§19.6 Special Cases ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals (original) (raw)

Contents
  1. §19.6(i) Complete Elliptic Integrals
  2. §19.6(ii) F⁡(ϕ,k)
  3. §19.6(iii) E⁡(ϕ,k)
  4. §19.6(iv) Π⁡(ϕ,α2,k)
  5. §19.6(v) RC⁡(x,y)

§19.6(i) Complete Elliptic Integrals

19.6.1 K⁡(0) =E⁡(0)=K′⁡(1)=E′⁡(1)=12⁢π,
K⁡(1) =K′⁡(0)=∞,
E⁡(1) =E′⁡(0)=1.
ⓘ 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 andE⁡(k): Legendre’s complete elliptic integral of the second kind Permalink: http://dlmf.nist.gov/19.6.E1 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §19.6(i),§19.6 andCh.19
19.6.2 Π⁡(k2,k) =E⁡(k)/k′2,
k2<1,
Π⁡(−k,k) =14⁢π⁢(1+k)−1+12⁢K⁡(k),
0≤k2<1.
ⓘ Symbols: π: the ratio of the circumference of a circle to its diameter,K⁡(k): Legendre’s complete elliptic integral of the first kind,E⁡(k): Legendre’s complete elliptic integral of the second kind,Π⁡(α2,k): Legendre’s complete elliptic integral of the third kind,k: real or complex modulus andk′: complementary modulus Referenced by: §19.6(i) Permalink: http://dlmf.nist.gov/19.6.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.6(i),§19.6 andCh.19
19.6.3 Π⁡(α2,0)=π/(2⁢1−α2),Π⁡(0,k)=K⁡(k),
−∞<α2<1.
ⓘ Symbols: π: the ratio of the circumference of a circle to its diameter,K⁡(k): Legendre’s complete elliptic integral of the first kind,Π⁡(α2,k): Legendre’s complete elliptic integral of the third kind,k: real or complex modulus andα2: real or complex parameter Permalink: http://dlmf.nist.gov/19.6.E3 Encodings: TeX, pMML, png See also: Annotations for §19.6(i),§19.6 andCh.19
19.6.4 Π⁡(α2,k) →+∞,
α2→1−,
Π⁡(α2,k) →∞⁢sign⁡(1−α2),
k2→1−.
ⓘ Symbols: Π⁡(α2,k): Legendre’s complete elliptic integral of the third kind,sign⁡x: sign of,k: real or complex modulus andα2: real or complex parameter Permalink: http://dlmf.nist.gov/19.6.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.6(i),§19.6 andCh.19

If 1<α2<∞, then the Cauchy principal value satisfies

19.6.5 Π⁡(α2,k)=K⁡(k)−Π⁡(k2/α2,k),

and

19.6.6 Π⁡(α2,0) =0,
Π⁡(α2,k) →K⁡(k)−(E⁡(k)/k′2),
α2→1+,
Π⁡(α2,k) →−∞,
k2→1−.
ⓘ Symbols: K⁡(k): Legendre’s complete elliptic integral of the first kind,E⁡(k): Legendre’s complete elliptic integral of the second kind,Π⁡(α2,k): Legendre’s complete elliptic integral of the third kind,k: real or complex modulus,k′: complementary modulus andα2: real or complex parameter Referenced by: Figure 19.3.6,Figure 19.3.6,Figure 19.3.6,§19.6(i) Permalink: http://dlmf.nist.gov/19.6.E6 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §19.6(i),§19.6 andCh.19

Exact values of K⁡(k) and E⁡(k) for various special values of k are given in Byrd and Friedman (1971, 111.10 and 111.11) andCooper et al. (2006).

§19.6(ii) F⁡(ϕ,k)

19.6.7 F⁡(0,k) =0,
F⁡(ϕ,0) =ϕ,
F⁡(12⁢π,1) =∞,
F⁡(12⁢π,k) =K⁡(k),
limϕ→0F⁡(ϕ,k)/ϕ =1.
ⓘ Symbols: π: the ratio of the circumference of a circle to its diameter,K⁡(k): Legendre’s complete elliptic integral of the first kind,F⁡(ϕ,k): Legendre’s incomplete elliptic integral of the first kind,ϕ: real or complex argument andk: real or complex modulus Permalink: http://dlmf.nist.gov/19.6.E7 Encodings: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png See also: Annotations for §19.6(ii),§19.6 andCh.19
19.6.8 F⁡(ϕ,1)=(sin⁡ϕ)⁢RC⁡(1,cos2⁡ϕ)=gd−1⁡(ϕ).

For the inverse Gudermannian function gd−1⁡(ϕ) see §4.23(viii). Compare also (19.10.2).

§19.6(iii) E⁡(ϕ,k)

19.6.9 E⁡(0,k) =0,
E⁡(ϕ,0) =ϕ,
E⁡(12⁢π,1) =1,
E⁡(ϕ,1) =sin⁡ϕ,
E⁡(12⁢π,k) =E⁡(k).
ⓘ Symbols: π: the ratio of the circumference of a circle to its diameter,E⁡(k): Legendre’s complete elliptic integral of the second kind,E⁡(ϕ,k): Legendre’s incomplete elliptic integral of the second kind,sin⁡z: sine function,ϕ: real or complex argument andk: real or complex modulus Permalink: http://dlmf.nist.gov/19.6.E9 Encodings: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png See also: Annotations for §19.6(iii),§19.6 andCh.19

§19.6(iv) Π⁡(ϕ,α2,k)

Circular and hyperbolic cases, including Cauchy principal values, are unified by using RC⁡(x,y). Let c=csc2⁡ϕ≠α2 andΔ=1−k2⁢sin2⁡ϕ. Then

19.6.11 Π⁡(0,α2,k) =0,
Π⁡(ϕ,0,0) =ϕ,
Π⁡(ϕ,1,0) =tan⁡ϕ.
ⓘ Symbols: Π⁡(ϕ,α2,k): Legendre’s incomplete elliptic integral of the third kind,tan⁡z: tangent function,ϕ: real or complex argument,k: real or complex modulus andα2: real or complex parameter Permalink: http://dlmf.nist.gov/19.6.E11 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §19.6(iv),§19.6 andCh.19
19.6.12 Π⁡(ϕ,α2,0) =RC⁡(c−1,c−α2),
Π⁡(ϕ,α2,1) =11−α2⁢(RC⁡(c,c−1)−α2⁢RC⁡(c,c−α2)),
Π⁡(ϕ,1,1) =12⁢(RC⁡(c,c−1)+c⁢(c−1)−1).
ⓘ Symbols: RC⁡(x,y): Carlson’s combination of inverse circular and inverse hyperbolic functions,Π⁡(ϕ,α2,k): Legendre’s incomplete elliptic integral of the third kind,ϕ: real or complex argument andα2: real or complex parameter Referenced by: §19.6(iv),§19.9(i),§19.9(ii) Permalink: http://dlmf.nist.gov/19.6.E12 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §19.6(iv),§19.6 andCh.19
19.6.13 Π⁡(ϕ,0,k) =F⁡(ϕ,k),
Π⁡(ϕ,k2,k) =1k′2⁢(E⁡(ϕ,k)−k2Δ⁢sin⁡ϕ⁢cos⁡ϕ),
Π⁡(ϕ,1,k) =F⁡(ϕ,k)−1k′2⁢(E⁡(ϕ,k)−Δ⁢tan⁡ϕ).
ⓘ Symbols: cos⁡z: cosine function,F⁡(ϕ,k): Legendre’s incomplete elliptic integral of the first kind,E⁡(ϕ,k): Legendre’s incomplete elliptic integral of the second kind,Π⁡(ϕ,α2,k): Legendre’s incomplete elliptic integral of the third kind,sin⁡z: sine function,tan⁡z: tangent function,ϕ: real or complex argument,k: real or complex modulus,k′: complementary modulus andΔ Referenced by: §19.36(ii),§19.6(iv) Permalink: http://dlmf.nist.gov/19.6.E13 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §19.6(iv),§19.6 andCh.19
19.6.14 Π⁡(12⁢π,α2,k) =Π⁡(α2,k),
limϕ→0Π⁡(ϕ,α2,k)/ϕ =1.
ⓘ Symbols: π: the ratio of the circumference of a circle to its diameter,Π⁡(α2,k): Legendre’s complete elliptic integral of the third kind,Π⁡(ϕ,α2,k): Legendre’s incomplete elliptic integral of the third kind,ϕ: real or complex argument,k: real or complex modulus andα2: real or complex parameter Permalink: http://dlmf.nist.gov/19.6.E14 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.6(iv),§19.6 andCh.19

For the Cauchy principal value of Π⁡(ϕ,α2,k) whenα2>c, see §19.7(iii).

§19.6(v) RC⁡(x,y)

19.6.15 RC⁡(x,x) =x−1/2,
RC⁡(λ⁢x,λ⁢y) =λ−1/2⁢RC⁡(x,y),
RC⁡(x,y) →+∞,
y→0+ or y→0−, x>0,
RC⁡(0,y) =12⁢π⁢y−1/2,
|ph⁡y <π,
RC⁡(0,y) =0,
y<0.
ⓘ Symbols: RC⁡(x,y): Carlson’s combination of inverse circular and inverse hyperbolic functions,π: the ratio of the circumference of a circle to its diameter andph: phase Referenced by: §19.2(iv),§19.2(iv),§19.20(iii),§19.6(i),§19.9(i) Permalink: http://dlmf.nist.gov/19.6.E15 Encodings: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png See also: Annotations for §19.6(v),§19.6 andCh.19