Legendre elliptic integrals
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31—40 of 76 matching pages
31: 19.12 Asymptotic Approximations
32: 22.14 Integrals
33: 22.4 Periods, Poles, and Zeros
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βΊFor example, the poles of , abbreviated as in the following tables, are at .
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βΊFigure 22.4.1 illustrates the locations in the -plane of the poles and zeros of the three principal Jacobian functions in the rectangle with vertices , , , .
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βΊThe set of points , , comprise the lattice for the 12 Jacobian functions; all other lattice unit cells are generated by translation of the fundamental unit cell by , where again .
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βΊThis half-period will be plus or minus a member of the triple ; the other two members of this triple are quarter periods of .
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βΊFor example, .
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34: 29.17 Other Solutions
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29.17.1
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35: 19.25 Relations to Other Functions
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§19.25(i) Legendre’s Integrals as Symmetric Integrals
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19.25.2
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βΊThus the five permutations induce five transformations of Legendre’s integrals (and also of the Jacobian elliptic functions).
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§19.25(iii) Symmetric Integrals as Legendre’s Integrals
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19.25.25
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36: 23.6 Relations to Other Functions
37: 23.4 Graphics
38: 19.30 Lengths of Plane Curves
39: 19.37 Tables
§19.37 Tables
… βΊ …40: 22.20 Methods of Computation
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βΊThis formula for becomes unstable near .
If only the value of at is required then the exact value is in the table 22.5.1.
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βΊIf are given with and , then can be found from
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βΊJacobi’s epsilon function can be computed from its representation (22.16.30) in terms of theta functions and complete elliptic integrals; compare §20.14.
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