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22 Jacobian Elliptic FunctionsProperties

§22.15 Inverse Functions

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§22.15(i) Definitions

The inverse Jacobian elliptic functions can be defined in an analogous manner to the inverse trigonometric functions (§4.23). With real variables, the solutions of the equations

22.15.1 sn(ξ,k)=x,
-1x1,
22.15.2 cn(η,k)=x,
-1x1,
22.15.3 dn(ζ,k)=x,
kx1,

are denoted respectively by

22.15.4 ξ =arcsn(x,k),
η =arccn(x,k),
ζ =arcdn(x,k).

Each of these inverse functions is multivalued. The principal values satisfy

22.15.5 -K arcsn(x,k)
K,
22.15.6 0 arccn(x,k)
2K,
22.15.7 0 arcdn(x,k)
K,

and unless stated otherwise it is assumed that the inverse functions assume their principal values. The general solutions of (22.15.1), (22.15.2), (22.15.3) are, respectively,

22.15.8 ξ =(-1)marcsn(x,k)+2mK,
22.15.9 η =±arccn(x,k)+4mK,
22.15.10 ζ =±arcdn(x,k)+2mK,

where m.

Equations (22.15.1) and (22.15.4), for arcsn(x,k), are equivalent to (22.15.12) and also to

22.15.11 x=0sn(x,k)dt(1-t2)(1-k2t2),
-1x1, 0k1.

Similarly with (22.15.13)–(22.15.23) and the other eleven Jacobian elliptic functions.

§22.15(ii) Representations as Elliptic Integrals

22.15.12 arcsn(x,k)=0xdt(1-t2)(1-k2t2),
-1x1,
22.15.13 arccn(x,k)=x1dt(1-t2)(k2+k2t2),
-1x1,
22.15.14 arcdn(x,k)=x1dt(1-t2)(t2-k2),
kx1 .
22.15.15 arccd(x,k)=x1dt(1-t2)(1-k2t2),
-1x1,
22.15.16 arcsd(x,k)=0xdt(1-k2t2)(1+k2t2),
-1/kx1/k,
22.15.17 arcnd(x,k)=1xdt(t2-1)(1-k2t2),
1x1/k,
22.15.18 arcdc(x,k)=1xdt(t2-1)(t2-k2),
1x<,
22.15.19 arcnc(x,k)=1xdt(t2-1)(k2+k2t2),
1x<,
22.15.20 arcsc(x,k)=0xdt(1+t2)(1+k2t2),
-<x<,
22.15.21 arcns(x,k)=xdt(t2-1)(t2-k2),
1x<,
22.15.22 arcds(x,k)=xdt(t2+k2)(t2-k2),
kx<,
22.15.23 arccs(x,k)=xdt(1+t2)(t2+k2),
-<x<.

The integrals (22.15.12)–(22.15.14) can be regarded as normal forms for representing the inverse functions. Other integrals, for example,

xbdt(a2+t2)(b2-t2)

can be transformed into normal form by elementary change of variables. Comprehensive treatments are given by Carlson (2005), Lawden (1989, pp. 52–55), Bowman (1953, Chapter IX), and Erdélyi et al. (1953b, pp. 296–301). See also Abramowitz and Stegun (1964, p. 596).

For representations of the inverse functions as symmetric elliptic integrals see §19.25(v). For power-series expansions see Carlson (2008).