Lagrange formula for equally-spaced nodes
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11: 9.19 Approximations
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Martín et al. (1992) provides two simple formulas for approximating to graphical accuracy, one for , the other for .
Corless et al. (1992) describe a method of approximation based on subdividing into a triangular mesh, with values of , stored at the nodes. and are then computed from Taylor-series expansions centered at one of the nearest nodes. The Taylor coefficients are generated by recursion, starting from the stored values of , at the node. Similarly for , .
12: 1.10 Functions of a Complex Variable
13: 27.20 Methods of Computation: Other Number-Theoretic Functions
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►The recursion formulas (27.14.6) and (27.14.7) can be used to calculate the partition function for .
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►A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function , and the values can be checked by the congruence (27.14.20).
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14: Howard S. Cohl
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►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.
15: Preface
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►Abramowitz and Stegun’s Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables is being completely rewritten with regard to the needs of today.
…The authors will review the relevant published literature and produce approximately twice the number of formulas that were contained in the original Handbook.
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16: 5.5 Functional Relations
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§5.5(ii) Reflection
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5.5.3
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§5.5(iii) Multiplication
►Duplication Formula
… ►Gauss’s Multiplication Formula
…17: 24.6 Explicit Formulas
18: 27.5 Inversion Formulas
§27.5 Inversion Formulas
… ►which, in turn, is the basis for the Möbius inversion formula relating sums over divisors: … ►Special cases of Möbius inversion pairs are: … ►Other types of Möbius inversion formulas include: …19: 18.38 Mathematical Applications
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►If the nodes in a quadrature formula with a positive weight function are chosen to be the zeros of the th degree OP with the same weight function, and the interval of orthogonality is the same as the integration range, then the weights in the quadrature formula can be chosen in such a way that the formula is exact for all polynomials of degree not exceeding .
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►See Koornwinder (2007a, (3.13), (4.9), (4.10)) for explicit formulas.
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