# 1. Differential Equation[-]

The function $${P_{v}^{u}\left(x\right)}$$ satisfies the differential equation$$\displaystyle \left( -{x}^{2}+1 \right) ^{2}{\frac {d^{2}}{d{x}^{2}}}y \left( x \right) -2\,x \left( -{x}^{2}+1 \right) {\frac {d}{dx}}y \left( x \right) + \left( v \left( v+1 \right) \left( -{x}^{2}+1 \right) -{u}^{2} \right) y \left( x \right) = 0$$ 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).

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

The differential equation above has 2 non-zero finite .

# 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 =