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11: 22.5 Special Values
§22.5 Special Values
Table 22.5.1 gives the value of each of the 12 Jacobian elliptic functions, together with its z -derivative (or at a pole, the residue), for values of z that are integer multiples of K , i K . …
Table 22.5.2: Other special values of Jacobian elliptic functions.
z
§22.5(ii) Limiting Values of k
12: 22.13 Derivatives and Differential Equations
§22.13(i) Derivatives
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
§22.13(ii) First-Order Differential Equations
22.13.1 ( d d z sn ( z , k ) ) 2 = ( 1 sn 2 ( z , k ) ) ( 1 k 2 sn 2 ( z , k ) ) ,
§22.13(iii) Second-Order Differential Equations
13: 22.10 Maclaurin Series
§22.10(i) Maclaurin Series in z
22.10.1 sn ( z , k ) = z ( 1 + k 2 ) z 3 3 ! + ( 1 + 14 k 2 + k 4 ) z 5 5 ! ( 1 + 135 k 2 + 135 k 4 + k 6 ) z 7 7 ! + O ( z 9 ) ,
§22.10(ii) Maclaurin Series in k and k
22.10.6 dn ( z , k ) = 1 k 2 2 sin 2 z + O ( k 4 ) ,
14: 22 Jacobian Elliptic Functions
Chapter 22 Jacobian Elliptic Functions
15: 22.3 Graphics
§22.3(i) Real Variables: Line Graphs
§22.3(iii) Complex z ; Real k
§22.3(iv) Complex k
See accompanying text
Figure 22.3.24: sn ( x + i y , k ) for 4 x 4 , 0 y 8 , k = 1 + 1 2 i . … Magnify 3D Help
See accompanying text
Figure 22.3.25: sn ( 5 , k ) as a function of complex k 2 , 1 ( k 2 ) 3.5 , 1 ( k 2 ) 1 . … Magnify 3D Help
16: 22.19 Physical Applications
§22.19(ii) Classical Dynamics: The Quartic Oscillator
§22.19(iii) Nonlinear ODEs and PDEs
§22.19(iv) Tops
§22.19(v) Other Applications
17: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
§22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
22.12.2 2 K k sn ( 2 K t , k ) = n = π sin ( π ( t ( n + 1 2 ) τ ) ) = n = ( m = ( 1 ) m t m ( n + 1 2 ) τ ) ,
22.12.8 2 K dc ( 2 K t , k ) = n = π sin ( π ( t + 1 2 n τ ) ) = n = ( m = ( 1 ) m t + 1 2 m n τ ) ,
22.12.11 2 K ns ( 2 K t , k ) = n = π sin ( π ( t n τ ) ) = n = ( m = ( 1 ) m t m n τ ) ,
22.12.13 2 K cs ( 2 K t , k ) = lim N n = N N ( 1 ) n π tan ( π ( t n τ ) ) = lim N n = N N ( 1 ) n ( lim M m = M M 1 t m n τ ) .
18: Sidebar 22.SB1: Decay of a Soliton in a Bose–Einstein Condensate
Jacobian elliptic functions arise as solutions to certain nonlinear Schrödinger equations, which model many types of wave propagation phenomena. …
19: 22.11 Fourier and Hyperbolic Series
§22.11 Fourier and Hyperbolic Series
In (22.11.7)–(22.11.12) the left-hand sides are replaced by their limiting values at the poles of the Jacobian functions. …
22.11.13 sn 2 ( z , k ) = 1 k 2 ( 1 E K ) 2 π 2 k 2 K 2 n = 1 n q n 1 q 2 n cos ( 2 n ζ ) .
A related hyperbolic series is …
20: 22.16 Related Functions
See Figure 22.16.2. …
22.16.18 ( x , k ) = k 2 0 x cd 2 ( t , k ) d t + x + k 2 sn ( x , k ) cd ( x , k ) ,
22.16.19 ( x , k ) = k 2 k 2 0 x sd 2 ( t , k ) d t + k 2 x + k 2 sn ( x , k ) cd ( x , k ) ,
22.16.21 ( x , k ) = 0 x dc 2 ( t , k ) d t + x + sn ( x , k ) dc ( x , k ) ,
22.16.26 ( x , k ) = 0 x ( cs 2 ( t , k ) t 2 ) d t + x 1 cn ( x , k ) ds ( x , k ) .