mathit{}\mathrm{(}x\mathrm{)}$. Tom I, Izdat. Akad. Nauk SSSR, Moscow, 1960.

External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

4th item.

Encodings:

BibTeX

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Tables of the Legendre functions $P_{-1/2+i\tau }(x)$. Part II. Translated by Prasenjit Basu. Mathematical Tables Series, Vol. 38. A Pergamon Press Book, The Macmillan Co., New York.

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Notes:

Translation of Žurina and Karmazina, Tablitsy funktsii Lezhandra $P_{\mathrm{-}\mathrm{1}\mathrm{/}\mathrm{2}\mathrm{+}i\mathit{}\tau }\mathit{}\mathrm{(}x\mathrm{)}$. Tom II, Izdat. Akad. Nauk SSSR, Moscow, 1962.

External Links:

MathReview, ZentralBlatt

Referenced by:

4th item.

Encodings:

BibTeX

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Tables and formulae for the spherical functions $P_{-1/2+i\tau }^m(z)$ . Translated by E. L. Albasiny, Pergamon Press, Oxford.

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Notes:

Translation of Žurina and Karmazina, Tablitsy i formuly dlya sfericheskikh funktsii $P_{\mathrm{-}\mathrm{1}\mathrm{/}\mathrm{2}\mathrm{+}i\mathit{}\tau }^m\mathit{}\mathrm{(}z\mathrm{)}$ , Vyčisl. Centr Akad Nauk SSSR, Moscow. 1962

External Links:

MathReview, ZentralBlatt

Referenced by:

§14.20(vii).

Encodings:

BibTeX

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