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1: 22.17 Moduli Outside the Interval [0,1]
§22.17 Moduli Outside the Interval [0,1]
22.17.6 sn ( z , i k ) = k 1 sd ( z / k 1 , k 1 ) ,
22.17.7 cn ( z , i k ) = cd ( z / k 1 , k 1 ) ,
22.17.8 dn ( z , i k ) = nd ( z / k 1 , k 1 ) .
For proofs of these results and further information see Walker (2003).
2: 19.7 Connection Formulas
§19.7(ii) Change of Modulus and Amplitude
Π ( ϕ , α 2 , k 1 ) = k Π ( β , k 2 α 2 , k ) , k 1 = 1 / k , sin β = k 1 sin ϕ 1 .
κ = k 1 + k 2 ,
κ = 1 1 + k 2 ,
19.7.8 Π ( ϕ , α 2 , k ) + Π ( ϕ , ω 2 , k ) = F ( ϕ , k ) + c R C ( ( c - 1 ) ( c - k 2 ) , ( c - α 2 ) ( c - ω 2 ) ) , α 2 ω 2 = k 2 .
3: 22.11 Fourier and Hyperbolic Series
22.11.1 sn ( z , k ) = 2 π K k n = 0 q n + 1 2 sin ( ( 2 n + 1 ) ζ ) 1 - q 2 n + 1 ,
22.11.5 sd ( z , k ) = 2 π K k k n = 0 ( - 1 ) n q n + 1 2 sin ( ( 2 n + 1 ) ζ ) 1 + q 2 n + 1 ,
22.11.6 nd ( z , k ) = π 2 K k + 2 π K k n = 1 ( - 1 ) n q n cos ( 2 n ζ ) 1 + q 2 n .
22.11.11 nc ( z , k ) - π 2 K k sec ζ = - 2 π K k n = 0 ( - 1 ) n q 2 n + 1 cos ( ( 2 n + 1 ) ζ ) 1 + q 2 n + 1 ,
22.11.12 sc ( z , k ) - π 2 K k tan ζ = 2 π K k n = 1 ( - 1 ) n q 2 n sin ( 2 n ζ ) 1 + q 2 n .
4: 22.2 Definitions
22.2.4 sn ( z , k ) = θ 3 ( 0 , q ) θ 2 ( 0 , q ) θ 1 ( ζ , q ) θ 4 ( ζ , q ) = 1 ns ( z , k ) ,
22.2.5 cn ( z , k ) = θ 4 ( 0 , q ) θ 2 ( 0 , q ) θ 2 ( ζ , q ) θ 4 ( ζ , q ) = 1 nc ( z , k ) ,
22.2.6 dn ( z , k ) = θ 4 ( 0 , q ) θ 3 ( 0 , q ) θ 3 ( ζ , q ) θ 4 ( ζ , q ) = 1 nd ( z , k ) ,
22.2.7 sd ( z , k ) = θ 3 2 ( 0 , q ) θ 2 ( 0 , q ) θ 4 ( 0 , q ) θ 1 ( ζ , q ) θ 3 ( ζ , q ) = 1 ds ( z , k ) ,
5: 22.7 Landen Transformations
22.7.2 sn ( z , k ) = ( 1 + k 1 ) sn ( z / ( 1 + k 1 ) , k 1 ) 1 + k 1 sn 2 ( z / ( 1 + k 1 ) , k 1 ) ,
22.7.4 dn ( z , k ) = dn 2 ( z / ( 1 + k 1 ) , k 1 ) - ( 1 - k 1 ) 1 + k 1 - dn 2 ( z / ( 1 + k 1 ) , k 1 ) .
22.7.6 sn ( z , k ) = ( 1 + k 2 ) sn ( z / ( 1 + k 2 ) , k 2 ) cn ( z / ( 1 + k 2 ) , k 2 ) dn ( z / ( 1 + k 2 ) , k 2 ) ,
22.7.7 cn ( z , k ) = ( 1 + k 2 ) ( dn 2 ( z / ( 1 + k 2 ) , k 2 ) - k 2 ) k 2 2 dn ( z / ( 1 + k 2 ) , k 2 ) ,
22.7.8 dn ( z , k ) = ( 1 - k 2 ) ( dn 2 ( z / ( 1 + k 2 ) , k 2 ) + k 2 ) k 2 2 dn ( z / ( 1 + k 2 ) , k 2 ) .
6: 22.6 Elementary Identities
§22.6(v) Change of Modulus
7: 19.8 Quadratic Transformations
k 2 = 2 k / ( 1 + k ) ,
k 1 = ( 1 - k ) / ( 1 + k ) ,
ρ = 1 - ( k 2 / α 2 ) ,
α 1 2 = α 2 ( 1 + ρ ) 2 / ( 1 + k ) 2 ,
8: 19.36 Methods of Computation
If (19.36.1) is used instead of its first five terms, then the factor ( 3 r ) - 1 / 6 in Carlson (1995, (2.2)) is changed to ( 3 r ) - 1 / 8 . For both R D and R J the factor ( r / 4 ) - 1 / 6 in Carlson (1995, (2.18)) is changed to ( r / 5 ) - 1 / 8 when the following polynomial of degree 7 (the same for both) is used instead of its first seven terms: …
9: 31.2 Differential Equations
31.2.8 d 2 w d ζ 2 + ( ( 2 γ - 1 ) cn ζ dn ζ sn ζ - ( 2 δ - 1 ) sn ζ dn ζ cn ζ - ( 2 ϵ - 1 ) k 2 sn ζ cn ζ dn ζ ) d w d ζ + 4 k 2 ( α β sn 2 ζ - q ) w = 0 .
10: 9.8 Modulus and Phase
In terms of Bessel functions, and with ξ = 2 3 | x | 3 / 2 ,
9.8.9 | x | 1 / 2 M 2 ( x ) = 1 2 ξ ( J 1 / 3 2 ( ξ ) + Y 1 / 3 2 ( ξ ) ) ,
9.8.10 | x | - 1 / 2 N 2 ( x ) = 1 2 ξ ( J 2 / 3 2 ( ξ ) + Y 2 / 3 2 ( ξ ) ) ,