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

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1. Differential Equation[-]

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

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