The Special Function {U_{n}\left(x\right)}


1. Differential Equation[-]

The function {U_{n}\left(x\right)} satisfies the differential equation with initial values y \left( 0 \right) =\cos \left( 1/2\,n\pi \right) and y' \left( 0 \right) =\left(n+1\right)\sin \left( 1/2\,n\pi \right) .
All formulas on this page are valid under the condition that 2\,n is not an integer (special values for parameters can be entered at the bottom).

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 2 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 \infty [+]

5. Hypergeometric Representation[+]

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

7. Parameters[-]

The Chebyshev function of the second kind {U_{n}\left(x\right)} depends on the parameter n. The box below can be used to rename or instantiate this parameter.
p1 =