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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: 10.35 Generating Function and Associated Series
10.35.2 e z cos θ = I 0 ( z ) + 2 k = 1 I k ( z ) cos ( k θ ) ,
10.35.4 1 = I 0 ( z ) - 2 I 2 ( z ) + 2 I 4 ( z ) - 2 I 6 ( z ) + ,
10.35.5 e ± z = I 0 ( z ) ± 2 I 1 ( z ) + 2 I 2 ( z ) ± 2 I 3 ( z ) + ,
cosh z = I 0 ( z ) + 2 I 2 ( z ) + 2 I 4 ( z ) + 2 I 6 ( z ) + ,
sinh z = 2 I 1 ( z ) + 2 I 3 ( z ) + 2 I 5 ( z ) + .
4: 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. …
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: 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 ) . …
7: 10.28 Wronskians and Cross-Products
10.28.1 𝒲 { I ν ( z ) , I - ν ( z ) } = I ν ( z ) I - ν - 1 ( z ) - I ν + 1 ( z ) I - ν ( z ) = - 2 sin ( ν π ) / ( π z ) ,
10.28.2 𝒲 { K ν ( z ) , I ν ( z ) } = I ν ( z ) K ν + 1 ( z ) + I ν + 1 ( z ) K ν ( z ) = 1 / z .
8: 14.10 Recurrence Relations and Derivatives
14.10.2 ( 1 - x 2 ) 1 / 2 P ν μ + 1 ( x ) - ( ν - μ + 1 ) P ν + 1 μ ( x ) + ( ν + μ + 1 ) x P ν μ ( x ) = 0 ,
14.10.3 ( ν - μ + 2 ) P ν + 2 μ ( x ) - ( 2 ν + 3 ) x P ν + 1 μ ( x ) + ( ν + μ + 1 ) P ν μ ( x ) = 0 ,
14.10.5 ( 1 - x 2 ) d P ν μ ( x ) d x = ( ν + μ ) P ν - 1 μ ( x ) - ν x P ν μ ( x ) .
Q ν μ ( x ) also satisfies (14.10.1)–(14.10.5). …In addition, P ν μ ( x ) and Q ν μ ( x ) satisfy (14.10.3)–(14.10.5).
9: 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 . …
10: 14.33 Tables
  • Abramowitz and Stegun (1964, Chapter 8) tabulates P n ( x ) for n = 0 ( 1 ) 3 , 9 , 10 , x = 0 ( .01 ) 1 , 5–8D; P n ( x ) for n = 1 ( 1 ) 4 , 9 , 10 , x = 0 ( .01 ) 1 , 5–7D; Q n ( x ) and Q n ( x ) for n = 0 ( 1 ) 3 , 9 , 10 , x = 0 ( .01 ) 1 , 6–8D; P n ( x ) and P n ( x ) for n = 0 ( 1 ) 5 , 9 , 10 , x = 1 ( .2 ) 10 , 6S; Q n ( x ) and Q n ( x ) for n = 0 ( 1 ) 3 , 9 , 10 , x = 1 ( .2 ) 10 , 6S. (Here primes denote derivatives with respect to x .)

  • Zhang and Jin (1996, Chapter 4) tabulates P n ( x ) for n = 2 ( 1 ) 5 , 10 , x = 0 ( .1 ) 1 , 7D; P n ( cos θ ) for n = 1 ( 1 ) 4 , 10 , θ = 0 ( 5 ) 90 , 8D; Q n ( x ) for n = 0 ( 1 ) 2 , 10 , x = 0 ( .1 ) 0.9 , 8S; Q n ( cos θ ) for n = 0 ( 1 ) 3 , 10 , θ = 0 ( 5 ) 90 , 8D; P n m ( x ) for m = 1 ( 1 ) 4 , n - m = 0 ( 1 ) 2 , n = 10 , x = 0 , 0.5 , 8S; Q n m ( x ) for m = 1 ( 1 ) 4 , n = 0 ( 1 ) 2 , 10 , 8S; P ν m ( cos θ ) for m = 0 ( 1 ) 3 , ν = 0 ( .25 ) 5 , θ = 0 ( 15 ) 90 , 5D; P n ( x ) for n = 2 ( 1 ) 5 , 10 , x = 1 ( 1 ) 10 , 7S; Q n ( x ) for n = 0 ( 1 ) 2 , 10 , x = 2 ( 1 ) 10 , 8S. Corresponding values of the derivative of each function are also included, as are 6D values of the first 5 ν -zeros of P ν m ( cos θ ) and of its derivative for m = 0 ( 1 ) 4 , θ = 10 , 30 , 150 .

  • Belousov (1962) tabulates P n m ( cos θ ) (normalized) for m = 0 ( 1 ) 36 , n - m = 0 ( 1 ) 56 , θ = 0 ( 2.5 ) 90 , 6D.

  • Žurina and Karmazina (1964, 1965) tabulate the conical functions P - 1 2 + i τ ( x ) for τ = 0 ( .01 ) 50 , x = - 0.9 ( .1 ) 0.9 , 7S; P - 1 2 + i τ ( x ) for τ = 0 ( .01 ) 50 , x = 1.1 ( .1 ) 2 ( .2 ) 5 ( .5 ) 10 ( 10 ) 60 , 7D. Auxiliary tables are included to facilitate computation for larger values of τ when - 1 < x < 1 .

  • Žurina and Karmazina (1963) tabulates the conical functions P - 1 2 + i τ 1 ( x ) for τ = 0 ( .01 ) 25 , x = - 0.9 ( .1 ) 0.9 , 7S; P - 1 2 + i τ 1 ( x ) for τ = 0 ( .01 ) 25 , x = 1.1 ( .1 ) 2 ( .2 ) 5 ( .5 ) 10 ( 10 ) 60 , 7S. Auxiliary tables are included to assist computation for larger values of τ when - 1 < x < 1 .