# §19.3 Graphics

## §19.3(i) Real Variables

See Figures 19.3.119.3.6 for complete and incomplete Legendre’s elliptic integrals. 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 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 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 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 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 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

## §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.