Short Question Asso Leg poly_Orthogonal ppt

3-Mark Question: Orthogonality of Associated Legendre Polynomials
3 Marks

Short Question: Orthogonality of Associated Legendre Polynomials

Question (3 Marks)

State the orthogonality property of associated Legendre polynomials \(P_\ell^{\,m}(x)\) for fixed \(m\) on the interval \([-1,1]\).

\[ \text{(Write the standard relation and define } \delta_{\ell n}\text{.)} \]

Expected Answer (Brief)

\[ \boxed{ \int_{-1}^{1} P_\ell^{\,m}(x)\,P_n^{\,m}(x)\,dx = \frac{2}{2\ell+1}\,\frac{(\ell+m)!}{(\ell-m)!}\,\delta_{\ell n} } \]

where \(\delta_{\ell n}=1\) if \(\ell=n\) and \(0\) if \(\ell\neq n\).

Scheme of Evaluation

Marking Scheme (Total: 3 Marks)

What the student should write Marks
Writes the correct orthogonality statement in integral form (must include limits \(-1\) to \(1\)). 1.5
Correct normalization factor (must have factorial ratio \(\dfrac{(\ell+m)!}{(\ell-m)!}\) and \(\dfrac{2}{2\ell+1}\)). 1.0
Correct meaning of \(\delta_{\ell n}\) (Kronecker delta) OR clearly states “integral is zero for \(\ell\neq n\)”. 0.5

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