DLMF:

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

Figure 19.3.3 (See in context.)

Figure 19.3.3: $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ as a function of $k^{2}$ and ${\mathop{\sin\/}\nolimits^{{2}}}\phi$ for $-1\leq k^{2}\leq 2$ , $0\leq{\mathop{\sin\/}\nolimits^{{2}}}\phi\leq 1$ . If ${\mathop{\sin\/}\nolimits^{{2}}}\phi=1$ ( $\geq k^{2}$ ), then the function reduces to $\mathop{K\/}\nolimits\!\left(k\right)$ , becoming infinite when $k^{2}=1$ . If ${\mathop{\sin\/}\nolimits^{{2}}}\phi=1/k^{2}$ (

), then it has the value $\mathop{K\/}\nolimits\!\left(1/k\right)/k$ : put $c=k^{2}$ in (19.25.5) and use (19.25.1).

Annotations:

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\mathop{\sin\/}\nolimits z$ : sine function, $\phi$ : real or complex argument and : real or complex modulus
Permalink:: http://dlmf.nist.gov/19.3.F3.mag
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