L’Hôpital rule
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11—20 of 134 matching pages
11: 18.41 Tables
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►Abramowitz and Stegun (1964, Tables 22.4, 22.6, 22.11, and 22.13) tabulates , , , and for .
The ranges of are for and , and for and .
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►For , , and see §3.5(v).
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12: Bibliography
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Numerical computation of Tricomi’s psi function by the trapezoidal rule.
Computing 39 (3), pp. 271–279.
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Numerical evaluation of the Kummer function with complex argument by the trapezoidal rule.
Rend. Sem. Mat. Univ. Politec. Torino 49 (3), pp. 315–327.
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Numerical calculation of incomplete gamma functions by the trapezoidal rule.
Numer. Math. 50 (4), pp. 419–428.
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Derivatives and integrals with respect to the order of the Struve functions and
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J. Math. Anal. Appl. 137 (1), pp. 17–36.
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Note on the trivial zeros of Dirichlet -functions.
Proc. Amer. Math. Soc. 94 (1), pp. 29–30.
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13: 19.33 Triaxial Ellipsoids
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►The external field and the induced magnetization together produce a uniform field inside the ellipsoid with strength , where is the demagnetizing factor, given in cgs units by
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19.33.7
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19.33.8
►where and are obtained from by permutation of , , and .
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14: 1.2 Elementary Algebra
15: 23.9 Laurent and Other Power Series
16: 11.15 Approximations
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Luke (1975, pp. 416–421) gives Chebyshev-series expansions for , , , and , , for ; , , , and , , ; the coefficients are to 20D.
MacLeod (1993) gives Chebyshev-series expansions for , , , and , , ; the coefficients are to 20D.
17: 23.7 Quarter Periods
18: 18.8 Differential Equations
19: 23.3 Differential Equations
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23.3.1
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23.3.2
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►Given and there is a unique lattice such that (23.3.1) and (23.3.2) are satisfied.
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►Conversely, , , and the set are determined uniquely by the lattice independently of the choice of generators.
However, given any pair of generators , of , and with defined by (23.2.1), we can identify the individually, via
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20: 3.11 Approximation Techniques
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►to the maximum error of the minimax polynomial is bounded by , where is the th Lebesgue constant for Fourier series; see §1.8(i).
Since , is a monotonically increasing function of , and (for example) , this means that in practice the gain in replacing a truncated Chebyshev-series expansion by the corresponding minimax polynomial approximation is hardly worthwhile.
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►The Padé approximants can be computed by Wynn’s cross rule.
Any five approximants arranged in the Padé table as
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