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22 Jacobian Elliptic FunctionsProperties

§22.16 Related Functions

  1. §22.16(i) Jacobi’s Amplitude (am) Function
  2. §22.16(ii) Jacobi’s Epsilon Function
  3. §22.16(iii) Jacobi’s Zeta Function
  4. §22.16(iv) Graphs

§22.16(i) Jacobi’s Amplitude (am) Function


22.16.1 am(x,k)=Arcsin(sn(x,k)),

where the inverse sine has its principal value when KxK and is defined by continuity elsewhere. See Figure 22.16.1. am(x,k) is an infinitely differentiable function of x.


Integral Representation

Special Values

22.16.4 am(x,0) =x,
22.16.5 am(x,1) =gd(x).

For the Gudermannian function gd(x) see §4.23(viii).

Approximation for Small x

22.16.6 am(x,k)=xk2x33!+k2(4+k2)x55!+O(x7).

Approximations for Small k, k

22.16.7 am(x,k)=x14k2(xsinxcosx)+O(k4),
22.16.8 am(x,k)=gdx14k2(xsinhxcoshx)sechx+O(k4).

Fourier Series

With q as in (22.2.1) and ζ=πx/(2K),

22.16.9 am(x,k)=π2Kx+2n=1qnsin(2nζ)n(1+q2n).

Relation to Elliptic Integrals

If KxK, then the following four equations are equivalent:

22.16.10 x=F(ϕ,k),
22.16.11 am(x,k)=ϕ,
22.16.12 sn(x,k)=sinϕ=sin(am(x,k)),
22.16.13 cn(x,k)=cosϕ=cos(am(x,k)).

For F(ϕ,k) see §19.2(ii).

§22.16(ii) Jacobi’s Epsilon Function

Integral Representations

For K<x<K,

22.16.14 (x,k)=0sn(x,k)1k2t21t2dt;

compare (19.2.5). See Figure 22.16.2.

22.16.15 (x,k) =k20xsn2(t,k)dt+x,
22.16.16 (x,k) =k20xcn2(t,k)dt+k2x,
22.16.17 (x,k) =0xdn2(t,k)dt.
22.16.18 (x,k) =k20xcd2(t,k)dt+x+k2sn(x,k)cd(x,k),
22.16.19 (x,k) =k2k20xsd2(t,k)dt+k2x+k2sn(x,k)cd(x,k),
22.16.20 (x,k) =k20xnd2(t,k)dt+k2sn(x,k)cd(x,k).

In Equations (22.16.21)–(22.16.23), K<x<K.

22.16.21 (x,k) =0xdc2(t,k)dt+x+sn(x,k)dc(x,k),
22.16.22 (x,k) =k20xnc2(t,k)dt+k2x+sn(x,k)dc(x,k),
22.16.23 (x,k) =k20xsc2(t,k)dt+sn(x,k)dc(x,k).

In Equations (22.16.24)–(22.16.26), 2K<x<2K.

22.16.24 (x,k) =0x(ns2(t,k)t2)dt+x1+xcn(x,k)ds(x,k),
22.16.25 (x,k) =0x(ds2(t,k)t2)dt+x1+k2xcn(x,k)ds(x,k),
22.16.26 (x,k) =0x(cs2(t,k)t2)dt+x1cn(x,k)ds(x,k).

Quasi-Addition and Quasi-Periodic Formulas

22.16.27 (x1+x2,k)=(x1,k)+(x2,k)k2sn(x1,k)sn(x2,k)sn(x1+x2,k),
22.16.28 (x+K,k)=(x,k)+E(k)k2sn(x,k)cd(x,k),
22.16.29 (x+2K,k)=(x,k)+2E(k).

For E(k) see §19.2(ii).

Relation to Theta Functions

22.16.30 (x,k)=1θ32(0,q)θ4(ξ,q)ddξθ4(ξ,q)+E(k)K(k)x,

where ξ=x/θ32(0,q). For θj see §20.2(i). For E(k) see §19.2(ii).

Relation to the Elliptic Integral E(ϕ,k)

§22.16(iii) Jacobi’s Zeta Function


With E(k) and K(k) as in §19.2(ii) and x,

22.16.32 Z(x|k)=(x,k)(E(k)/K(k))x.

See Figure 22.16.3. (Sometimes in the literature Z(x|k) is denoted by Z(am(x,k),k2).)


Z(x|k) satisfies the same quasi-addition formula as the function (x,k), given by (22.16.27). Also,

22.16.33 Z(x+K|k)=Z(x|k)k2sn(x,k)cd(x,k),
22.16.34 Z(x+2K|k)=Z(x|k).

§22.16(iv) Graphs

See accompanying text
Figure 22.16.1: Jacobi’s amplitude function am(x,k) for 0x10π and k=0.4,0.7,0.99,0.999999. Values of k greater than 1 are illustrated in Figure 22.19.1. Magnify
See accompanying text
Figure 22.16.2: Jacobi’s epsilon function (x,k) for 0x10π and k=0.4,0.7,0.99,0.999999. (These graphs are similar to those in Figure 22.16.1; compare (22.16.3), (22.16.17), and the graphs of dn(x,k) in §22.3(i).) Magnify
See accompanying text
Figure 22.16.3: Jacobi’s zeta function Z(x|k) for 0x10π and k=0.4,0.7,0.99,0.999999. Magnify