# §19.17 Graphics

See Figures 19.17.119.17.8 for symmetric elliptic integrals with real arguments.

Because the -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 , , and , which are symmetric in , we may further assume that is the largest of if the variables are real, then choose , and consider only and . The cases or correspond to the complete integrals. The case corresponds to elementary functions.

To view and for complex , put , use (19.25.1), and see Figures 19.3.719.3.12.

 Figure 19.17.1: for , . corresponds to . Symbols: : Carlson’s combination of inverse circular and inverse hyperbolic functions and : symmetric elliptic integral of first kind Referenced by: §19.17 Permalink: http://dlmf.nist.gov/19.17.F1 Encodings: pdf, png Figure 19.17.2: for , . corresponds to . Symbols: : Carlson’s combination of inverse circular and inverse hyperbolic functions and : symmetric elliptic integral of second kind Permalink: http://dlmf.nist.gov/19.17.F2 Encodings: pdf, png

To view and for complex , put , use (19.25.1), and see Figures 19.3.719.3.12.

 Figure 19.17.3: for , . corresponds to , . Symbols: : Carlson’s combination of inverse circular and inverse hyperbolic functions and : elliptic integral symmetric in only two variables Permalink: http://dlmf.nist.gov/19.17.F3 Encodings: pdf, png Figure 19.17.4: for , . corresponds to . Symbols: : Carlson’s combination of inverse circular and inverse hyperbolic functions and : symmetric elliptic integral of third kind Permalink: http://dlmf.nist.gov/19.17.F4 Encodings: pdf, png
 Figure 19.17.5: for , . corresponds to . Symbols: : Carlson’s combination of inverse circular and inverse hyperbolic functions and : symmetric elliptic integral of third kind Permalink: http://dlmf.nist.gov/19.17.F5 Encodings: pdf, png Figure 19.17.6: Cauchy principal value of for , . corresponds to . Symbols: : Carlson’s combination of inverse circular and inverse hyperbolic functions and : symmetric elliptic integral of third kind Permalink: http://dlmf.nist.gov/19.17.F6 Encodings: pdf, png
 Figure 19.17.7: Cauchy principal value of for , . corresponds to . As the curve for has the finite limit ; see (19.20.10). Symbols: : Carlson’s combination of inverse circular and inverse hyperbolic functions and : symmetric elliptic integral of third kind Permalink: http://dlmf.nist.gov/19.17.F7 Encodings: pdf, png Figure 19.17.8: , , . Cauchy principal values are shown when . The function is asymptotic to as , and to as . As it has the limit . When , it reduces to . If , then it has the value when , and when . See (19.20.10), (19.20.11), and (19.20.8) for the cases , , and , respectively. Symbols: : elliptic integral symmetric in only two variables, : symmetric elliptic integral of second kind, : symmetric elliptic integral of third kind and : principal branch of logarithm function Referenced by: §19.17 Permalink: http://dlmf.nist.gov/19.17.F8 Encodings: VRML, X3D, pdf, png
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