# Elliptic function

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Related to Elliptic function: Theta function, Elliptic integrals

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Also found in: Encyclopedia, Wikipedia.

Related to Elliptic function: Theta function, Elliptic integrals

where E([THETA] | m) is the elliptic integral of the second kind, am(x | m) is the Jacobi elliptic function amplitude, and E(m) is the complete elliptic integral of the second kind.

in which [DELTA]([g.sub.2], [g.sub.3]) [not equal to] 0, [g.sub.2] = 60[h.sub.4], and [g.sub.3] = 140[h.sub.6], is called the Weierstrass elliptic function.

[5]) which is based on the elliptic relaxation concept, The variable [zeta] represents the ratio [[upsilon].sup.2] / k ([[upsilon].sup.2] is a scalar property in the Durbin's v2 - f model [6]), which reduces to the wall-normal stress in the near-wall region, thus providing more convenient formulation of the equation for [zeta] and especially of the wall boundary conditions for the elliptic function f).

It can be observed from (15)-(24) and Figures 1-6 that triangular periodic solution, bell-shaped solitary wave solution, kink-shaped solitary wave solution, and Jacobi elliptic function solution of the time fractional simplified MCH are obtained.

where sn is the Jacobi elliptic function of module s [19], [tau] is the normalized time, and and are constants specific for the release system.

where [g.sub.2]([tau]) = 60[G.sub.4]([tau]), [g.sub.3]([tau]) = 140[G.sub.6]([tau]) for the Eisenstein series [G.sub.2k]([tau]) of weight 2k, p(z, [tau]) := p(z, [L.sub.[tau]]) is the Weierstrass elliptic function, and b([tau]), c([tau]) are the coefficients of the Tate normal form contained in [C/[L.sub.[tau]], 1/N + [L.sub.[tau]]].

p(z; [g.sub.2], [g.sub.3]) is the Weierstrass elliptic function with invariants [g.sub.2] and [g.sub.3].

Although intensive investigations have made significant progress in recent years, many methods have been proposed to construct exact solutions, such as Weierstrass elliptic function method, the homogeneous balance method, sine-cosine method, the nonlinear transformation method, the hyperbolic tangent functions finite expansion, improved mapping approach, and further extended tanh method.

where [C.sub.0] is a constant, sn([xi]), cn([xi]), and dn([xi]) denote the Jacobi elliptic sine function, Jacobi elliptic cosine function, and the Jacobi elliptic function of the third kind, respectively, m is the modulus, and

The sum of all the residues of an elliptic function in the period parallelogram is zero.

On the other hand, the BPF with elliptic function response is also used widely because of advantages of sharp frequency characteristic and good out-of-band performance.

Liu: Jacobi elliptic function solutions of the Ablowitz-Ladik discrete nonlinear Schrodinger system, Chaos, Solitons and Fractals, 40(2009), 786-792.

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