§19.2 Definitions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals (original) (raw)

Contents

1. §19.2(i) General Elliptic Integrals
2. §19.2(ii) Legendre’s Integrals
3. §19.2(iii) Bulirsch’s Integrals
4. §19.2(iv) A Related Function: RC⁡(x,y)

§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 tcontaining at least one odd power of s. Then

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

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(ii) Legendre’s Integrals

Assume 1−sin2⁡ϕ∈ℂ∖(−∞,0] and1−k2⁢sin2⁡ϕ∈ℂ∖(−∞,0], except that one of them may be 0, and 1−α2⁢sin2⁡ϕ∈ℂ∖{0}. Then

The paths of integration are the line segments connecting the limits of integration. The integral for E⁡(ϕ,k) is well defined ifk2=sin2⁡ϕ=1, and the Cauchy principal value (§1.4(v)) of Π⁡(ϕ,α2,k) is taken if1−α2⁢sin2⁡ϕ vanishes at an interior point of the integration path. Also, if k2 and α2 are real, thenΠ⁡(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:

The principal branch of K⁡(k) and E⁡(k) is|ph⁡(1−k2)|≤π, that is, the branch-cuts are (−∞,−1]∪[1,+∞). The principal values of K⁡(k) and E⁡(k) are even functions.

Legendre’s complementary complete elliptic integrals are defined via

with a branch point at k=0 and principal branch|ph⁡k|≤π. Let k′=1−k2. Then

| 19.2.9 | K′⁡(k) | ={K⁡(k′),|ph⁡k|≤12⁢π,K⁡(k′)∓2⁢i⁢K⁡(−k),12⁢π<±ph⁡k<π, | | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------- | ------------------------------------------------------ | | E′⁡(k) | ={E⁡(k′),|ph⁡k|≤12⁢π,E⁡(k′)∓2⁢i⁢(K⁡(−k)−E⁡(−k)),12⁢π<±ph⁡k<π. | | | ⓘ Symbols: π: the ratio of the circumference of a circle to its diameter,K′⁡(k): Legendre’s complementary complete elliptic integral of the first kind,E′⁡(k): Legendre’s complementary complete elliptic integral of the second kind,K⁡(k): Legendre’s complete elliptic integral of the first kind,E⁡(k): Legendre’s complete elliptic integral of the second kind,i: imaginary unit,ph: phase,k: real or complex modulus andk′: complementary modulus Referenced by: §19.2(ii),§22.11,Erratum (V1.1.10) for Subsection 19.2(ii) and Equation (19.2.9),Erratum (V1.1.10) for Subsection 19.2(ii) and Equation (19.2.9) Permalink: http://dlmf.nist.gov/19.2.E9 Encodings: TeX, TeX, pMML, pMML, png, png Correction (effective with 1.1.10): This equation has been updated so that it has correct analytic continuation. See also: Annotations for §19.2(ii),§19.2 andCh.19 | | |

For more details on the analytical continuation of these complete elliptic integrals see Lawden (1989, §§8.12–8.14).

If m is an integer, then

§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

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

With

special cases include

and

The integrals are complete if x=∞. If 1<k≤1/sin⁡ϕ, then kc is pure imaginary.

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

§19.2(iv) A Related Function: RC⁡(x,y)

Let x∈ℂ∖(−∞,0) and y∈ℂ∖{0}. We define

where the Cauchy principal value is taken if y<0. Formulas involvingΠ⁡(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) and4.37(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:

The Cauchy principal value is hyperbolic:

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 inℂ∖(−∞,0], then