19 Elliptic Integrals Symmetric Integrals 19.16 Definitions 19.18 Derivatives and Differential Equations

§19.17 Graphics

Notes:: These graphics were produced at NIST.
Keywords:: symmetric elliptic integrals
Permalink:: http://dlmf.nist.gov/19.17

See Figures 19.17.1–19.17.8 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 $\mathop{R_{F}\/}\nolimits$ , $\mathop{R_{G}\/}\nolimits$ , and $\mathop{R_{J}\/}\nolimits$ , which are symmetric in , we may further assume that is the largest of if the variables are real, then choose , and consider only $0\leq x\leq 1$ and $0\leq y\leq 1$ . The cases or correspond to the complete integrals. The case corresponds to elementary functions.

To view $\mathop{R_{F}\/}\nolimits\!\left(0,y,1\right)$ and $2\!\mathop{R_{G}\/}\nolimits\!\left(0,y,1\right)$ for complex , put $y=1-k^{2}$ , use (19.25.1), and see Figures 19.3.7–19.3.12.

Figure 19.17.1: $\mathop{R_{F}\/}\nolimits\!\left(x,y,1\right)$ for $0\leq x\leq 1$ , $y=0,\,0.1,\,0.5,\,1$ .

corresponds to $\mathop{R_{C}\/}\nolimits\!\left(x,1\right)$ .

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions and $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : 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: $\mathop{R_{G}\/}\nolimits\!\left(x,y,1\right)$ for $0\leq x\leq 1$ , $y=0,\,0.1,\,0.5,\,1$ .

corresponds to $\frac{1}{2}(\mathop{R_{C}\/}\nolimits\!\left(x,1\right)+\sqrt{x})$ .

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions and $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind
Permalink:: http://dlmf.nist.gov/19.17.F2
Encodings:: pdf, png

To view $\mathop{R_{F}\/}\nolimits\!\left(0,y,1\right)$ and $2\!\mathop{R_{G}\/}\nolimits\!\left(0,y,1\right)$ for complex , put $y=1-k^{2}$ , use (19.25.1), and see Figures 19.3.7–19.3.12.

Figure 19.17.3: $\mathop{R_{D}\/}\nolimits\!\left(x,y,1\right)$ for $0\leq x\leq 2$ , $y=0,\,0.1,\,1,\,5,\,25$ .

corresponds to $\frac{3}{2}(\mathop{R_{C}\/}\nolimits\!\left(x,1\right)-\sqrt{x})/(1-x)$ , $x\neq 1$ .

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions and $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables
Permalink:: http://dlmf.nist.gov/19.17.F3
Encodings:: pdf, png

Figure 19.17.4: $\mathop{R_{J}\/}\nolimits\!\left(x,y,1,2\right)$ for $0\leq x\leq 1$ , $y=0,\,0.1,\,0.5,\,1$ .

corresponds to $3(\mathop{R_{C}\/}\nolimits\!\left(x,1\right)-\mathop{R_{C}\/}\nolimits\!\left% (x,2\right))$ .

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions and $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind
Permalink:: http://dlmf.nist.gov/19.17.F4
Encodings:: pdf, png

Figure 19.17.5: $\mathop{R_{J}\/}\nolimits\!\left(x,y,1,0.5\right)$ for $0\leq x\leq 1$ , $y=0,\,0.1,\,0.5,\,1$ .

corresponds to $6(\mathop{R_{C}\/}\nolimits\!\left(x,0.5\right)-\mathop{R_{C}\/}\nolimits\!% \left(x,1\right))$ .

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions and $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : 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 $\mathop{R_{J}\/}\nolimits\!\left(x,y,1,-0.5\right)$ for $0\leq x\leq 1$ , $y=0,\,0.1,\,0.5,\,1$ .

corresponds to $2(\mathop{R_{C}\/}\nolimits\!\left(x,-0.5\right)-\mathop{R_{C}\/}\nolimits\!% \left(x,1\right))$ .

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions and $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : 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 $\mathop{R_{J}\/}\nolimits\!\left(0.5,y,1,p\right)$ for $y=0,\,0.01,\,0.05,\,0.2,\,1$ , $-1\leq p<0$ .

corresponds to $3(\mathop{R_{C}\/}\nolimits\!\left(0.5,p\right)-(\pi/\sqrt{8}))/(1-p)$ . As $p\to 0$ the curve for

has the finite limit $-8.10386\dots$ ; see (19.20.10).

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions and $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind
Permalink:: http://dlmf.nist.gov/19.17.F7
Encodings:: pdf, png

Visualization Help

Figure 19.17.8: $\mathop{R_{J}\/}\nolimits\!\left(0,y,1,p\right)$ , $0\leq y\leq 1$ , $-1\leq p\leq 2$ . Cauchy principal values are shown when

. The function is asymptotic to $\frac{3}{2}\pi/\sqrt{yp}$ as $p\to 0+$ , and to $(\frac{3}{2}/p)\mathop{\ln\/}\nolimits\!\left(16/y\right)$ as $y\to 0+$ . As $p\to 0-$ it has the limit $(-6/y)\mathop{R_{G}\/}\nolimits\!\left(0,y,1\right)$ . When