# Barycentric form

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##### 1: 1.13 Differential Equations
###### §1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms
This is the Sturm-Liouville form of a second order differential equation, where denotes $\frac{\mathrm{d}}{\mathrm{d}x}$. Assuming that $u(x)$ satisfies un-mixed boundary conditions of the form
##### 2: 3.3 Interpolation
The final expression in (3.3.1) is the Barycentric form of the Lagrange interpolation formula. … … The $(n+1)$-point formula (3.3.4) can be written in the form
3.3.35 $\left[z_{0},z_{1},\dots,z_{n}\right]f=\sum_{k=0}^{n}\left(\ifrac{f(z_{k})}{% \prod_{\begin{subarray}{c}0\leq j\leq n\\ j\neq k\end{subarray}}(z_{k}-z_{j})}\right).$
For example, for $k+1$ coincident points the limiting form is given by $\left[z_{0},z_{0},\dots,z_{0}\right]f=f^{(k)}(z_{0})/k!$. …
##### 3: 31.13 Asymptotic Approximations
For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999). For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).
##### 4: Bibliography B
• J. Berrut and L. N. Trefethen (2004) Barycentric Lagrange interpolation. SIAM Rev. 46 (3), pp. 501–517.
• W. S. Burnside and A. W. Panton (1960) The Theory of Equations: With an Introduction to the Theory of Binary Algebraic Forms. Dover Publications, New York.
##### 6: 31.12 Confluent Forms of Heun’s Equation
###### §31.12 Confluent Forms of Heun’s Equation
Confluent forms of Heun’s differential equation (31.2.1) arise when two or more of the regular singularities merge to form an irregular singularity. …There are four standard forms, as follows: … This has regular singularities at $z=0$ and $1$, and an irregular singularity of rank 1 at $z=\infty$. …
##### 7: Frank Garvan
His research is in the areas of $q$-series and modular forms, and he enjoys using MAPLE in his research. …
##### 9: Bruce R. Miller
He is the developer of the tools used to process the DLMF into both book and web forms. …
##### 10: 30.2 Differential Equations
###### §30.2(ii) Other Forms
The Liouville normal form of equation (30.2.1) is …