The Special Function {P_{v}^{u}\left(x\right)}

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1. Differential Equation[-]

The function {P_{v}^{u}\left(x\right)} satisfies the differential equation with initial values y \left( 0 \right) =\frac{{2}^{u}\sin \left( \pi \, \left( 1/2-1/2\,u-1/2\,v \right) \right) \,\Gamma \left( 1/2+1/2\,u+1/2\,v \right) }{\sqrt {\pi } \left( -1 \right) ^{1/2\,u}\,\Gamma \left( 1+1/2\,v-1/2\,u \right) } and y' \left( 0 \right) =-\frac{{2}^{u}\left(1-u+v\right)\sin \left( \pi \, \left( -1/2\,u-1/2\,v \right) \right) \,\Gamma \left( 1+1/2\,u+1/2\,v \right) }{\sqrt {\pi } \left( -1 \right) ^{1/2\,u}\,\Gamma \left( 3/2+1/2\,v-1/2\,u \right) }.
All formulas on this page are valid under the condition that u and 2\,v are not integers (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. Chebyshev Expansion over [-1,1][+]

6. Parameters[-]

The associated Legendre function of the first kind {P_{v}^{u}\left(x\right)} depends on the parameters v and u. The boxes below can be used to rename or instantiate these parameters.
p1 =  p2 =