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19 Elliptic Integrals Legendre’s Integrals 19.3 Graphics

Figure 19.3.6 (See in context.)

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-.
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