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1: 22.7 Landen Transformations
22.7.1 k 1 = 1 k 1 + k ,
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 ) ,
k 2 = 2 k 1 + k ,
k 2 = 1 k 1 + k ,
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: 22.6 Elementary Identities
22.6.4 k 2 k 2 sd 2 ( z , k ) = k 2 ( cd 2 ( z , k ) 1 ) = k 2 ( 1 nd 2 ( z , k ) ) .
22.6.8 cd ( 2 z , k ) = cd 2 ( z , k ) k 2 sd 2 ( z , k ) nd 2 ( z , k ) 1 + k 2 k 2 sd 4 ( z , k ) ,
22.6.10 nd ( 2 z , k ) = nd 2 ( z , k ) + k 2 sd 2 ( z , k ) cd 2 ( z , k ) 1 + k 2 k 2 sd 4 ( z , k ) ,
22.6.20 cn 2 ( 1 2 z , k ) = k 2 + dn ( z , k ) + k 2 cn ( z , k ) k 2 ( 1 + cn ( z , k ) ) = k 2 ( 1 dn ( z , k ) ) k 2 ( dn ( z , k ) cn ( z , k ) ) = k 2 ( 1 + cn ( z , k ) ) k 2 + dn ( z , k ) k 2 cn ( z , k ) ,
22.6.21 dn 2 ( 1 2 z , k ) = k 2 cn ( z , k ) + dn ( z , k ) + k 2 1 + dn ( z , k ) = k 2 ( 1 cn ( z , k ) ) dn ( z , k ) cn ( z , k ) = k 2 ( 1 + dn ( z , k ) ) k 2 + dn ( z , k ) k 2 cn ( z , k ) .
3: 19.4 Derivatives and Differential Equations
d ( E ( k ) k 2 K ( k ) ) d k = k K ( k ) ,
Let D k = / k . Then …If ϕ = π / 2 , then these two equations become hypergeometric differential equations (15.10.1) for K ( k ) and E ( k ) . An analogous differential equation of third order for Π ( ϕ , α 2 , k ) is given in Byrd and Friedman (1971, 118.03).
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. Lawden (1989, pp. 280–284 and 293–297) tabulates sn ( x , k ) , cn ( x , k ) , dn ( x , k ) , ( x , k ) , Z ( x | k ) to 5D for k = 0.1 ( .1 ) 0.9 , x = 0 ( .1 ) X , where X ranges from 1. … 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: 22.17 Moduli Outside the Interval [0,1]
k 1 k 1 = k 1 + k 2 ,
In terms of the coefficients of the power series of §22.10(i), the above equations are polynomial identities in k . … When z is fixed each of the twelve Jacobian elliptic functions is a meromorphic function of k 2 . …In consequence, the formulas in this chapter remain valid when k is complex. In particular, the Landen transformations in §§22.7(i) and 22.7(ii) are valid for all complex values of k , irrespective of which values of k and k = 1 k 2 are chosen—as long as they are used consistently. …
6: 22.13 Derivatives and Differential Equations
(The modulus k is suppressed throughout the table.) …
22.13.1 ( d d z sn ( z , k ) ) 2 = ( 1 sn 2 ( z , k ) ) ( 1 k 2 sn 2 ( z , k ) ) ,
22.13.14 d 2 d z 2 cn ( z , k ) = ( k 2 k 2 ) cn ( z , k ) 2 k 2 cn 3 ( z , k ) ,
22.13.17 d 2 d z 2 sd ( z , k ) = ( k 2 k 2 ) sd ( z , k ) 2 k 2 k 2 sd 3 ( z , k ) ,
22.13.20 d 2 d z 2 nc ( z , k ) = ( k 2 k 2 ) nc ( z , k ) + 2 k 2 nc 3 ( z , k ) ,
7: 22.5 Special Values
For example, at z = K + i K , sn ( z , k ) = 1 / k , d sn ( z , k ) / d z = 0 . (The modulus k is suppressed throughout the table.) … Table 22.5.2 gives sn ( z , k ) , cn ( z , k ) , dn ( z , k ) for other special values of z . …
§22.5(ii) Limiting Values of k
Expansions for K , K as k 0 or 1 are given in §§19.5, 19.12. …
8: 29.14 Orthogonality
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 ) ,
29.14.5 𝑐𝐸 2 n + 1 m ( s , k 2 ) 𝑐𝐸 2 n + 1 m ( K + i t , k 2 ) ,
29.14.6 𝑑𝐸 2 n + 1 m ( s , k 2 ) 𝑑𝐸 2 n + 1 m ( K + i t , k 2 ) ,
29.14.7 𝑠𝑐𝐸 2 n + 2 m ( s , k 2 ) 𝑠𝑐𝐸 2 n + 2 m ( K + i t , k 2 ) ,
9: 22.16 Related Functions
am ( x , k ) is an infinitely differentiable function of x . …
Approximations for Small k , k
In Equations (22.16.21)–(22.16.23), K < x < K . In Equations (22.16.24)–(22.16.26), 2 K < x < 2 K . … For E ( k ) see §19.2(ii). …
10: 26.1 Special Notation
x

real variable.

j | k

j divides k .

( h , k )

greatest common divisor of positive integers h and k .

( m n )

binomial coefficient.

p k ( n )

number of partitions of n into at most k parts.

Other notations for s ( n , k ) , the Stirling numbers of the first kind, include S n ( k ) (Abramowitz and Stegun (1964, Chapter 24), Fort (1948)), S n k (Jordan (1939), Moser and Wyman (1958a)), ( n 1 k 1 ) B n k ( n ) (Milne-Thomson (1933)), ( 1 ) n k S 1 ( n 1 , n k ) (Carlitz (1960), Gould (1960)), ( 1 ) n k [ n k ] (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)). Other notations for S ( n , k ) , the Stirling numbers of the second kind, include 𝒮 n ( k ) (Fort (1948)), 𝔖 n k (Jordan (1939)), σ n k (Moser and Wyman (1958b)), ( n k ) B n k ( k ) (Milne-Thomson (1933)), S 2 ( k , n k ) (Carlitz (1960), Gould (1960)), { n k } (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)), and also an unconventional symbol in Abramowitz and Stegun (1964, Chapter 24).