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1: 22.7 Landen Transformations
22.7.1 k 1 = 1 k 1 + k ,
k 2 = 2 k 1 + k ,
k 2 = 1 k 1 + k ,
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 ) ,
22.7.8 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 ) .
2: 23.21 Physical Applications
Another form is obtained by identifying e 1 , e 2 , e 3 as lattice roots (§23.3(i)), and setting …
23.21.5 ( ( v ) ( w ) ) ( ( w ) ( u ) ) ( ( u ) ( v ) ) 2 = ( ( w ) ( v ) ) 2 u 2 + ( ( u ) ( w ) ) 2 v 2 + ( ( v ) ( u ) ) 2 w 2 .
3: 22.17 Moduli Outside the Interval [0,1]
k 1 = k 1 + k 2 ,
k 1 k 1 = k 1 + k 2 ,
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 ) .
4: 19.8 Quadratic Transformations
k 1 = 1 k 1 + k ,
c 1 = csc 2 ϕ 1 .
k 2 = 2 k / ( 1 + k ) ,
k 1 = ( 1 k ) / ( 1 + k ) ,
c = csc 2 ϕ .
5: 19.7 Connection Formulas
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 ,
With sinh ϕ = tan ψ , … The first of the three relations maps each circular region onto itself and each hyperbolic region onto the other; in particular, it gives the Cauchy principal value of Π ( ϕ , α 2 , k ) when α 2 > csc 2 ϕ (see (19.6.5) for the complete case). …
6: 1.13 Differential Equations
§1.13(iv) Change of Variables
Transformation of the Point at Infinity
Elimination of First Derivative by Change of Dependent Variable
Elimination of First Derivative by Change of Independent Variable
Liouville Transformation
7: 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 ,
22.11.2 cn ( z , k ) = 2 π K k n = 0 q n + 1 2 cos ( ( 2 n + 1 ) ζ ) 1 + q 2 n + 1 ,
22.11.3 dn ( z , k ) = π 2 K + 2 π K n = 1 q n cos ( 2 n ζ ) 1 + q 2 n .
22.11.4 cd ( z , k ) = 2 π K k n = 0 ( 1 ) n q n + 1 2 cos ( ( 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 ,
8: 9.7 Asymptotic Expansions
9.7.1 ζ = 2 3 z 3 / 2 .
9.7.18 Ai ( z ) = e ζ 2 π z 1 / 4 ( k = 0 n 1 ( 1 ) k u k ζ k + R n ( z ) ) ,
9.7.19 Ai ( z ) = z 1 / 4 e ζ 2 π ( k = 0 n 1 ( 1 ) k v k ζ k + S n ( z ) ) ,
9.7.20 R n ( z ) = ( 1 ) n k = 0 m 1 ( 1 ) k u k G n k ( 2 ζ ) ζ k + R m , n ( z ) ,
9.7.21 S n ( z ) = ( 1 ) n 1 k = 0 m 1 ( 1 ) k v k G n k ( 2 ζ ) ζ k + S m , n ( z ) ,
9: 7.16 Generalized Error Functions
These functions can be expressed in terms of the incomplete gamma function γ ( a , z ) 8.2(i)) by change of integration variable.
10: 7.5 Interrelations
7.5.7 ζ = 1 2 π ( 1 i ) z ,
7.5.8 C ( z ) ± i S ( z ) = 1 2 ( 1 ± i ) erf ζ .
7.5.9 C ( z ) ± i S ( z ) = 1 2 ( 1 ± i ) ( 1 e ± 1 2 π i z 2 w ( i ζ ) ) .
7.5.10 g ( z ) ± i f ( z ) = 1 2 ( 1 ± i ) e ζ 2 erfc ζ .