DLMF:

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

Figure 19.3.5 (See in context.)

Figure 19.3.5: $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)$ 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 $\mathop{K\/}\nolimits\!\left(k\right)-(\mathop{E\/}\nolimits\!\left(k\right)/{% k^{{\prime}}}^{2})$ . If $\alpha^{2}=0$ , then it reduces to $\mathop{K\/}\nolimits\!\left(k\right)$ . 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).

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, $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)$ : Legendre’s complete elliptic integral of the third kind, : real or complex modulus, $k^{{\prime}}$ : complementary modulus and $\alpha^{2}$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/19.3.F5.mag
Encodings:: Magnified png, VRML, X3D, pdf

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