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Legendre elliptic integrals

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1: 19.2 Definitions
§19.2(ii) Legendre’s Integrals
19.2.5 E ( ϕ , k ) = 0 ϕ 1 - k 2 sin 2 θ d θ = 0 sin ϕ 1 - k 2 t 2 1 - t 2 d t .
19.2.6 D ( ϕ , k ) = 0 ϕ sin 2 θ d θ 1 - k 2 sin 2 θ = 0 sin ϕ t 2 d t 1 - t 2 1 - k 2 t 2 = ( F ( ϕ , k ) - E ( ϕ , k ) ) / k 2 .
Also, if k 2 and α 2 are real, then Π ( ϕ , α 2 , k ) is called a circular or hyperbolic case according as α 2 ( α 2 - k 2 ) ( α 2 - 1 ) is negative or positive. The circular and hyperbolic cases alternate in the four intervals of the real line separated by the points α 2 = 0 , k 2 , 1 . …
2: 19.35 Other Applications
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 π to high precision (Borwein and Borwein (1987, p. 26)).
§19.35(ii) Physical
3: 29.14 Orthogonality
29.14.2 g , h = 0 K 0 K w ( s , t ) g ( s , t ) h ( s , t ) d t d s ,
29.14.3 w ( s , t ) = sn 2 ( K + i t , k ) - sn 2 ( s , k ) .
29.14.4 sE 2 n + 1 m ( s , k 2 ) sE 2 n + 1 m ( K + i t , k 2 ) ,
29.14.5 cE 2 n + 1 m ( s , k 2 ) cE 2 n + 1 m ( K + i t , k 2 ) ,
29.14.11 g , h = 0 4 K 0 2 K w ( s , t ) g ( s , t ) h ( s , t ) d t d s ,
4: 19.4 Derivatives and Differential Equations
§19.4(i) Derivatives
19.4.3 d 2 E ( k ) d k 2 = - 1 k d K ( k ) d k = k 2 K ( k ) - E ( k ) k 2 k 2 ,
§19.4(ii) Differential Equations
5: 19.13 Integrals of Elliptic Integrals
§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
22.11.1 sn ( z , k ) = 2 π K k n = 0 q n + 1 2 sin ( ( 2 n + 1 ) ζ ) 1 - q 2 n + 1 ,
22.11.2 cn ( z , k ) = 2 π K k n = 0 q n + 1 2 cos ( ( 2 n + 1 ) ζ ) 1 + q 2 n + 1 ,
22.11.3 dn ( z , k ) = π 2 K + 2 π K n = 1 q n cos ( 2 n ζ ) 1 + q 2 n .
7: 19.3 Graphics
§19.3 Graphics
See Figures 19.3.119.3.6 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. …
See accompanying text
Figure 19.3.12: ( E ( k ) ) as a function of complex k 2 for - 2 ( k 2 ) 2 , - 2 ( k 2 ) 2 . … Magnify 3D Help
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 Π ( ϕ , α 2 , k )
9: 19.9 Inequalities
§19.9(i) Complete Integrals
19.9.9 L ( a , b ) = 4 a E ( k ) , k 2 = 1 - ( b 2 / a 2 ) , a > b .
§19.9(ii) Incomplete Integrals
10: 19.6 Special Cases
§19.6 Special Cases
19.6.10 lim ϕ 0 E ( ϕ , k ) / ϕ = 1 .