Legendre elliptic integrals
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1: 19.2 Definitions
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§19.2(ii) Legendre’s Integrals
… ►Also, if and are real, then is called a circular or hyperbolic case according as is negative or positive. The circular and hyperbolic cases alternate in the four intervals of the real line separated by the points . … ►Legendre’s complementary complete elliptic integrals are defined via ►
19.2.8_1
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2: 19.35 Other Applications
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►Generalizations of elliptic integrals appear in analysis of modular theorems of Ramanujan (Anderson et al. (2000)); analysis of Selberg integrals (Van Diejen and Spiridonov (2001)); use of Legendre’s relation (19.7.1) to compute to high precision (Borwein and Borwein (1987, p. 26)).
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§19.35(ii) Physical
…3: 29.14 Orthogonality
4: 19.4 Derivatives and Differential Equations
5: 19.13 Integrals of Elliptic Integrals
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§19.13(i) Integration with Respect to the Modulus
… ►§19.13(ii) Integration with Respect to the Amplitude
… ►§19.13(iii) Laplace Transforms
…6: 22.11 Fourier and Hyperbolic Series
7: 19.3 Graphics
§19.3 Graphics
… ►See Figures 19.3.1–19.3.6 for complete and incomplete Legendre’s elliptic integrals. … ►In Figures 19.3.7 and 19.3.8 for complete Legendre’s elliptic integrals with complex arguments, height corresponds to the absolute value of the function and color to the phase. … ►8: 19.7 Connection Formulas
§19.7 Connection Formulas
… ►Reciprocal-Modulus Transformation
… ►Imaginary-Modulus Transformation
… ►Imaginary-Argument Transformation
… ►§19.7(iii) Change of Parameter of
…9: 19.9 Inequalities
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§19.9(i) Complete Integrals
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19.9.6
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19.9.8
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19.9.9
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