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Jacobian elliptic-function form

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21: 22 Jacobian Elliptic Functions
Chapter 22 Jacobian Elliptic Functions
22: 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
23: 23.21 Physical Applications
In §22.19(ii) it is noted that Jacobian elliptic functions provide a natural basis of solutions for problems in Newtonian classical dynamics with quartic potentials in canonical form ( 1 x 2 ) ( 1 k 2 x 2 ) . The Weierstrass function plays a similar role for cubic potentials in canonical form g 3 + g 2 x 4 x 3 . … Another form is obtained by identifying e 1 , e 2 , e 3 as lattice roots (§23.3(i)), and setting …
24: 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
25: 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 ) .
26: 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. …
27: 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 …
28: Bibliography C
  • B. C. Carlson (2004) Symmetry in c, d, n of Jacobian elliptic functions. J. Math. Anal. Appl. 299 (1), pp. 242–253.
  • B. C. Carlson (2005) Jacobian elliptic functions as inverses of an integral. J. Comput. Appl. Math. 174 (2), pp. 355–359.
  • B. C. Carlson (2006a) Some reformulated properties of Jacobian elliptic functions. J. Math. Anal. Appl. 323 (1), pp. 522–529.
  • B. C. Carlson (2006b) Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric R -functions. Math. Comp. 75 (255), pp. 1309–1318.
  • B. C. Carlson (2008) Power series for inverse Jacobian elliptic functions. Math. Comp. 77 (263), pp. 1615–1621.
  • 29: 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 ) .
    30: 1.5 Calculus of Two or More Variables
    If D can be represented in both forms (1.5.30) and (1.5.33), and f ( x , y ) is continuous on D , then … In case of triple integrals the ( x , y , z ) sets are of the form
    §1.5(vi) Jacobians and Change of Variables
    Jacobian
    1.5.38 ( f , g ) ( x , y ) = | f / x f / y g / x g / y | ,