DLMF: Figure 19.3.12 ‣ §19.3 Graphics ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals

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DLMF: Figure 19.3.12 ‣ §19.3 Graphics ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals

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

Figure 19.3.12 (See in context.)

Figure 19.3.12:

\Im\left(E\left(k\right)\right)

as a function of complex

k^{2}

for

-2\leq\Re\left(k^{2}\right)\leq 2

,

-2\leq\Im\left(k^{2}\right)\leq 2

. The imaginary part is 0 for

k^{2}\leq 1

and is antisymmetric under reflection in the real axis. On the upper edge of the branch cut (

k^{2}>1

) it has the (negative) value

K\left(k^{\prime}\right)-E\left(k^{\prime}\right)

, with limit 0 as

k^{2}\to 1+

.

ⓘ

Annotations:

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, $\Im$ : imaginary part, $\Re$ : real part, $k$ : real or complex modulus and $k^{\prime}$ : complementary modulus
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19.3 Graphics 19.4 Derivatives and Differential Equations

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