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1: 22.8 Addition Theorems
22.8.5 sd ( u + v ) = sd u cd v nd v + sd v cd u nd u 1 + k 2 k 2 sd 2 u sd 2 v ,
22.8.9 sc ( u + v ) = sc u dc v nc v + sc v dc u nc u 1 k 2 sc 2 u sc 2 v ,
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 .
2: 22.6 Elementary Identities
22.6.2 1 + cs 2 ( z , k ) = k 2 + ds 2 ( z , k ) = ns 2 ( z , k ) ,
22.6.9 sd ( 2 z , k ) = 2 sd ( z , k ) cd ( z , k ) nd ( z , k ) 1 + k 2 k 2 sd 4 ( 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 ) ,
22.6.15 ds ( 2 z , k ) = k 2 k 2 + ds 4 ( z , k ) 2 cs ( z , k ) ds ( z , k ) ns ( z , k ) ,
22.6.16 cs ( 2 z , k ) = cs 4 ( z , k ) k 2 2 cs ( z , k ) ds ( z , k ) ns ( z , k ) .
3: 22.13 Derivatives and Differential Equations
22.13.5 ( d d z sd ( z , k ) ) 2 = ( 1 k 2 sd 2 ( z , k ) ) ( 1 + k 2 sd 2 ( z , k ) ) ,
22.13.9 ( d d z sc ( z , k ) ) 2 = ( 1 + sc 2 ( z , k ) ) ( 1 + k 2 sc 2 ( z , k ) ) ,
22.13.12 ( d d z cs ( z , k ) ) 2 = ( 1 + cs 2 ( z , k ) ) ( k 2 + cs 2 ( 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 ) ,
22.13.21 d 2 d z 2 sc ( z , k ) = ( 1 + k 2 ) sc ( z , k ) + 2 k 2 sc 3 ( z , k ) ,
4: 22.4 Periods, Poles, and Zeros
For example, the poles of sn ( z , k ) , abbreviated as sn in the following tables, are at z = 2 m K + ( 2 n + 1 ) i K . … For example, sn ( z + K , k ) = cd ( z , k ) . …
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
ns u dc z k cd z k sn z ns z ns z ns z
cs u k sc z i k nd z i dn z cs z cs z cs z
5: 27.9 Quadratic Characters
If an odd integer P has prime factorization P = r = 1 ν ( n ) p r a r , then the Jacobi symbol ( n | P ) is defined by ( n | P ) = r = 1 ν ( n ) ( n | p r ) a r , with ( n | 1 ) = 1 . The Jacobi symbol ( n | P ) is a Dirichlet character (mod P ). …
6: 22.14 Integrals
22.14.4 cd ( x , k ) d x = k 1 ln ( nd ( x , k ) + k sd ( x , k ) ) ,
22.14.5 sd ( x , k ) d x = ( k k ) 1 Arcsin ( k cd ( x , 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 . … Table 22.5.2 gives sn ( z , k ) , cn ( z , k ) , dn ( z , k ) for other special values of z . For example, sn ( 1 2 K , k ) = ( 1 + k ) 1 / 2 . …
Table 22.5.3: Limiting forms of Jacobian elliptic functions as k 0 .
sn ( z , k ) sin z cd ( z , k ) cos z dc ( z , k ) sec z ns ( z , k ) csc z
dn ( z , k ) 1 nd ( z , k ) 1 sc ( z , k ) tan z cs ( z , k ) cot z
Table 22.5.4: Limiting forms of Jacobian elliptic functions as k 1 .
sn ( z , k ) tanh z cd ( z , k ) 1 dc ( z , k ) 1 ns ( z , k ) coth z
dn ( z , k ) sech z nd ( z , k ) cosh z sc ( z , k ) sinh z cs ( z , k ) csch z
8: 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). …
9: 22.3 Graphics
Line graphs of the functions sn ( x , k ) , cn ( x , k ) , dn ( x , k ) , cd ( x , k ) , sd ( x , k ) , nd ( x , k ) , dc ( x , k ) , nc ( x , k ) , sc ( x , k ) , ns ( x , k ) , ds ( x , k ) , and cs ( x , k ) for representative values of real x and real k illustrating the near trigonometric ( k = 0 ), and near hyperbolic ( k = 1 ) limits. … sn ( x , k ) , cn ( x , k ) , and dn ( x , k ) as functions of real arguments x and k . …
See accompanying text
Figure 22.3.13: sn ( x , k ) for k = 1 e n , n = 0 to 20, 5 π x 5 π . Magnify 3D Help
See accompanying text
Figure 22.3.21: ns ( x + i y , k ) for k = 0.99 , 3 K x 3 K , 0 y 4 K . … Magnify 3D Help
See accompanying text
Figure 22.3.22: sn ( x , k ) , x = 120 , as a function of k 2 = i κ 2 , 0 κ 4 . Magnify
10: 27.1 Special Notation
d , k , m , n positive integers (unless otherwise indicated).
( n | P ) Jacobi symbol; see §27.9.