DLMF: Figure 19.3.4 ‣ §19.3 Graphics ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals

Figure 19.3.4 (See in context.)

Figure 19.3.4:

E\left(\phi,k\right)

as a function of

k^{2}

and

{\sin}^{2}\phi

for

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

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

. If

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

(

\geq k^{2}

), then the function reduces to

E\left(k\right)

, with value 1 at

k^{2}=1

. If

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

(

<1

), then 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+

: put

c=k^{2}

in (19.25.7) and use (19.25.1).

ⓘ

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, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus and $k^{\prime}$ : complementary modulus
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