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elliptic integrals

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1: 19.16 Definitions
§19.16(i) Symmetric Integrals
All elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.23.6) and (19.23.6_5), are special cases of a multivariate hypergeometric function …The R -function is often used to make a unified statement of a property of several elliptic integrals. … …
§19.16(iii) Various Cases of R - a ( b ; z )
2: 19.2 Definitions
§19.2(i) General Elliptic Integrals
is called an elliptic integral. …
§19.2(ii) Legendre’s Integrals
K ( k ) = K ( k ) ,
§19.2(iii) Bulirsch’s Integrals
3: 36.2 Catastrophes and Canonical Integrals
Canonical Integrals
Ψ ( E ) ( 0 ) = 1 3 π Γ ( 1 6 ) = 3.28868 ,
36.2.25 Ψ ( E ) ( x , - y , z ) = Ψ ( E ) ( x , y , z ) .
36.2.26 Ψ ( E ) ( - 1 2 x 3 2 y , ± 3 2 x - 1 2 y , z ) = Ψ ( E ) ( x , y , z ) ,
36.2.28 Ψ ( E ) ( 0 , 0 , z ) = Ψ ( E ) ( 0 , 0 , - z ) ¯ = 2 π π z 27 exp ( 2 27 i z 3 ) ( J - 1 / 6 ( 2 27 z 3 ) + i J 1 / 6 ( 2 27 z 3 ) ) , z 0 ,
4: 29.10 Lamé Functions with Imaginary Periods
Ec ν 2 m ( i ( z - K - i K ) , k 2 ) ,
Ec ν 2 m + 1 ( i ( z - K - i K ) , k 2 ) ,
Es ν 2 m + 1 ( i ( z - K - i K ) , k 2 ) ,
The first and the fourth functions have period 2 i K ; the second and the third have period 4 i K . …
5: 29.13 Graphics
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Figure 29.13.5: uE 4 m ( x , 0.1 ) for - 2 K x 2 K , m = 0 , 1 , 2 . K = 1.61244 . Magnify
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Figure 29.13.6: uE 4 m ( x , 0.9 ) for - 2 K x 2 K , m = 0 , 1 , 2 . K = 2.57809 . Magnify
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Figure 29.13.21: | uE 4 1 ( x + i y , 0.1 ) | for - 3 K x 3 K , 0 y 2 K . K = 1.61244 , K = 2.57809 . Magnify 3D Help
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Figure 29.13.22: | uE 4 1 ( x + i y , 0.5 ) | for - 3 K x 3 K , 0 y 2 K . K = K = 1.85407 . Magnify 3D Help
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Figure 29.13.23: | uE 4 1 ( x + i y , 0.9 ) | for - 3 K x 3 K , 0 y 2 K . K = 2.57809 , K = 1.61244 . Magnify 3D Help
6: 22.21 Tables
Spenceley and Spenceley (1947) tabulates sn ( K x , k ) , cn ( K x , k ) , dn ( K x , k ) , am ( K x , k ) , ( K x , k ) for arcsin k = 1 ( 1 ) 89 and x = 0 ( 1 90 ) 1 to 12D, or 12 decimals of a radian in the case of am ( K x , k ) . Curtis (1964b) tabulates sn ( m K / n , k ) , cn ( m K / n , k ) , dn ( m K / n , k ) for n = 2 ( 1 ) 15 , m = 1 ( 1 ) n - 1 , and q (not k ) = 0 ( .005 ) 0.35 to 20D. … Zhang and Jin (1996, p. 678) tabulates sn ( K x , k ) , cn ( K x , k ) , dn ( K x , k ) for k = 1 4 , 1 2 and x = 0 ( .1 ) 4 to 7D. …
7: 19.6 Special Cases
§19.6(i) Complete Elliptic Integrals
K ( 0 ) = E ( 0 ) = K ( 1 ) = E ( 1 ) = 1 2 π ,
K ( 1 ) = K ( 0 ) = ,
§19.6(ii) F ( ϕ , k )
§19.6(iii) E ( ϕ , k )
8: 19.7 Connection Formulas
K ( 1 / k ) = k ( K ( k ) i K ( k ) ) ,
K ( 1 / k ) = k ( K ( k ) ± i K ( k ) ) ,
E ( ϕ , k 1 ) = ( E ( β , k ) - k 2 F ( β , k ) ) / k ,
§19.7(iii) Change of Parameter of Π ( ϕ , α 2 , k )
9: 19.35 Other Applications
§19.35(i) Mathematical
§19.35(ii) Physical
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)).
10: 22.3 Graphics
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Figure 22.3.16: sn ( x + i y , k ) for k = 0.99 , - 3 K x 3 K , 0 y 4 K . K = 3.3566 , K = 1.5786 . Magnify 3D Help
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Figure 22.3.17: cn ( x + i y , k ) for k = 0.99 , - 3 K x 3 K , 0 y 4 K . K = 3.3566 , K = 1.5786 . Magnify 3D Help
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Figure 22.3.18: dn ( x + i y , k ) for k = 0.99 , - 3 K x 3 K , 0 y 4 K . K = 3.3566 , K = 1.5786 . Magnify 3D Help
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Figure 22.3.19: cd ( x + i y , k ) for k = 0.99 , - 3 K x 3 K , 0 y 4 K . K = 3.3566 , K = 1.5786 . Magnify 3D Help
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Figure 22.3.20: dc ( x + i y , k ) for k = 0.99 , - 3 K x 3 K , 0 y 4 K . K = 3.3566 , K = 1.5786 . Magnify 3D Help