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1: 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 . …
2: 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 29.13.7: sE 5 m ( x , 0.1 ) for - 2 K x 2 K , m = 0 , 1 , 2 . K = 1.61244 . Magnify
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Figure 29.13.8: sE 5 m ( x , 0.9 ) for - 2 K x 2 K , m = 0 , 1 , 2 . K = 2.57809 . Magnify
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Figure 29.13.9: cE 5 m ( x , 0.1 ) for - 2 K x 2 K , m = 0 , 1 , 2 . K = 1.61244 . Magnify
3: 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
4: 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 , where E = E ( k ) is defined by §19.2.9. …
5: 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. …
6: 22.4 Periods, Poles, and Zeros
The other poles are at congruent points, which is the set of points obtained by making translations by 2 m K + 2 n i K , where m , n . … 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 . The set of points z = m K + n i K , m , n , comprise the lattice for the 12 Jacobian functions; all other lattice unit cells are generated by translation of the fundamental unit cell by m K + n i K , where again m , n . … 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 ) . …
7: 29.14 Orthogonality
is orthogonal and complete with respect to the inner product
29.14.2 g , h = 0 K 0 K w ( s , t ) g ( s , t ) h ( s , t ) d t d s ,
Each of the following seven systems is orthogonal and complete with respect to the inner product (29.14.2): …When combined, all eight systems (29.14.1) and (29.14.4)–(29.14.10) form an orthogonal and complete system with respect to the inner product
29.14.11 g , h = 0 4 K 0 2 K w ( s , t ) g ( s , t ) h ( s , t ) d t d s ,
8: 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 . …
9: 19.3 Graphics
See Figures 19.3.119.3.6 for complete and incomplete Legendre’s elliptic integrals.
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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
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. …
10: 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. … Approximations for Legendre’s complete or incomplete integrals of all three kinds, derived by Padé approximation of the square root in the integrand, are given in Luke (1968, 1970). …