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elliptic integrals

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11: 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)).
12: 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
13: 22.4 Periods, Poles, and Zeros
Table 22.4.1: Periods and poles of Jacobian elliptic functions.
Periods z -Poles
i K K + i K K 0
Table 22.4.2: Periods and zeros of Jacobian elliptic functions.
Periods z -Zeros
0 K K + i K i K
Figure 22.4.1 illustrates the locations in the z -plane of the poles and zeros of the three principal Jacobian functions in the rectangle with vertices 0 , 2 K , 2 K + 2 i K , 2 i K . … Figure 22.4.2 depicts the fundamental unit cell in the z -plane, with vertices s = 0 , c = K , d = K + i K , n = i K . … This half-period will be plus or minus a member of the triple K , i K , K + i K ; the other two members of this triple are quarter periods of p q ( z , k ) . …
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 F ( ϕ , k ) and E ( ϕ , k ) , together with special cases.
§19.13(iii) Laplace Transforms
For direct and inverse Laplace transforms for the complete elliptic integrals K ( k ) , E ( k ) , and D ( k ) see Prudnikov et al. (1992a, §3.31) and Prudnikov et al. (1992b, §§3.29 and 4.3.33), respectively.
15: 29.17 Other Solutions
29.17.1 F ( z ) = E ( z ) i K z d u ( E ( u ) ) 2 .
They are algebraic functions of sn ( z , k ) , cn ( z , k ) , and dn ( z , k ) , and have primitive period 8 K . … Lamé–Wangerin functions are solutions of (29.2.1) with the property that ( sn ( z , k ) ) 1 / 2 w ( z ) is bounded on the line segment from i K to 2 K + i K . …
16: 36.3 Visualizations of Canonical Integrals
Figure 36.3.6: Modulus of elliptic umbilic canonical integral function | Ψ ( E ) ( x , y , 0 ) | . …
Figure 36.3.7: Modulus of elliptic umbilic canonical integral function | Ψ ( E ) ( x , y , 2 ) | . …
Figure 36.3.15: Phase of elliptic umbilic canonical integral ph Ψ ( E ) ( x , y , 0 ) . …
Figure 36.3.16: Phase of elliptic umbilic canonical integral ph Ψ ( E ) ( x , y , 2 ) . …
Figure 36.3.17: Phase of elliptic umbilic canonical integral ph Ψ ( E ) ( x , y , 4 ) . …
17: 22.11 Fourier and Hyperbolic Series
22.11.3 dn ( z , k ) = π 2 K + 2 π K n = 1 q n cos ( 2 n ζ ) 1 + q 2 n .
Next, with E = E ( k ) denoting the complete elliptic integral of the second kind (§19.2(ii)) and q exp ( 2 | ζ | ) < 1 ,
22.11.13 sn 2 ( z , k ) = 1 k 2 ( 1 E K ) 2 π 2 k 2 K 2 n = 1 n q n 1 q 2 n cos ( 2 n ζ ) .
where E = E ( k ) is defined by §19.2.9. …
18: 19.3 Graphics
§19.3 Graphics
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Figure 19.3.1: K ( k ) and E ( k ) as functions of k 2 for 2 k 2 1 . Graphs of K ( k ) and E ( k ) are the mirror images in the vertical line k 2 = 1 2 . Magnify
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Figure 19.3.5: Π ( α 2 , k ) as a function of k 2 and α 2 for 2 k 2 < 1 , 2 α 2 2 . …If α 2 = 0 , then it reduces to K ( k ) . … Magnify 3D Help
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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
19: 19.38 Approximations
§19.38 Approximations
Minimax polynomial approximations (§3.11(i)) for K ( k ) and E ( k ) in terms of m = k 2 with 0 m < 1 can be found in Abramowitz and Stegun (1964, §17.3) with maximum absolute errors ranging from 4×10⁻⁵ to 2×10⁻⁸. Approximations of the same type for K ( k ) and E ( k ) for 0 < k 1 are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. …
20: 36.10 Differential Equations
In terms of the normal forms (36.2.2) and (36.2.3), the Ψ ( U ) ( 𝐱 ) satisfy the following operator equations …
36.10.14 3 ( 2 Ψ ( E ) x 2 2 Ψ ( E ) y 2 ) + 2 i z Ψ ( E ) x x Ψ ( E ) = 0 .