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1: 22.16 Related Functions
§22.16(i) Jacobi’s Amplitude ( am ) Function
Definition
Quasi-Periodicity
Integral Representation
Special Values
2: 22.8 Addition Theorems
22.8.10 ns ( u + v ) = ns u ds v cs v - ns v ds u cs u cs 2 v - cs 2 u ,
22.8.11 ds ( u + v ) = ds u cs v ns v - ds v cs u ns u cs 2 v - cs 2 u ,
22.8.12 cs ( u + v ) = cs u ds v ns v - cs v ds u ns u cs 2 v - cs 2 u .
22.8.14 sn ( u + v ) = sn u cn u dn v + sn v cn v dn u cn u cn v + sn u dn u sn v dn v ,
22.8.23 | sn z 1 cn z 1 cn z 1 dn z 1 cn z 1 dn z 1 sn z 2 cn z 2 cn z 2 dn z 2 cn z 2 dn z 2 sn z 3 cn z 3 cn z 3 dn z 3 cn z 3 dn z 3 sn z 4 cn z 4 cn z 4 dn z 4 cn z 4 dn z 4 | = 0 .
3: 22.6 Elementary Identities
22.6.2 1 + cs 2 ( z , k ) = k 2 + ds 2 ( z , k ) = ns 2 ( z , k ) ,
22.6.14 ns ( 2 z , k ) = ns 4 ( z , k ) - k 2 2 cs ( z , k ) ds ( z , k ) ns ( z , k ) ,
Table 22.6.1: Jacobi’s imaginary transformation of Jacobian elliptic functions.
sn ( i z , k ) = i sc ( z , k ) dc ( i z , k ) = dn ( z , k )
cn ( i z , k ) = nc ( z , k ) nc ( i z , k ) = cn ( z , k )
dn ( i z , k ) = dc ( z , k ) sc ( i z , k ) = i sn ( z , k )
4: 22.4 Periods, Poles, and Zeros
Table 22.4.3: Half- or quarter-period shifts of variable for the Jacobian elliptic functions.
u
sn u cd z k - 1 dc z k - 1 ns z - sn z - sn z sn z
dn u k nd z i k sc z - i cs z dn z - dn z - dn z
cd u - sn z - k - 1 ns z k - 1 dc z - cd z - cd z cd z
dc u - ns z - k sn z k cd z - dc z - dc z dc z
ns u dc z k cd z k sn z - ns z - ns z ns z
5: 22.13 Derivatives and Differential Equations
Table 22.13.1: Derivatives of Jacobian elliptic functions with respect to variable.
d d z ( sn z ) = cn z dn z d d z ( dc z )  = k 2 sc z nc z
d d z ( cn z ) = - sn z dn z d d z ( nc z )  = sc z dc z
d d z ( dn z ) = - k 2 sn z cn z d d z ( sc z )  = dc z nc z
d d z ( sd z ) = cd z nd z d d z ( ds z )  = - cs z ns z
d d z ( nd z ) = k 2 sd z cd z d d z ( cs z )  = - ns z ds z
6: 22.21 Tables
§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. …
7: 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 ) ,
22.7.3 cn ( z , k ) = cn ( z / ( 1 + k 1 ) , k 1 ) dn ( 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 ) ,
22.7.8 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 ) .
8: 22.14 Integrals
See §22.16(i) for am ( z , k ) . …
22.14.15 cn ( x , k ) d x sn 2 ( x , k ) = - dn ( x , k ) sn ( x , k ) .
9: 22.1 Special Notation
The functions treated in this chapter are the three principal Jacobian elliptic functions sn ( z , k ) , cn ( z , k ) , dn ( z , k ) ; the nine subsidiary Jacobian elliptic functions cd ( z , k ) , sd ( z , k ) , nd ( z , k ) , dc ( z , k ) , nc ( z , k ) , sc ( z , k ) , ns ( z , k ) , ds ( z , k ) , cs ( z , k ) ; the amplitude function am ( x , k ) ; Jacobi’s epsilon and zeta functions ( x , k ) and Z ( x | k ) . … The notation sn ( z , k ) , cn ( z , k ) , dn ( z , k ) is due to Gudermann (1838), following Jacobi (1827); that for the subsidiary functions is due to Glaisher (1882). Other notations for sn ( z , k ) are sn ( z | m ) and sn ( z , m ) with m = k 2 ; see Abramowitz and Stegun (1964) and Walker (1996). …
10: 22.17 Moduli Outside the Interval [0,1]
22.17.1 p q ( z , k ) = p q ( z , - k ) ,
22.17.2 sn ( z , 1 / k ) = k sn ( z / k , k ) ,
22.17.3 cn ( z , 1 / k ) = dn ( z / k , k ) ,
22.17.4 dn ( z , 1 / k ) = cn ( z / k , k ) .
22.17.7 cn ( z , i k ) = cd ( z / k 1 , k 1 ) ,