§ 4 Legendre function
First,
the definition of Legendre function
[ Legendre functions of the first kind ]
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It resolves singlevalued in the removed plane ._{}_{}
[ Legendre functions of the second kind ]
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It resolves singlevalued in the removed plane ._{}_{}
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It resolves singlevalued in the removed plane ._{}_{}
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[ General Legendre function ]
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They are singlevalued analytically in the removed plane and are Legendre differential equations_{}_{}
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two linearly independent solutions of .
At that time , they were Legendre functions of the first and second kinds, respectively ._{}
when a positive integer), there are_{}
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for having_{}
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( At that time , the Legendre polynomial_{}_{}
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2.
Other expressions of Legendre function
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where is a forward simple closed curve on the plane (Fig. 12.2 ), the enclosing point is the sum , but not the enclosing point ._{}_{}_{}_{}_{}
When (or when an integer),_{}_{} _{}
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The integral route is shown in Figure 12.3. At that time ,_{}_{}_{}
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3.
The recurrence formula and related formulas of the Legendre function
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The above formula is also applicable to , just replace P in the formula with . Use_{}_{}
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The corresponding recursive formula on the interval can be obtained , and there are similar formulas for ._{}_{}
4.
Orthogonality of Legendre functions
Only the orthogonality of the function is a positive integer, and the formula is as follows_{}_{}_{}
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5.
Asymptotic expressions and inequalities of Legendre functions
[ asymptotic expression ]
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[ inequality ]
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The inequalities are real numbers and positive integers ._{}_{}