DLMF: Figure 19.3.6 ‣ §19.3 Graphics ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals

Figure 19.3.6 (See in context.)

Figure 19.3.6:

\Pi\left(\phi,2,k\right)

as a function of

k^{2}

and

{\sin}^{2}\phi

for

-1\leq k^{2}\leq 3

0\leq{\sin}^{2}\phi<1

. Cauchy principal values are shown when

{\sin}^{2}\phi>\frac{1}{2}

. The function tends to

+\infty

{\sin}^{2}\phi\to\frac{1}{2}

, except in the last case below. If

{\sin}^{2}\phi=1

(

>k^{2}

), then the function reduces to

\Pi\left(2,k\right)

with Cauchy principal value

K\left(k\right)-\Pi\left(\frac{1}{2}k^{2},k\right)

, which tends to

-\infty

k^{2}\to 1-

. See (19.6.5) and (19.6.6). If

{\sin}^{2}\phi=1/k^{2}

(

<1

), then by (19.7.4) it reduces to

\Pi\left(2/k^{2},1/k\right)/k

k^{2}\neq 2

, with Cauchy principal value

(K\left(1/k\right)-\Pi\left(\frac{1}{2},1/k\right))/k

1<k^{2}<2

, by (19.6.5). Its value tends to

-\infty

k^{2}\to 1+

by (19.6.6), and to the negative of the second lemniscate constant (see (19.20.22)) as

k^{2}(={\csc}^{2}\phi)\to 2-

ⓘ

Symbols:: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s complete elliptic integral of the third kind, $\csc\NVar{z}$ : cosecant function, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument and $k$ : real or complex modulus
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