# Lagrange formula for reversion of series

(0.002 seconds)

## 1—10 of 402 matching pages

##### 1: 2.2 Transcendental Equations
2.2.6 $t=y^{\frac{1}{2}}\left(1+\tfrac{1}{4}y^{-1}\ln y+o\left(y^{-1}\right)\right),$ $y\to\infty$.
An important case is the reversion of asymptotic expansions for zeros of special functions. …where $F_{0}=f_{0}$ and $sF_{s}$ ($s\geq 1$) is the coefficient of $x^{-1}$ in the asymptotic expansion of $(f(x))^{s}$ (Lagrange’s formula for the reversion of series). Conditions for the validity of the reversion process in $\mathbb{C}$ are derived in Olver (1997b, pp. 14–16). …
##### 2: 3.3 Interpolation
###### §3.3(i) Lagrange Interpolation
The final expression in (3.3.1) is the Barycentric form of the Lagrange interpolation formula. … With an error term the Lagrange interpolation formula for $f$ is given by …
##### 3: 27.13 Functions
Lagrange (1770) proves that $g\left(2\right)=4$, and during the next 139 years the existence of $g\left(k\right)$ was shown for $k=3,4,5,6,7,8,10$. …A general formula states that … Explicit formulas for $r_{k}\left(n\right)$ have been obtained by similar methods for $k=6,8,10$, and $12$, but they are more complicated. Exact formulas for $r_{k}\left(n\right)$ have also been found for $k=3,5$, and $7$, and for all even $k\leq 24$. …Also, Milne (1996, 2002) announce new infinite families of explicit formulas extending Jacobi’s identities. …
##### 4: Bibliography D
• P. Dienes (1931) The Taylor Series. Oxford University Press, Oxford.
• A. M. Din (1981) A simple sum formula for Clebsch-Gordan coefficients. Lett. Math. Phys. 5 (3), pp. 207–211.
• J. J. Duistermaat (1974) Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math. 27, pp. 207–281.
• T. M. Dunster (2001c) Uniform asymptotic expansions for the reverse generalized Bessel polynomials, and related functions. SIAM J. Math. Anal. 32 (5), pp. 987–1013.
• L. Durand (1978) Product formulas and Nicholson-type integrals for Jacobi functions. I. Summary of results. SIAM J. Math. Anal. 9 (1), pp. 76–86.
The nodes $x_{1},x_{2},\dots,x_{n}$ are prescribed, and the weights $w_{k}$ and error term $E_{n}(f)$ are found by integrating the product of the Lagrange interpolation polynomial of degree $n-1$ and $w(x)$. …
###### Gauss–Laguerre Formula
a complex Gauss quadrature formula is available. …
##### 6: 3.4 Differentiation
###### §3.4(i) Equally-Spaced Nodes
The Lagrange $(n+1)$-point formula is …
##### 7: Bibliography G
• F. Gao and V. J. W. Guo (2013) Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.
• G. Gasper (1975) Formulas of the Dirichlet-Mehler Type. In Fractional Calculus and its Applications, B. Ross (Ed.), Lecture Notes in Math., Vol. 457, pp. 207–215.
• W. Gautschi (1983) How and how not to check Gaussian quadrature formulae. BIT 23 (2), pp. 209–216.
• W. Gautschi (1992) On mean convergence of extended Lagrange interpolation. J. Comput. Appl. Math. 43 (1-2), pp. 19–35.
• H. W. Gould (1972) Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.
• ##### 8: 1.10 Functions of a Complex Variable
###### §1.10(iii) Laurent Series
The series (1.10.6) converges uniformly and absolutely on compact sets in the annulus. …
##### 9: Bibliography B
• W. Barrett (1981) Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V. Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.
• B. C. Berndt (1975a) Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications. J. Number Theory 7 (4), pp. 413–445.
• B. C. Berndt (1975b) Periodic Bernoulli numbers, summation formulas and applications. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pp. 143–189.
• J. Berrut and L. N. Trefethen (2004) Barycentric Lagrange interpolation. SIAM Rev. 46 (3), pp. 501–517.
• R. Bo and R. Wong (1999) A uniform asymptotic formula for orthogonal polynomials associated with $\exp(-x^{4})$ . J. Approx. Theory 98, pp. 146–166.
• ##### 10: 27.20 Methods of Computation: Other Number-Theoretic Functions
The recursion formulas (27.14.6) and (27.14.7) can be used to calculate the partition function $p\left(n\right)$ for $n. …To compute a particular value $p\left(n\right)$ it is better to use the Hardy–Ramanujan–Rademacher series (27.14.9). … A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function $\tau\left(n\right)$, and the values can be checked by the congruence (27.14.20). …