elliptic integrals
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11: 19.35 Other Applications
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§19.35(i) Mathematical
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►Elliptic integrals appear in lattice models of critical phenomena (Guttmann and Prellberg (1993)); theories of layered materials (Parkinson (1969)); fluid dynamics (Kida (1981)); string theory (Arutyunov and Staudacher (2004)); astrophysics (Dexter and Agol (2009)).12: 22.3 Graphics
13: 22.4 Periods, Poles, and Zeros
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►Figure 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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►Figure 22.4.2 depicts the fundamental unit cell in the -plane, with vertices , , , .
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►This 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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14: 19.13 Integrals of Elliptic Integrals
§19.13 Integrals of Elliptic Integrals
►§19.13(i) Integration with Respect to the Modulus
… ►Cvijović and Klinowski (1994) contains fractional integrals (with free parameters) for and , together with special cases. ►§19.13(iii) Laplace Transforms
►For direct and inverse Laplace transforms for the complete elliptic integrals , , and see Prudnikov et al. (1992a, §3.31) and Prudnikov et al. (1992b, §§3.29 and 4.3.33), respectively.15: 29.17 Other Solutions
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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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►Lamé–Wangerin functions are solutions of (29.2.1) with the property that is bounded on the line segment from to .
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16: 36.3 Visualizations of Canonical Integrals
17: 22.11 Fourier and Hyperbolic Series
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22.11.3
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►Next, with denoting the complete elliptic integral of the second kind (§19.2(ii)) and ,
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22.11.13
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22.11.14
►where is defined by §19.2.9.
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