q-Leibniz rule

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1: 3.5 Quadrature

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§3.5(i) Trapezoidal Rules

… ►The composite trapezoidal rule is … ►

§3.5(ii) Simpson’s Rule

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§3.5(iv) Interpolatory Quadrature Rules

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2: 3.8 Nonlinear Equations

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§3.8(ii) Newton’s Rule

… ► … ►Newton’s rule is given by … ►Another iterative method is Halley’s rule: …The rule converges locally and is cubically convergent. …

3: 8.25 Methods of Computation

… ►See Allasia and Besenghi (1987b) for the numerical computation of $\mathrm{\Gamma }(a,z)$ from (8.6.4) by means of the trapezoidal rule. … ►A numerical inversion procedure is also given for calculating the value of $x$ (with 10S accuracy), when $a$ and $P(a,x)$ are specified, based on Newton’s rule (§3.8(ii)). …

4: 9.17 Methods of Computation

… ►The trapezoidal rule (§3.5(i)) is then applied. … ►Zeros of the Airy functions, and their derivatives, can be computed to high precision via Newton’s rule (§3.8(ii)) or Halley’s rule (§3.8(v)), using values supplied by the asymptotic expansions of §9.9(iv) as initial approximations. …

5: 7.22 Methods of Computation

… ►Additional references are Matta and Reichel (1971) for the application of the trapezoidal rule, for example, to the first of (7.7.2), and Gautschi (1970) and Cuyt et al. (2008) for continued fractions. …

6: About MathML

… ►As a general rule, using the latest available version of your chosen browser, plugins and an updated operating system is helpful. …

7: 10.74 Methods of Computation

… ►Newton’s rule (§3.8(i)) or Halley’s rule (§3.8(v)) can be used to compute to arbitrarily high accuracy the real or complex zeros of all the functions treated in this chapter. …Newton’s rule is quadratically convergent and Halley’s rule is cubically convergent. …

8: 29.20 Methods of Computation

… ►A second approach is to solve the continued-fraction equations typified by (29.3.10) by Newton’s rule or other iterative methods; see §3.8. …

9: 34.7 Basic Properties: $\mathit{9}j$ Symbol

… ►This equation is the sum rule. It constitutes an addition theorem for the $\mathit{9}j$ symbol. …

10: 1.4 Calculus of One Variable

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Chain Rule

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L’Hôpital’s Rule

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