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21—30 of 769 matching pages
21: 29.17 Other Solutions
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►If (29.2.1) admits a Lamé polynomial solution , then a second linearly independent solution is given by
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29.17.1
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►They are algebraic functions of , , and , and have primitive period .
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22: 19.30 Lengths of Plane Curves
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19.30.3
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►Cancellation on the second right-hand side of (19.30.3) can be avoided by use of (19.25.10).
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19.30.5
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►Let and
be replaced respectively by and , where , to produce a family of confocal ellipses.
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►For in terms of , , and an algebraic term, see Byrd and Friedman (1971, p. 3).
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23: 16.25 Methods of Computation
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►There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations).
…Instead a boundary-value problem needs to be formulated and solved.
See §§3.6(vii), 3.7(iii), Olde Daalhuis and Olver (1998), Lozier (1980), and Wimp (1984, Chapters 7, 8).
24: 17.16 Mathematical Applications
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►More recent applications are given in Gasper and Rahman (2004, Chapter 8) and Fine (1988, Chapters 1 and 2).
25: 20.11 Generalizations and Analogs
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►As in §20.11(ii), the modulus of elliptic integrals (§19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in -series via (20.9.1).
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►The first of equations (20.9.2) can also be written
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20.11.5
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►Similar identities can be constructed for , , and .
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►The importance of these combined theta functions is that sets of twelve equations for the theta functions often can be replaced by corresponding sets of three equations of the combined theta functions, plus permutation symmetry.
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26: 19.1 Special Notation
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►We use also the function , introduced by Jahnke et al. (1966, p. 43).
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►In Abramowitz and Stegun (1964, Chapter 17) the functions (19.1.1) and (19.1.2) are denoted, in order, by , , , , , and , where and is the (not related to ) in (19.1.1) and (19.1.2).
…However, it should be noted that in Chapter 8 of Abramowitz and Stegun (1964) the notation used for elliptic integrals differs from Chapter 17 and is consistent with that used in the present chapter and the rest of the NIST Handbook and DLMF.
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, , and are the symmetric (in , , and ) integrals of the first, second, and third kinds; they are complete if exactly one of , , and is identically 0.
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27: 2.10 Sums and Sequences
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►For extensions to , higher terms, and other examples, see Olver (1997b, Chapter 8).
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►Hence
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►For generalizations and other examples see Olver (1997b, Chapter 8), Ford (1960), and Berndt and Evans (1984).
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►For examples see Olver (1997b, Chapters 8, 9).
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►For other examples and extensions see Olver (1997b, Chapter 8), Olver (1970), Wong (1989, Chapter 2), and Wong and Wyman (1974).
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28: 9.4 Maclaurin Series
29: 19.27 Asymptotic Approximations and Expansions
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§19.27(ii)
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19.27.2
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§19.27(iv)
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19.27.7
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►These series converge but not fast enough, given the complicated nature of their terms, to be very useful in practice.
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30: 19.9 Inequalities
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►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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►Even for the extremely eccentric ellipse with and , this is correct within 0.
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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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►Other inequalities for can be obtained from inequalities for given in Carlson (1966, (2.15)) and Carlson (1970) via (19.25.5).