Lagrange formula for reversion of series

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11: 30.10 Series and Integrals

§30.10 Series and Integrals

… ►For product formulas and convolutions see Connett et al. (1993). …

12: Howard S. Cohl

… ►Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and $q$-series. Howard is the project leader for the NIST Digital Repository of Mathematical Formulae seeding and development projects. In this regard, he has been exploring mathematical knowledge management and the digital expression of mostly unambiguous context-free full semantic information for mathematical formulae.

13: 1.8 Fourier Series

§1.8 Fourier Series

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Uniqueness of Fourier Series

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Parseval’s Formula

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Poisson’s Summation Formula

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14: Bibliography F

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On the reversion of an asymptotic expansion and the zeros of the Airy functions. SIAM Rev. 41 (4), pp. 762–773.

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External Links:

ISSN 1095-7200, Document, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§2.2, §2.2, §3.8(v), §9.9(iv), §9.9(iv).

Encodings:

BibTeX

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The Edmonds asymptotic formulas for the $3j$ and $6j$ symbols. J. Math. Phys. 39 (7), pp. 3906–3915.

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External Links:

ISSN 0022-2488, Document, MathReview (A. J. Coleman), ZentralBlatt

Referenced by:

§34.8.

Encodings:

BibTeX

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Studies on Divergent Series and Summability & The Asymptotic Developments of Functions Defined by Maclaurin Series. Chelsea Publishing Co., New York.

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Notes:

Reprint with corrections of two books published originally in 1916 and 1936.

External Links:

MathReview

Referenced by:

§2.10(iii).

Encodings:

BibTeX

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On the coefficients in the recursion formulae of orthogonal polynomials. Proc. Roy. Irish Acad. Sect. A 76 (1), pp. 1–6.

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External Links:

MathReview (A. G. Law), ZentralBlatt

Referenced by:

§32.15.

Encodings:

BibTeX

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Series expansions of symmetric elliptic integrals. Math. Comp. 81 (278), pp. 957–990.

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External Links:

Document

Referenced by:

§20.7(ii), §20.7(ii).

Encodings:

BibTeX

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15: 3.11 Approximation Techniques

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§3.11(ii) Chebyshev-Series Expansions

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Summation of Chebyshev Series: Clenshaw’s Algorithm

… ►be a formal power series. … ►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 $[a,b]$. …

16: 29.20 Methods of Computation

… ►Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6. …The Fourier series may be summed using Clenshaw’s algorithm; see §3.11(ii). … ►The corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. §29.15(i) includes formulas for normalizing the eigenvectors. …

17: 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:

BibTeX

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Integral and series representations of the Dirac delta function. Commun. Pure Appl. Anal. 7 (2), pp. 229–247.

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External Links:

ISSN 1534-0392, MathReview Entry, ZentralBlatt

Referenced by:

§1.17(iv).

Encodings:

BibTeX

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New series expansions for the confluent hypergeometric function $M(a,b,z)$ . Appl. Math. Comput. 235, pp. 26–31.

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External Links:

ISSN 0096-3003, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§13.11, §13.11.

Encodings:

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New series expansions of the Gauss hypergeometric function. Adv. Comput. Math. 39 (2), pp. 349–365.

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External Links:

ISSN 1019-7168, Document, FullText, MathReview (Jochen Denzler), ZentralBlatt

Referenced by:

§15.19(i), §15.19(i).

Encodings:

BibTeX

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Expansion of the confluent hypergeometric function in series of Bessel functions. Math. Tables Aids Comput. 13 (68), pp. 261–271.

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External Links:

FullText, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§13.24(ii).

Encodings:

BibTeX

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18: 36.15 Methods of Computation

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§36.15(i) Convergent Series

►Close to the origin $\mathbf{x}=\mathbf{0}$ of parameter space, the series in §36.8 can be used. … ►Far from the bifurcation set, the leading-order asymptotic formulas of §36.11 reproduce accurately the form of the function, including the geometry of the zeros described in §36.7. … ►This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of $\mathrm{\Phi }$, with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. …

19: 14.28 Sums

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§14.28(ii) Heine’s Formula

… ►The series converges uniformly for $z_1$ outside or on ${ℰ}_1$, and $z_2$ within or on ${ℰ}_2$. …

20: 15.6 Integral Representations

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15.6.1 $$\mathbf{F}(a,b;c;z)=\frac{1}{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-b\right)}{\int }_0^1\frac{t^{b-1}{(1-t)}^{c-b-1}}{{(1-zt)}^a}dt,$$ ; $\mathrm{\Re }c>\mathrm{\Re }b>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Keywords:

Mellin transform

Referenced by:

§15.19(iii), (15.4.34), §15.6, §15.6, Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9), Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9)

Permalink:

http://dlmf.nist.gov/15.6.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.23):

The Olver hypergeometric function $\mathbf{F}(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.

See also:

Annotations for §15.6 and Ch.15

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15.6.2 $$\mathbf{F}(a,b;c;z)=\frac{\mathrm{\Gamma }\left(1+b-c\right)}{2\pi \mathrm{i}\mathrm{\Gamma }\left(b\right)}{\int }_0^{(1+)}\frac{t^{b-1}{(t-1)}^{c-b-1}}{{(1-zt)}^a}dt,$$ ; $c-b\ne 1,2,3,\mathrm{\dots }$, $\mathrm{\Re }b>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Keywords:

Mellin transform

Referenced by:

§15.6

Permalink:

http://dlmf.nist.gov/15.6.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.0.23):

The Olver hypergeometric function $\mathbf{F}(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.

See also:

Annotations for §15.6 and Ch.15

►

15.6.3 $$\mathbf{F}(a,b;c;z)={\mathrm{e}}^{-b\pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(1-b\right)}{2\pi \mathrm{i}\mathrm{\Gamma }\left(c-b\right)}{\int }_{\mathrm{\infty }}^{(0+)}\frac{t^{b-1}{(t+1)}^{a-c}}{{(t-zt+1)}^a}dt,$$ ; $b\ne 1,2,3,\mathrm{\dots }$, $\mathrm{\Re }\left(c-b\right)>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Keywords:

Mellin transform

Referenced by:

§15.6, §15.6

Permalink:

http://dlmf.nist.gov/15.6.E3

Encodings:

pMML, png, TeX

Addition (effective with 1.0.23):

The Olver hypergeometric function $\mathbf{F}(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.

See also:

Annotations for §15.6 and Ch.15

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15.6.8 $$\mathbf{F}(a,b;c;z)=\frac{1}{\mathrm{\Gamma }\left(c-d\right)}{\int }_0^1\mathbf{F}(a,b;d;zt)t^{d-1}{(1-t)}^{c-d-1}dt,$$ ; $\mathrm{\Re }c>\mathrm{\Re }d>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Keywords:

Mellin transform

Source:

Follows from (15.6.9) with $\lambda =a+b$.

Referenced by:

(15.6.9), §15.6, §15.6, §15.6, Erratum (V1.0.14) for Equation (15.6.8), Erratum (V1.0.14) for Equation (15.6.8)

Permalink:

http://dlmf.nist.gov/15.6.E8

Encodings:

pMML, png, TeX

Addition (effective with 1.0.23):

The Olver hypergeometric function $\mathbf{F}(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.

See also:

Annotations for §15.6 and Ch.15

… ►However, this reverses the direction of the integration contour, and in consequence (15.6.5) would need to be multiplied by $-1$. …