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22 Jacobian Elliptic FunctionsProperties

§22.16 Related Functions


§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

22.16.3 am(x,k)=0xdn(t,k)dt.

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)=x-k2x33!+k2(4+k2)x55!+O(x7).

Approximations for Small k, k

22.16.7 am(x,k)=x-14k2(x-sinxcosx)+O(k4),
22.16.8 am(x,k)=gdx-14k2(x-sinhxcoshx)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)1-k2t21-t2dt;

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)-t-2)dt+x-1+x-cn(x,k)ds(x,k),
22.16.25 (x,k) =-0x(ds2(t,k)-t-2)dt+x-1+k2x-cn(x,k)ds(x,k),
22.16.26 (x,k) =-0x(cs2(t,k)-t-2)dt+x-1-cn(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