Formula:6401

Digital Library of Mathematical Functions ID 18.15.E12

${\displaystyle{\displaystyle P_{n}\left(\cos\theta\right)=\left(\frac{2}{\sin\theta}\right)^{\frac{1}{2}}\sum_{m=0}^{M-1}\genfrac{(}{)}{0.0pt}{}{-\tfrac{1}{2}}{m}\genfrac{(}{)}{0.0pt}{}{m-\frac{1}{2}}{n}\frac{\cos\alpha_{n,m}}{(2\sin\theta)^{m}}+O\left(\frac{1}{n^{M+\frac{1}{2}}}\right),}}$

Symbols List

• ${\displaystyle{\displaystyle O\left(\NVar{x}\right)}}$: order not exceeding
• ${\displaystyle{\displaystyle P_{\NVar{n}}\left(\NVar{x}\right)}}$: Legendre polynomial
• ${\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}$:
• ${\displaystyle{\displaystyle\cos\NVar{z}}}$: cosine function
• ${\displaystyle{\displaystyle\sin\NVar{z}}}$: sine function
• ${\displaystyle{\displaystyle m}}$: nonnegative integer
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
• ${\displaystyle{\displaystyle\alpha_{n,m}}}$: 18.15.E13
 

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