Aquileo | DLMF: Figure 19.3.11 ‣ §19.3 Graphics ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals

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 $k^{2}\to 1+$ .

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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.F11.viz