elliptic integrals
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21—30 of 132 matching pages
21: 29.4 Graphics
22: 29.16 Asymptotic Expansions
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►The approximations for Lamé polynomials hold uniformly on the rectangle , , when and assume large real values.
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23: 19.1 Special Notation
24: 19.39 Software
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►Unless otherwise stated, the functions are and , with .
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►For other software, sometimes with and complex variables, see the Software Index.
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►Unless otherwise stated, the variables are real, and the functions are and .
►For research software see Bulirsch (1965b, function ), Bulirsch (1969b, function ), Jefferson (1961), and Neuman (1969a, functions and ).
For other software, sometimes with and complex variables, see the Software Index.
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25: 19.9 Inequalities
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19.9.5
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19.9.8
►Further inequalities for and can be found in Alzer and Qiu (2004), Anderson et al. (1992a, b, 1997), and Qiu and Vamanamurthy (1996).
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►Sharper inequalities for are:
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►Inequalities for both and involving inverse circular or inverse hyperbolic functions are given in Carlson (1961b, §4).
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26: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
27: 22.5 Special Values
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►Table 22.5.1 gives the value of each of the 12 Jacobian elliptic functions, together with its -derivative (or at a pole, the residue), for values of that are integer multiples of , .
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Table 22.5.1: Jacobian elliptic function values, together with derivatives or residues, for special values of the variable.
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►Expansions for as or are given in §§19.5, 19.12.
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28: 19.37 Tables
29: 22.1 Special Notation
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real variables. | |
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, | , (complete elliptic integrals of the first kind (§19.2(ii))). |
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30: 19.5 Maclaurin and Related Expansions
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19.5.5
, .
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►Coefficients of terms up to are given in Lee (1990), along with tables of fractional errors in and , , obtained by using 12 different truncations of (19.5.6) in (19.5.8) and (19.5.9).
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►An infinite series for is equivalent to the infinite product
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►Series expansions of and are surveyed and improved in Van de Vel (1969), and the case of is summarized in Gautschi (1975, §1.3.2).
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