The Special Function {F\left(z,k\right)}

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1. Differential Equation[-]

The function {F\left(z,k\right)} satisfies the differential equation 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).

2. Derivative in Terms of Lower-Order Derivatives[+]

3. Expansion at 0[+]

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

The differential equation above has 4 non-zero finite singular pointsA complex numberĀ z_0 is a singular point (or singularity of a linear ordinary differential equation with polynomial coefficients, if the leading coefficient of the equation vanishes atĀ z_0..

4.1. Expansion at -1[+]

4.2. Expansion at 1[+]

4.3. Expansion at {z}^{-1}[+]

4.4. Expansion at -{z}^{-1}[+]

4.5. Expansion at \infty [+]

5. Chebyshev Expansion over [-1,1][+]

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 =