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DLMF: §19.3 Graphics ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals
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19 Elliptic IntegralsLegendre’s Integrals

§19.3 Graphics

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Keywords:
Legendre’s elliptic integrals, graphics
Permalink:
http://dlmf.nist.gov/19.3
See also:
Annotations for Ch.19
Contents
  1. §19.3(i) Real Variables
  2. §19.3(ii) Complex Variables

§19.3(i) Real Variables

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Notes:
These graphics were produced at NIST.
Permalink:
http://dlmf.nist.gov/19.3.i
See also:
Annotations for §19.3 and Ch.19

See Figures 19.3.1–19.3.6 for complete and incomplete Legendre’s elliptic integrals.

See accompanying text
Figure 19.3.1: K⁡(k) and E⁡(k) as functions of k2 for −2≤k2≤1. Graphs of K′⁡(k) and E′⁡(k) are the mirror images in the vertical line k2=12. Magnify
See accompanying text
Figure 19.3.2: RC⁡(x,1) and the Cauchy principal value of RC⁡(x,−1) for 0≤x≤5. Both functions are asymptotic to ln⁡(4⁢x)/4⁢x as x→∞; see (19.2.19) and (19.2.20). Note that RC⁡(x,±y)=y−1/2⁢RC⁡(x/y,±1), y>0. Magnify
See accompanying text
Figure 19.3.3: F⁡(ϕ,k) as a function of k2 and sin2⁡ϕ for −1≤k2≤2, 0≤sin2⁡ϕ≤1. If sin2⁡ϕ=1 (≥k2), then the function reduces to K⁡(k), becoming infinite when k2=1. If sin2⁡ϕ=1/k2 (<1), then it has the value K⁡(1/k)/k: put c=k2 in (19.25.5) and use (19.25.1). Magnify 3D Help
See accompanying text
Figure 19.3.4: E⁡(ϕ,k) as a function of k2 and sin2⁡ϕ for −1≤k2≤2, 0≤sin2⁡ϕ≤1. If sin2⁡ϕ=1 (≥k2), then the function reduces to E⁡(k), with value 1 at k2=1. If sin2⁡ϕ=1/k2 (<1), then it has the value k⁢E⁡(1/k)+(k′2/k)⁢K⁡(1/k), with limit 1 as k2→1+: put c=k2 in (19.25.7) and use (19.25.1). Magnify 3D Help
See accompanying text
Figure 19.3.5: Π⁡(α2,k) as a function of k2 and α2 for −2≤k2<1, −2≤α2≤2. Cauchy principal values are shown when α2>1. The function is unbounded as α2→1−, and also (with the same sign as 1−α2) as k2→1−. As α2→1+ it has the limit K⁡(k)−(E⁡(k)/k′2). If α2=0, then it reduces to K⁡(k). If k2=0, then it has the value 12⁢π/1−α2 when α2<1, and 0 when α2>1. See §19.6(i). Magnify 3D Help
See accompanying text
Figure 19.3.6: Π⁡(ϕ,2,k) as a function of k2 and sin2⁡ϕ for −1≤k2≤3, 0≤sin2⁡ϕ<1. Cauchy principal values are shown when sin2⁡ϕ>12. The function tends to +∞ as sin2⁡ϕ→12, except in the last case below. If sin2⁡ϕ=1 (>k2), then the function reduces to Π⁡(2,k) with Cauchy principal value K⁡(k)−Π⁡(12⁢k2,k), which tends to −∞ as k2→1−. See (19.6.5) and (19.6.6). If sin2⁡ϕ=1/k2 (<1), then by (19.7.4) it reduces to Π⁡(2/k2,1/k)/k, k2≠2, with Cauchy principal value (K⁡(1/k)−Π⁡(12,1/k))/k, 1<k2<2, by (19.6.5). Its value tends to −∞ as k2→1+ by (19.6.6), and to the negative of the second lemniscate constant (see (19.20.22)) as k2(=csc2⁡ϕ)→2−. Magnify 3D Help

§19.3(ii) Complex Variables

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Keywords:
Legendre’s elliptic integrals, graphics
Notes:
These graphics were produced at NIST.
Permalink:
http://dlmf.nist.gov/19.3.ii
See also:
Annotations for §19.3 and Ch.19

In Figures 19.3.7 and 19.3.8 for complete Legendre’s elliptic integrals with complex arguments, height corresponds to the absolute value of the function and color to the phase. See also About Color Map.

See accompanying text
Figure 19.3.7: K⁡(k) as a function of complex k2 for −2≤ℜ⁡(k2)≤2, −2≤ℑ⁡(k2)≤2. There is a branch cut where 1<k2<∞. Magnify 3D Help
See accompanying text
Figure 19.3.8: E⁡(k) as a function of complex k2 for −2≤ℜ⁡(k2)≤2, −2≤ℑ⁡(k2)≤2. There is a branch cut where 1<k2<∞. Magnify 3D Help
See accompanying text
Figure 19.3.9: ℜ⁡(K⁡(k)) as a function of complex k2 for −2≤ℜ⁡(k2)≤2, −2≤ℑ⁡(k2)≤2. The real part is symmetric under reflection in the real axis. On the branch cut (k2≥1) it is infinite at k2=1, and has the value K⁡(1/k)/k when k2>1. Magnify 3D Help
See accompanying text
Figure 19.3.10: ℑ⁡(K⁡(k)) as a function of complex k2 for −2≤ℜ⁡(k2)≤2, −2≤ℑ⁡(k2)≤2. The imaginary part is 0 for k2<1, and is antisymmetric under reflection in the real axis. On the upper edge of the branch cut (k2≥1) it has the value K⁡(k′) if k2>1, and 14⁢π if k2=1. Magnify 3D Help
See accompanying text
Figure 19.3.11: ℜ⁡(E⁡(k)) as a function of complex k2 for −2≤ℜ⁡(k2)≤2, −2≤ℑ⁡(k2)≤2. The real part is symmetric under reflection in the real axis. On the branch cut (k2>1) it has the value k⁢E⁡(1/k)+(k′2/k)⁢K⁡(1/k), with limit 1 as k2→1+. Magnify 3D Help
See accompanying text
Figure 19.3.12: ℑ⁡(E⁡(k)) as a function of complex k2 for −2≤ℜ⁡(k2)≤2, −2≤ℑ⁡(k2)≤2. The imaginary part is 0 for k2≤1 and is antisymmetric under reflection in the real axis. On the upper edge of the branch cut (k2>1) it has the (negative) value K⁡(k′)−E⁡(k′), with limit 0 as k2→1+. Magnify 3D Help
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