Digital Library of Mathematical Functions
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19 Elliptic IntegralsSymmetric Integrals

§19.17 Graphics

See Figures 19.17.119.17.8 for symmetric elliptic integrals with real arguments.

Because the R-function is homogeneous, there is no loss of generality in giving one variable the value 1 or -1 (as in Figure 19.3.2). For RF, RG, and RJ, which are symmetric in x,y,z, we may further assume that z is the largest of x,y,z if the variables are real, then choose z=1, and consider only 0x1 and 0y1. The cases x=0 or y=0 correspond to the complete integrals. The case y=1 corresponds to elementary functions.

To view RF(0,y,1) and 2RG(0,y,1) for complex y, put y=1-k2, use (19.25.1), and see Figures 19.3.719.3.12.

See accompanying text
Figure 19.17.1: RF(x,y,1) for 0x1, y=0, 0.1, 0.5, 1. y=1 corresponds to RC(x,1). Magnify
See accompanying text
Figure 19.17.2: RG(x,y,1) for 0x1, y=0, 0.1, 0.5, 1. y=1 corresponds to 12(RC(x,1)+x). Magnify

To view RF(0,y,1) and 2RG(0,y,1) for complex y, put y=1-k2, use (19.25.1), and see Figures 19.3.719.3.12.

See accompanying text
Figure 19.17.3: RD(x,y,1) for 0x2, y=0, 0.1, 1, 5, 25. y=1 corresponds to 32(RC(x,1)-x)/(1-x), x1. Magnify
See accompanying text
Figure 19.17.4: RJ(x,y,1,2) for 0x1, y=0, 0.1, 0.5, 1. y=1 corresponds to 3(RC(x,1)-RC(x,2)). Magnify
See accompanying text
Figure 19.17.5: RJ(x,y,1,0.5) for 0x1, y=0, 0.1, 0.5, 1. y=1 corresponds to 6(RC(x,0.5)-RC(x,1)). Magnify
See accompanying text
Figure 19.17.6: Cauchy principal value of RJ(x,y,1,-0.5) for 0x1, y=0, 0.1, 0.5, 1. y=1 corresponds to 2(RC(x,-0.5)-RC(x,1)). Magnify
See accompanying text
Figure 19.17.7: Cauchy principal value of RJ(0.5,y,1,p) for y=0, 0.01, 0.05, 0.2, 1, -1p<0. y=1 corresponds to 3(RC(0.5,p)-(π/8))/(1-p). As p0 the curve for y=0 has the finite limit -8.10386; see (19.20.10). Magnify
Figure 19.17.8: RJ(0,y,1,p), 0y1, -1p2. Cauchy principal values are shown when p<0. The function is asymptotic to 32π/yp as p0+, and to (32/p)ln(16/y) as y0+. As p0- it has the limit (-6/y)RG(0,y,1). When p=1, it reduces to RD(0,y,1). If y=1, then it has the value 32π/(p+p) when p>0, and 32π/(p-1) when p<0. See (19.20.10), (19.20.11), and (19.20.8) for the cases p0±, y0+, and y=1, respectively. Magnify