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19 Elliptic IntegralsLegendre’s Integrals

§19.3 Graphics

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Keywords:
Legendre’s elliptic integrals, graphics
Permalink:
http://dlmf.nist.gov/19.3
See also:
Annotations for Ch.19
Contents
  1. §19.3(i) Real Variables
  2. §19.3(ii) Complex Variables

§19.3(i) Real Variables

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Notes:
These graphics were produced at NIST.
Permalink:
http://dlmf.nist.gov/19.3.i
See also:
Annotations for §19.3 and Ch.19

See Figures 19.3.1–19.3.6 for complete and incomplete Legendre’s elliptic integrals.

See accompanying text
Figure 19.3.2: R_{C}\left(x,1\right) and the Cauchy principal value of R_{C}\left(x,-1\right) for 0\leq x\leq 5. Both functions are asymptotic to \ln\left(4x\right)/\sqrt{4x} as x\to\infty; see (19.2.19) and (19.2.20). Note that R_{C}\left(x,\pm y\right)=y^{-1/2}R_{C}\left(x/y,\pm 1\right), y>0. Magnify
See accompanying text
Figure 19.3.3: F\left(\phi,k\right) as a function of k^{2} and {\sin}^{2}\phi for -1\leq k^{2}\leq 2, 0\leq{\sin}^{2}\phi\leq 1. If {\sin}^{2}\phi=1 (\geq k^{2}), then the function reduces to K\left(k\right), becoming infinite when k^{2}=1. If {\sin}^{2}\phi=1/k^{2} (<1), then it has the value K\left(1/k\right)/k: put c=k^{2} in (19.25.5) and use (19.25.1). Magnify 3D Help
See accompanying text
Figure 19.3.4: E\left(\phi,k\right) as a function of k^{2} and {\sin}^{2}\phi for -1\leq k^{2}\leq 2, 0\leq{\sin}^{2}\phi\leq 1. If {\sin}^{2}\phi=1 (\geq k^{2}), then the function reduces to E\left(k\right), with value 1 at k^{2}=1. If {\sin}^{2}\phi=1/k^{2} (<1), then it has the value kE\left(1/k\right)+({k^{\prime}}^{2}/k)K\left(1/k\right), with limit 1 as k^{2}\to 1+: put c=k^{2} in (19.25.7) and use (19.25.1). Magnify 3D Help
See accompanying text
Figure 19.3.5: \Pi\left(\alpha^{2},k\right) as a function of k^{2} and \alpha^{2} for -2\leq k^{2}<1, -2\leq\alpha^{2}\leq 2. Cauchy principal values are shown when \alpha^{2}>1. The function is unbounded as \alpha^{2}\to 1-, and also (with the same sign as 1-\alpha^{2}) as k^{2}\to 1-. As \alpha^{2}\to 1+ it has the limit K\left(k\right)-(E\left(k\right)/{k^{\prime}}^{2}). If \alpha^{2}=0, then it reduces to K\left(k\right). If k^{2}=0, then it has the value \frac{1}{2}\pi/\sqrt{1-\alpha^{2}} when \alpha^{2}<1, and 0 when \alpha^{2}>1. See §19.6(i). Magnify 3D Help
See accompanying text
Figure 19.3.6: \Pi\left(\phi,2,k\right) as a function of k^{2} and {\sin}^{2}\phi for -1\leq k^{2}\leq 3, 0\leq{\sin}^{2}\phi<1. Cauchy principal values are shown when {\sin}^{2}\phi>\frac{1}{2}. The function tends to +\infty as {\sin}^{2}\phi\to\frac{1}{2}, except in the last case below. If {\sin}^{2}\phi=1 (>k^{2}), then the function reduces to \Pi\left(2,k\right) with Cauchy principal value K\left(k\right)-\Pi\left(\frac{1}{2}k^{2},k\right), which tends to -\infty as k^{2}\to 1-. See (19.6.5) and (19.6.6). If {\sin}^{2}\phi=1/k^{2} (<1), then by (19.7.4) it reduces to \Pi\left(2/k^{2},1/k\right)/k, k^{2}\neq 2, with Cauchy principal value (K\left(1/k\right)-\Pi\left(\frac{1}{2},1/k\right))/k, 1<k^{2}<2, by (19.6.5). Its value tends to -\infty as k^{2}\to 1+ by (19.6.6), and to the negative of the second lemniscate constant (see (19.20.22)) as k^{2}(={\csc}^{2}\phi)\to 2-. Magnify 3D Help

§19.3(ii) Complex Variables

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Keywords:
Legendre’s elliptic integrals, graphics
Notes:
These graphics were produced at NIST.
Permalink:
http://dlmf.nist.gov/19.3.ii
See also:
Annotations for §19.3 and Ch.19

In Figures 19.3.7 and 19.3.8 for complete Legendre’s elliptic integrals with complex arguments, height corresponds to the absolute value of the function and color to the phase. See also About Color Map.

See accompanying text
Figure 19.3.7: K\left(k\right) as a function of complex k^{2} for -2\leq\Re\left(k^{2}\right)\leq 2, -2\leq\Im\left(k^{2}\right)\leq 2. There is a branch cut where 1<k^{2}<\infty. Magnify 3D Help
See accompanying text
Figure 19.3.8: E\left(k\right) as a function of complex k^{2} for -2\leq\Re\left(k^{2}\right)\leq 2, -2\leq\Im\left(k^{2}\right)\leq 2. There is a branch cut where 1<k^{2}<\infty. Magnify 3D Help
See accompanying text
Figure 19.3.9: \Re\left(K\left(k\right)\right) as a function of complex k^{2} for -2\leq\Re\left(k^{2}\right)\leq 2, -2\leq\Im\left(k^{2}\right)\leq 2. The real part is symmetric under reflection in the real axis. On the branch cut (k^{2}\geq 1) it is infinite at k^{2}=1, and has the value K(1/k)/k when k^{2}>1. Magnify 3D Help
See accompanying text
Figure 19.3.10: \Im\left(K\left(k\right)\right) as a function of complex k^{2} for -2\leq\Re\left(k^{2}\right)\leq 2, -2\leq\Im\left(k^{2}\right)\leq 2. The imaginary part is 0 for k^{2}<1, and is antisymmetric under reflection in the real axis. On the upper edge of the branch cut (k^{2}\geq 1) it has the value K\left(k^{\prime}\right) if k^{2}>1, and \frac{1}{4}\pi if k^{2}=1. Magnify 3D Help
See accompanying text
Figure 19.3.11: \Re\left(E\left(k\right)\right) as a function of complex k^{2} for -2\leq\Re\left(k^{2}\right)\leq 2, -2\leq\Im\left(k^{2}\right)\leq 2. The real part is symmetric under reflection in the real axis. On the branch cut (k^{2}>1) it has the value kE\left(1/k\right)+({k^{\prime}}^{2}/k)K\left(1/k\right), with limit 1 as k^{2}\to 1+. Magnify 3D Help
See accompanying text
Figure 19.3.12: \Im\left(E\left(k\right)\right) as a function of complex k^{2} for -2\leq\Re\left(k^{2}\right)\leq 2, -2\leq\Im\left(k^{2}\right)\leq 2. The imaginary part is 0 for k^{2}\leq 1 and is antisymmetric under reflection in the real axis. On the upper edge of the branch cut (k^{2}>1) it has the (negative) value K\left(k^{\prime}\right)-E\left(k^{\prime}\right), with limit 0 as k^{2}\to 1+. Magnify 3D Help
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