The Special Function  {V\left(a,x\right)}

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1. Differential Equation[-]

The function  {V\left(a,x\right)} satisfies the differential equation with initial values y \left( 0 \right) =\frac{{2}^{1/2\,a-1/4}\left(\sin \left( 1/2\,a\pi  \right) +\cos \left( 1/2\,a\pi  \right) \right)}{\Gamma  \left( 3/4-1/2\,a \right) } and y' \left( 0 \right) =\frac{{2}^{1/2\,a+1/4}\left(\cos \left( 1/2\,a\pi  \right) -\sin \left( 1/2\,a\pi  \right) \right)}{\Gamma  \left( 1/4-1/2\,a \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 0 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 \infty [+]

5. Hypergeometric Representation[+]

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

7. Laplace Transform[+]

8. Parameters[-]

The parabolic cylinder function {V\left(a,x\right)} depends on the parameter a . The box below can be used to rename or instantiate this parameter.
p1 =