1. Differential Equation[-]

The function $${E\left(z,k\right)}$$ satisfies the differential equation$$\displaystyle -{k}^{2} \left( {k}^{2}-1 \right) \left( {k}^{2}{z}^{2}-1 \right) {\frac {d^{3}}{d{k}^{3}}}y \left( k \right) -k \left( 4\,{k}^{4}{z}^{2}-2\,{k}^{2}{z}^{2}-3\,{k}^{2}+1 \right) {\frac {d^{2}}{d{k}^{2}}}y \left( k \right) - \left( {k}^{4}{z}^{2}-1 \right) {\frac {d}{dk}}y \left( k \right) +{k}^{3}{z}^{2}y \left( k \right) = 0$$ with initial values $$y \left( 0 \right) =\arcsin \left( z \right)$$, $$y'' \left( 0 \right) =1/2\,z\sqrt {-{z}^{2}+1}-1/2\,\arcsin \left( z \right)$$, and $$y^{(4)} \left( 0 \right) =24\, \left( 1/32\,{z}^{3}+{\frac {3}{64}}\,z \right) \sqrt {-{z}^{2}+1}-{\frac {9}{8}}\,\arcsin \left( z \right)$$.

4. Local Expansions at Singularities and at Infinity[-]

The differential equation above has 4 non-zero finite .

6. Parameters[-]

The incomplete elliptic integral of the second kind $${E\left(z,k\right)}$$ depends on the parameter $$z$$. The box below can be used to rename or instantiate this parameter.
p1 =