What's New
About the Project
NIST
19 Elliptic IntegralsLegendre’s Integrals19.3 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-.
Viewpoint
Scale Figure
Cutting Control

Your browser does not support WebGL. Please choose an alternative format for 3D interactive visualization.

Format

Please see Vizualization Help for more details, and Customize to change your choice or for other customizations.