DLMF: Figure 19.3.5 ‣ §19.3 Graphics ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals

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:

\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).

ⓘ

Annotations:

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s complete elliptic integral of the third kind, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus and $\alpha^{2}$ : real or complex parameter
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19.3 Graphics 19.4 Derivatives and Differential Equations

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