DLMF:

19 Elliptic Integrals Legendre’s Integrals 19.3 Graphics 19.3 Graphics 19.4 Derivatives and Differential Equations

Figure 19.3.11 (See in context.)

Figure 19.3.11: $\realpart{(\mathop{E\/}\nolimits\!\left(k\right))}$ as a function of complex $k^{2}$ for $-2\leq\realpart{(k^{2})}\leq 2$ , $-2\leq\imagpart{(k^{2})}\leq 2$ . The real part is symmetric under reflection in the real axis. On the branch cut ( $k^{2}>1$ ) it has the value $k\mathop{E\/}\nolimits\!\left(1/k\right)+({k^{{\prime}}}^{2}/k)\mathop{K\/}% \nolimits\!\left(1/k\right)$ , with limit 1 as $k^{2}\to 1+$ .

Annotations:

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the second kind, $\imagpart{}$ : imaginary part, $\realpart{}$ : real part, : real or complex modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/19.3.F11.mag
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