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Ferrers functions

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1: 14.1 Special Notation
§14.1 Special Notation
β–ΊThe main functions treated in this chapter are the Legendre functions 𝖯 Ξ½ ⁑ ( x ) , 𝖰 Ξ½ ⁑ ( x ) , P Ξ½ ⁑ ( z ) , Q Ξ½ ⁑ ( z ) ; Ferrers functions 𝖯 Ξ½ ΞΌ ⁑ ( x ) , 𝖰 Ξ½ ΞΌ ⁑ ( x ) (also known as the Legendre functions on the cut); associated Legendre functions P Ξ½ ΞΌ ⁑ ( z ) , Q Ξ½ ΞΌ ⁑ ( z ) , 𝑸 Ξ½ ΞΌ ⁑ ( z ) ; conical functions 𝖯 1 2 + i ⁒ Ο„ ΞΌ ⁑ ( x ) , 𝖰 1 2 + i ⁒ Ο„ ΞΌ ⁑ ( x ) , 𝖰 ^ 1 2 + i ⁒ Ο„ ΞΌ ⁑ ( x ) , P 1 2 + i ⁒ Ο„ ΞΌ ⁑ ( x ) , Q 1 2 + i ⁒ Ο„ ΞΌ ⁑ ( x ) (also known as Mehler functions). …
2: 14.32 Methods of Computation
§14.32 Methods of Computation
3: 14.7 Integer Degree and Order
§14.7 Integer Degree and Order
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§14.7(ii) Rodrigues-Type Formulas
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§14.7(iii) Reflection Formulas
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§14.7(iv) Generating Functions
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4: 14.17 Integrals
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§14.17(i) Indefinite Integrals
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§14.17(ii) Barnes’ Integral
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§14.17(iii) Orthogonality Properties
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§14.17(v) Laplace Transforms
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§14.17(vi) Mellin Transforms
5: 14.5 Special Values
§14.5 Special Values
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14.5.7 𝖰 0 ⁑ ( x ) = 1 2 ⁒ ln ⁑ ( 1 + x 1 x ) ,
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14.5.8 𝖰 1 ⁑ ( x ) = x 2 ⁒ ln ⁑ ( 1 + x 1 x ) 1 .
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§14.5(v) ΞΌ = 0 , Ξ½ = ± 1 2
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6: 14.10 Recurrence Relations and Derivatives
§14.10 Recurrence Relations and Derivatives
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14.10.1 𝖯 Ξ½ ΞΌ + 2 ⁑ ( x ) + 2 ⁒ ( ΞΌ + 1 ) ⁒ x ⁒ ( 1 x 2 ) 1 / 2 ⁒ 𝖯 Ξ½ ΞΌ + 1 ⁑ ( x ) + ( Ξ½ ΞΌ ) ⁒ ( Ξ½ + ΞΌ + 1 ) ⁒ 𝖯 Ξ½ ΞΌ ⁑ ( x ) = 0 ,
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14.10.2 ( 1 x 2 ) 1 / 2 ⁒ 𝖯 Ξ½ ΞΌ + 1 ⁑ ( x ) ( Ξ½ ΞΌ + 1 ) ⁒ 𝖯 Ξ½ + 1 ΞΌ ⁑ ( x ) + ( Ξ½ + ΞΌ + 1 ) ⁒ x ⁒ 𝖯 Ξ½ ΞΌ ⁑ ( x ) = 0 ,
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14.10.3 ( Ξ½ ΞΌ + 2 ) ⁒ 𝖯 Ξ½ + 2 ΞΌ ⁑ ( x ) ( 2 ⁒ Ξ½ + 3 ) ⁒ x ⁒ 𝖯 Ξ½ + 1 ΞΌ ⁑ ( x ) + ( Ξ½ + ΞΌ + 1 ) ⁒ 𝖯 Ξ½ ΞΌ ⁑ ( x ) = 0 ,
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14.10.4 ( 1 x 2 ) ⁒ d 𝖯 Ξ½ ΞΌ ⁑ ( x ) d x = ( ΞΌ Ξ½ 1 ) ⁒ 𝖯 Ξ½ + 1 ΞΌ ⁑ ( x ) + ( Ξ½ + 1 ) ⁒ x ⁒ 𝖯 Ξ½ ΞΌ ⁑ ( x ) ,
7: 14.4 Graphics
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§14.4(i) Ferrers Functions: 2D Graphs
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§14.4(ii) Ferrers Functions: 3D Surfaces
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See accompanying text
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Figure 14.4.15: 𝖯 0 ΞΌ ⁑ ( x ) , 0 ΞΌ 10 , 1 < x < 1 . Magnify 3D Help
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See accompanying text
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Figure 14.4.16: 𝖰 0 ΞΌ ⁑ ( x ) , 0 ΞΌ 6.2 , 1 < x < 1 . Magnify 3D Help
8: 14.18 Sums
§14.18 Sums
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§14.18(ii) Addition Theorems
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14.18.3 𝖰 Ξ½ ⁑ ( cos ⁑ ΞΈ 1 ⁒ cos ⁑ ΞΈ 2 + sin ⁑ ΞΈ 1 ⁒ sin ⁑ ΞΈ 2 ⁒ cos ⁑ Ο• ) = 𝖯 Ξ½ ⁑ ( cos ⁑ ΞΈ 1 ) ⁒ 𝖰 Ξ½ ⁑ ( cos ⁑ ΞΈ 2 ) + 2 ⁒ m = 1 ( 1 ) m ⁒ 𝖯 Ξ½ m ⁑ ( cos ⁑ ΞΈ 1 ) ⁒ 𝖰 Ξ½ m ⁑ ( cos ⁑ ΞΈ 2 ) ⁒ cos ⁑ ( m ⁒ Ο• ) .
β–ΊThe formulas are also valid with the Ferrers functions as in §14.3(i) with ΞΌ = 0 . … β–Ί
9: 14.9 Connection Formulas
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§14.9(i) Connections Between 𝖯 Ξ½ ± ΞΌ ⁑ ( x ) , 𝖯 Ξ½ 1 ± ΞΌ ⁑ ( x ) , 𝖰 Ξ½ ± ΞΌ ⁑ ( x ) , 𝖰 Ξ½ 1 ΞΌ ⁑ ( x )
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14.9.3 𝖯 Ξ½ m ⁑ ( x ) = ( 1 ) m ⁒ Ξ“ ⁑ ( Ξ½ m + 1 ) Ξ“ ⁑ ( Ξ½ + m + 1 ) ⁒ 𝖯 Ξ½ m ⁑ ( x ) ,
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14.9.4 𝖰 Ξ½ m ⁑ ( x ) = ( 1 ) m ⁒ Ξ“ ⁑ ( Ξ½ m + 1 ) Ξ“ ⁑ ( Ξ½ + m + 1 ) ⁒ 𝖰 Ξ½ m ⁑ ( x ) , Ξ½ m 1 , m 2 , .
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§14.9(ii) Connections Between 𝖯 Ξ½ ± ΞΌ ⁑ ( ± x ) , 𝖰 Ξ½ ΞΌ ⁑ ( ± x ) , 𝖰 Ξ½ ΞΌ ⁑ ( x )
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14.9.9 2 Ξ“ ⁑ ( Ξ½ + ΞΌ + 1 ) ⁒ Ξ“ ⁑ ( ΞΌ Ξ½ ) ⁒ 𝖰 Ξ½ ΞΌ ⁑ ( x ) = cos ⁑ ( Ξ½ ⁒ Ο€ ) ⁒ 𝖯 Ξ½ ΞΌ ⁑ ( x ) + cos ⁑ ( ΞΌ ⁒ Ο€ ) ⁒ 𝖯 Ξ½ ΞΌ ⁑ ( x ) ,
10: 14.6 Integer Order
§14.6 Integer Order
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14.6.1 𝖯 Ξ½ m ⁑ ( x ) = ( 1 ) m ⁒ ( 1 x 2 ) m / 2 ⁒ d m 𝖯 Ξ½ ⁑ ( x ) d x m ,
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14.6.2 𝖰 Ξ½ m ⁑ ( x ) = ( 1 ) m ⁒ ( 1 x 2 ) m / 2 ⁒ d m 𝖰 Ξ½ ⁑ ( x ) d x m .
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14.6.6 𝖯 Ξ½ m ⁑ ( x ) = ( 1 x 2 ) m / 2 ⁒ x 1 ⁒ x 1 𝖯 Ξ½ ⁑ ( x ) ⁒ ( d x ) m .
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