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19 Elliptic IntegralsLegendre’s Integrals

§19.2 Definitions


§19.2(i) General Elliptic Integrals

Let s2(t) be a cubic or quartic polynomial in t with simple zeros, and let r(s,t) be a rational function of s and t containing at least one odd power of s. Then

19.2.1 r(s,t)dt

is called an elliptic integral. Because s2 is a polynomial, we have

19.2.2 r(s,t)=(p1+p2s)(p3-p4s)s(p3+p4s)(p3-p4s)s=ρs+σ,

where pj is a polynomial in t while ρ and σ are rational functions of t. Thus the elliptic part of (19.2.1) is

19.2.3 ρ(t)s(t)dt.

§19.2(ii) Legendre’s Integrals

Assume 1-sin2ϕ(-,0] and 1-k2sin2ϕ(-,0], except that one of them may be 0, and 1-α2sin2ϕ{0}. Then

19.2.4 F(ϕ,k) =0ϕdθ1-k2sin2θ=0sinϕdt1-t21-k2t2,
19.2.5 E(ϕ,k) =0ϕ1-k2sin2θdθ=0sinϕ1-k2t21-t2dt.
19.2.6 D(ϕ,k) =0ϕsin2θdθ1-k2sin2θ=0sinϕt2dt1-t21-k2t2=(F(ϕ,k)-E(ϕ,k))/k2.
19.2.7 Π(ϕ,α2,k)=0ϕdθ1-k2sin2θ(1-α2sin2θ)=0sinϕdt1-t21-k2t2(1-α2t2).

The paths of integration are the line segments connecting the limits of integration. The integral for E(ϕ,k) is well defined if k2=sin2ϕ=1, and the Cauchy principal value (§1.4(v)) of Π(ϕ,α2,k) is taken if 1-α2sin2ϕ vanishes at an interior point of the integration path. Also, if k2 and α2 are real, then Π(ϕ,α2,k) is called a circular or hyperbolic case according as α2(α2-k2)(α2-1) is negative or positive. The circular and hyperbolic cases alternate in the four intervals of the real line separated by the points α2=0,k2,1.

The cases with ϕ=π/2 are the complete integrals:

19.2.8 K(k) =F(π/2,k),
E(k) =E(π/2,k),
D(k) =D(π/2,k)=(K(k)-E(k))/k2,
Π(α2,k) =Π(π/2,α2,k),
19.2.9 K(k) =K(k),
E(k) =E(k),
k =1-k2.

§19.2(iii) Bulirsch’s Integrals

Bulirsch’s integrals are linear combinations of Legendre’s integrals that are chosen to facilitate computational application of Bartky’s transformation (Bartky (1938)). Three are defined by

19.2.11 cel(kc,p,a,b)=0π/2acos2θ+bsin2θcos2θ+psin2θdθcos2θ+kc2sin2θ,
19.2.11_5 el1(x,kc)=0arctanx1cos2θ+kc2sin2θdθ,
19.2.12 el2(x,kc,a,b)=0arctanxa+btan2θ(1+tan2θ)(1+kc2tan2θ)dθ.

Here a,b,p are real parameters, and kc and x are real or complex variables, with p0, kc0. If -<p<0, then the integral in (19.2.11) is a Cauchy principal value.


19.2.13 kc =k,
p =1-α2,
x =tanϕ,

special cases include

19.2.14 K(k) =cel(kc,1,1,1),
E(k) =cel(kc,1,1,kc2),
D(k) =cel(kc,1,0,1),
(E(k)-k2K(k))/k2 =cel(kc,1,1,0),
Π(α2,k) =cel(kc,p,1,1),


19.2.15 F(ϕ,k) =el1(x,kc)=el2(x,kc,1,1),
E(ϕ,k) =el2(x,kc,1,kc2),
D(ϕ,k) =el2(x,kc,0,1).

The integrals are complete if x=. If 1<k1/sinϕ, then kc is pure imaginary.

Lastly, corresponding to Legendre’s incomplete integral of the third kind we have

19.2.16 el3(x,kc,p)=0arctanxdθ(cos2θ+psin2θ)cos2θ+kc2sin2θ=Π(arctanx,1-p,k),

§19.2(iv) A Related Function: RC(x,y)

Let x(-,0) and y{0}. We define

19.2.17 RC(x,y)=120dtt+x(t+y),

where the Cauchy principal value is taken if y<0. Formulas involving Π(ϕ,α2,k) that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using RC(x,y).

In (19.2.18)–(19.2.22) the inverse trigonometric and hyperbolic functions assume their principal values (§§4.23(ii) and 4.37(ii)). When x and y are positive, RC(x,y) is an inverse circular function if x<y and an inverse hyperbolic function (or logarithm) if x>y:

19.2.18 RC(x,y)=1y-xarctany-xx=1y-xarccosx/y,
19.2.19 RC(x,y)=1x-yarctanhx-yx=1x-ylnx+x-yy,

The Cauchy principal value is hyperbolic:

19.2.20 RC(x,y)=xx-yRC(x-y,-y)=1x-yarctanhxx-y=1x-ylnx+x-y-y,

For the special cases of RC(x,x) and RC(0,y) see (19.6.15).

If the line segment with endpoints x and y lies in (-,0], then

19.2.21 RC(x,y)=01(v2x+(1-v2)y)-1/2dv,
19.2.22 RC(x,y)=2π0π/2RC(y,xcos2θ+ysin2θ)dθ.