n-tuple%20formulas

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1: 23.10 Addition Theorems and Other Identities

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§23.10(ii) Duplication Formulas

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§23.10(iii) $n$-Tuple Formulas

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2: 19.29 Reduction of General Elliptic Integrals

… ►Cubic cases of these formulas are obtained by setting one of the factors in (19.29.3) equal to 1. … ►(19.29.7) subsumes all 72 formulas in Gradshteyn and Ryzhik (2000, 3.168), and its cubic cases similarly replace the $18+36+18=72$ formulas in Gradshteyn and Ryzhik (2000, 3.133, 3.142, and 3.141(1-18)). … ►where ${𝐞}_j$ is an $n$-tuple with 1 in the $j$th position and 0’s elsewhere. … ►The first choice gives a formula that includes the 18+9+18 = 45 formulas in Gradshteyn and Ryzhik (2000, 3.133, 3.156, 3.158), and the second choice includes the 8+8+8+12 = 36 formulas in Gradshteyn and Ryzhik (2000, 3.151, 3.149, 3.137, 3.157) (after setting $x^2=t$ in some cases). … ►The first formula replaces (19.14.4)–(19.14.10). …

3: 19.19 Taylor and Related Series

… ►and define the $n$-tuple $\frac{\mathrm{𝟏}}{\mathrm{𝟐}}=(\frac{1}{2},\mathrm{\dots },\frac{1}{2})$. …

4: 19.18 Derivatives and Differential Equations

… ►Let ${\partial }_j=\partial /\partial z_j$, and ${𝐞}_j$ be an $n$-tuple with 1 in the $j$th place and 0’s elsewhere. …

5: Gergő Nemes

… ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …

6: Wolter Groenevelt

… ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …

7: 20 Theta Functions

Chapter 20 Theta Functions

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8: Foreword

… ►In 1964 the National Institute of Standards and Technology published the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, edited by Milton Abramowitz and Irene A. … ►November 20, 2009 …

9: 36.4 Bifurcation Sets

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§36.4(i) Formulas

… ► $K=2$, cusp bifurcation set: … ► $K=3$, swallowtail bifurcation set: … ►Elliptic umbilic bifurcation set (codimension three): for fixed $z$, the section of the bifurcation set is a three-cusped astroid … ►Hyperbolic umbilic bifurcation set (codimension three): …

10: 3.4 Differentiation

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Two-Point Formula

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Three-Point Formula

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Four-Point Formula

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Five-Point Formula

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Six-Point Formula

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