Aquileo | DLMF: Figure 19.3.12 ‣ §19.3 Graphics ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals

Figure 19.3.12 (See in context.) 3D Help

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 $k^{2}\to 1+$ . 3D Help

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Annotations:

Symbols:: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $\Im$ : imaginary part, $\Re$ : real part, : real or complex modulus and $k^{\prime}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/19.3.F12.mag
Encodings:: Magnified png, Vizualization, pdf
See also:: Annotations for §19.3 and Ch.19