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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: 10.35 Generating Function and Associated Series
10.35.1 e 1 2 z ( t + t 1 ) = m = t m I m ( z ) .
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: 29.14 Orthogonality
First, the orthogonality relations (29.3.19) apply; see §29.12(i). …
29.14.2 g , h = 0 K 0 K w ( s , t ) g ( s , t ) h ( s , t ) d t d s ,
29.14.3 w ( s , t ) = sn 2 ( K + i t , k ) sn 2 ( s , k ) .
29.14.4 𝑠𝐸 2 n + 1 m ( s , k 2 ) 𝑠𝐸 2 n + 1 m ( K + i t , k 2 ) ,
29.14.11 g , h = 0 4 K 0 2 K w ( s , t ) g ( s , t ) h ( s , t ) d t d s ,
5: 14.10 Recurrence Relations and Derivatives
14.10.1 𝖯 ν μ + 2 ( x ) + 2 ( μ + 1 ) x ( 1 x 2 ) 1 / 2 𝖯 ν μ + 1 ( x ) + ( ν μ ) ( ν + μ + 1 ) 𝖯 ν μ ( x ) = 0 ,
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 ,
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).
6: 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 .
7: 10.33 Continued Fractions
Assume I ν 1 ( z ) 0 . …
10.33.1 I ν ( z ) I ν 1 ( z ) = 1 2 ν z 1 + 1 2 ( ν + 1 ) z 1 + 1 2 ( ν + 2 ) z 1 + , z 0 ,
10.33.2 I ν ( z ) I ν 1 ( z ) = 1 2 z / ν 1 + 1 4 z 2 / ( ν ( ν + 1 ) ) 1 + 1 4 z 2 / ( ( ν + 1 ) ( ν + 2 ) ) 1 + , ν 0 , 1 , 2 , .
8: 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
9: 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. …
10: 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 ) . …