Digital Library of Mathematical Functions
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19 Elliptic IntegralsLegendre’s Integrals

§19.3 Graphics

Contents

§19.3(i) Real Variables

See Figures 19.3.119.3.6 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 -2k21. 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 0x5. Both functions are asymptotic to ln(4x)/4x as x; see (19.2.19) and (19.2.20). Note that RC(x,±y)=y-1/2RC(x/y,±1), y>0. Magnify
Figure 19.3.3: F(ϕ,k) as a function of k2 and sin2ϕ for -1k22, 0sin2ϕ1. If sin2ϕ=1 (k2), then the function reduces to K(k), becoming infinite when k2=1. If sin2ϕ=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
Figure 19.3.4: E(ϕ,k) as a function of k2 and sin2ϕ for -1k22, 0sin2ϕ1. If sin2ϕ=1 (k2), then the function reduces to E(k), with value 1 at k2=1. If sin2ϕ=1/k2 (<1), then it has the value kE(1/k)+(k2/k)K(1/k), with limit 1 as k21+: put c=k2 in (19.25.7) and use (19.25.1). Magnify
Figure 19.3.5: Π(α2,k) as a function of k2 and α2 for -2k2<1, -2α22. Cauchy principal values are shown when α2>1. The function is unbounded as α21-, and also (with the same sign as 1-α2) as k21-. As α21+ it has the limit K(k)-(E(k)/k2). If α2=0, then it reduces to K(k). If k2=0, then it has the value 12π/1-α2 when α2<1, and 0 when α2>1. See §19.6(i). Magnify
Figure 19.3.6: Π(ϕ,2,k) as a function of k2 and sin2ϕ for -1k23, 0sin2ϕ<1. Cauchy principal values are shown when sin2ϕ>12. The function tends to + as sin2ϕ12, except in the last case below. If sin2ϕ=1 (>k2), then the function reduces to Π(2,k) with Cauchy principal value K(k)-Π(12k2,k), which tends to - as k21-. See (19.6.5) and (19.6.6). If sin2ϕ=1/k2 (<1), then by (19.7.4) it reduces to Π(2/k2,1/k)/k, k22, with Cauchy principal value (K(1/k)-Π(12,1/k))/k, 1<k2<2, by (19.6.5). Its value tends to - as k21+ by (19.6.6), and to the negative of the second lemniscate constant (see (19.20.22)) as k2(=csc2ϕ)2-. Magnify

§19.3(ii) Complex Variables

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.

Figure 19.3.7: K(k) as a function of complex k2 for -2(k2)2, -2(k2)2. There is a branch cut where 1<k2<. Magnify
Figure 19.3.8: E(k) as a function of complex k2 for -2(k2)2, -2(k2)2. There is a branch cut where 1<k2<. Magnify
Figure 19.3.9: (K(k)) as a function of complex k2 for -2(k2)2, -2(k2)2. The real part is symmetric under reflection in the real axis. On the branch cut (k21) it is infinite at k2=1, and has the value K(1/k)/k when k2>1. Magnify
Figure 19.3.10: (K(k)) as a function of complex k2 for -2(k2)2, -2(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 (k21) it has the value K(k) if k2>1, and 14π if k2=1. Magnify
Figure 19.3.11: (E(k)) as a function of complex k2 for -2(k2)2, -2(k2)2. The real part is symmetric under reflection in the real axis. On the branch cut (k2>1) it has the value kE(1/k)+(k2/k)K(1/k), with limit 1 as k21+. Magnify
Figure 19.3.12: (E(k)) as a function of complex k2 for -2(k2)2, -2(k2)2. The imaginary part is 0 for k21 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 k21+. Magnify