DLMF: Figure 19.3.9 ‣ §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 19.3.9 (See in context.)

Figure 19.3.9:

\Re\left(K\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 real part is symmetric under reflection in the real axis. On the branch cut (

k^{2}\geq 1

) it is infinite at

k^{2}=1

, and has the value

K(1/k)/k

when

k^{2}>1

.

ⓘ

Annotations:

Symbols:: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\Im$ : imaginary part, $\Re$ : real part and $k$ : real or complex modulus
Permalink:: http://dlmf.nist.gov/19.3.F9.viz

19.3 Graphics 19.4 Derivatives and Differential Equations

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