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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 29.13.21: | 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
3: 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. …
4: 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 . … 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 ) . …
Table 22.4.3: Half- or quarter-period shifts of variable for the Jacobian elliptic functions.
u
z + K z + K + i K z + i K z + 2 K z + 2 K + 2 i K z + 2 i K
5: 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
6: 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 . …
7: 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. …
8: 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 . …As α 2 1 + it has the limit K ( k ) ( E ( k ) / k 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 . …On the upper edge of the branch cut ( k 2 > 1 ) it has the (negative) value K ( k ) E ( k ) , with limit 0 as k 2 1 + . Magnify 3D Help
9: 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. …
10: 29.16 Asymptotic Expansions
The approximations for Lamé polynomials hold uniformly on the rectangle 0 z K , 0 z K , when n k and n k assume large real values. …