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21: 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).
22: 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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23: 26.13 Permutations: Cycle Notation
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►An explicit representation of can be given by the matrix:
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►is in cycle notation.
…In consequence, (26.13.2) can also be written as .
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►For the example (26.13.2), this decomposition is given by
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►Again, for the example (26.13.2) a minimal decomposition into adjacent transpositions is given by : .
24: 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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25: 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).
26: 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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27: 9.4 Maclaurin Series
28: 10.62 Graphs
29: 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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30: 31.2 Differential Equations
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►All other homogeneous linear differential equations of the second order having four regular singularities in the extended complex plane, , can be transformed into (31.2.1).
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