# 1. Differential Equation[-]

The function $${F\left(z,k\right)}$$ satisfies the differential equation$$\displaystyle {k}^{2} \left( {k}^{2}-1 \right) \left( {k}^{2}{z}^{2}-1 \right) {\frac {d^{3}}{d{k}^{3}}}y \left( k \right) +k \left( 8\,{k}^{4}{z}^{2}-4\,{k}^{2}{z}^{2}-5\,{k}^{2}+1 \right) {\frac {d^{2}}{d{k}^{2}}}y \left( k \right) + \left( 13\,{k}^{4}{z}^{2}-2\,{k}^{2}{z}^{2}-4\,{k}^{2}-1 \right) {\frac {d}{dk}}y \left( k \right) +3\,{k}^{3}{z}^{2}y \left( k \right) = 0$$ with initial values $$y \left( 0 \right) =\arcsin \left( z \right)$$, $$y'' \left( 0 \right) =\frac{1}{3} {z}^{3}\,{{}_{2}F_{1}\!\left(1/2,3/2;\,5/2;\,{z}^{2}\right)}$$, and $$y^{(4)} \left( 0 \right) =\frac{3}{4} {z}^{2}\left(2\,{z}^{3}{{}_{2}F_{1}\!\left(1/2,3/2;\,5/2;\,{z}^{2}\right)}+3\,z{{}_{2}F_{1}\!\left(1/2,3/2;\,5/2;\,{z}^{2}\right)}-3\,\arcsin \left( z \right) \right)$$.

# 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 first kind $${F\left(z,k\right)}$$ depends on the parameter $$z$$. The box below can be used to rename or instantiate this parameter.
p1 =