Formula:7376

Digital Library of Mathematical Functions ID 19.30.E6

${\displaystyle{\displaystyle\frac{\partial s}{\partial(1/k)}=\sqrt{a^{2}-b^{2}}F\left(\phi,k\right)=\sqrt{a^{2}-b^{2}}R_{F}\left(c-1,c-k^{2},c\right),}}$

Constraint(s)

${\displaystyle{\displaystyle c={\csc^{2}}\phi}}$
${\displaystyle{\displaystyle k^{2}=(a^{2}-b^{2})/(a^{2}+\lambda)}}$
${\displaystyle{\displaystyle c={{\csc^{2}}}\phi}}$

Symbols List

• ${\displaystyle{\displaystyle R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)}}$: symmetric elliptic integral of first kind
• ${\displaystyle{\displaystyle\csc\NVar{z}}}$: cosecant function
• ${\displaystyle{\displaystyle F\left(\NVar{\phi},\NVar{k}\right)}}$: Legendre’s incomplete elliptic integral of the first kind
• ${\displaystyle{\displaystyle\frac{\partial\NVar{f}}{\partial\NVar{x}}}}$: partial derivative of $$f$$ with respect to $$x$$
• ${\displaystyle{\displaystyle\partial\NVar{x}}}$: partial differential of $$x$$
• ${\displaystyle{\displaystyle\phi}}$: real or complex argument
• ${\displaystyle{\displaystyle k}}$: real or complex modulus
• ${\displaystyle{\displaystyle s}}$: arclength
 

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