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11: 28.5 Second Solutions fe n , ge n
β–Ί
Wronskians
β–Ί
28.5.8 𝒲 ⁑ { ce n , fe n } = ce n ⁑ ( 0 , q ) ⁒ fe n ⁑ ( 0 , q ) ,
β–Ί
28.5.9 𝒲 ⁑ { se n , ge n } = se n ⁑ ( 0 , q ) ⁒ ge n ⁑ ( 0 , q ) .
12: 30.6 Functions of Complex Argument
β–Ί
Wronskian
β–Ί
30.6.3 𝒲 ⁑ { 𝑃𝑠 n m ⁑ ( z , Ξ³ 2 ) , 𝑄𝑠 n m ⁑ ( z , Ξ³ 2 ) } = ( 1 ) m ⁒ ( n + m ) ! ( 1 z 2 ) ⁒ ( n m ) ! ⁒ A n m ⁑ ( Ξ³ 2 ) ⁒ A n m ⁑ ( Ξ³ 2 ) ,
13: 13.2 Definitions and Basic Properties
β–Ί
§13.2(vi) Wronskians
β–Ί
13.2.33 𝒲 ⁑ { 𝐌 ⁑ ( a , b , z ) , z 1 b ⁒ 𝐌 ⁑ ( a b + 1 , 2 b , z ) } = sin ⁑ ( Ο€ ⁒ b ) ⁒ z b ⁒ e z / Ο€ ,
β–Ί
13.2.34 𝒲 ⁑ { 𝐌 ⁑ ( a , b , z ) , U ⁑ ( a , b , z ) } = z b ⁒ e z / Ξ“ ⁑ ( a ) ,
β–Ί
13.2.35 𝒲 ⁑ { 𝐌 ⁑ ( a , b , z ) , e z ⁒ U ⁑ ( b a , b , e ± Ο€ ⁒ i ⁒ z ) } = e βˆ“ b ⁒ Ο€ ⁒ i ⁒ z b ⁒ e z / Ξ“ ⁑ ( b a ) ,
β–Ί
13.2.36 𝒲 ⁑ { z 1 b ⁒ 𝐌 ⁑ ( a b + 1 , 2 b , z ) , U ⁑ ( a , b , z ) } = z b ⁒ e z / Ξ“ ⁑ ( a b + 1 ) ,
14: 31.9 Orthogonality
β–Ί
31.9.3 ΞΈ m = ( 1 e 2 ⁒ Ο€ ⁒ i ⁒ Ξ³ ) ⁒ ( 1 e 2 ⁒ Ο€ ⁒ i ⁒ Ξ΄ ) ⁒ ΞΆ Ξ³ ⁒ ( 1 ΞΆ ) Ξ΄ ⁒ ( ΞΆ a ) Ο΅ ⁒ f 0 ⁑ ( q , ΞΆ ) f 1 ⁑ ( q , ΞΆ ) ⁒ q ⁑ 𝒲 ⁑ { f 0 ⁑ ( q , ΞΆ ) , f 1 ⁑ ( q , ΞΆ ) } | q = q m ,
β–Ίand 𝒲 denotes the Wronskian1.13(i)). …
15: 14.21 Definitions and Basic Properties
β–Ί
§14.21(iii) Properties
β–ΊThis includes, for example, the Wronskian relations (14.2.7)–(14.2.11); hypergeometric representations (14.3.6)–(14.3.10) and (14.3.15)–(14.3.20); results for integer orders (14.6.3)–(14.6.5), (14.6.7), (14.6.8), (14.7.6), (14.7.7), and (14.7.11)–(14.7.16); behavior at singularities (14.8.7)–(14.8.16); connection formulas (14.9.11)–(14.9.16); recurrence relations (14.10.3)–(14.10.7). …
16: 33.2 Definitions and Basic Properties
β–Ί
§33.2(iv) Wronskians and Cross-Product
β–Ί
17: 33.14 Definitions and Basic Properties
β–Ί
§33.14(v) Wronskians
β–Ί
𝒲 ⁑ { h , f } = 2 / Ο€ ,
β–Ί
𝒲 ⁑ { c , s } = 1 / Ο€ .
18: 29.8 Integral Equations
β–ΊLet w ⁑ ( z ) be any solution of (29.2.1) of period 4 ⁒ K ⁑ , w 2 ⁑ ( z ) be a linearly independent solution, and 𝒲 ⁑ { w , w 2 } denote their Wronskian. … β–Ί
29.8.3 ΞΌ = 2 ⁒ Οƒ ⁒ Ο„ 𝒲 ⁑ { w , w 2 } ,
19: 28.20 Definitions and Basic Properties
β–Ί
§28.20(vi) Wronskians
β–Ί
𝒲 ⁑ { M Ξ½ ( 1 ) , M Ξ½ ( 2 ) } = 𝒲 ⁑ { M Ξ½ ( 2 ) , M Ξ½ ( 3 ) } = 𝒲 ⁑ { M Ξ½ ( 2 ) , M Ξ½ ( 4 ) } = 2 / Ο€ ,
β–Ί
𝒲 ⁑ { M Ξ½ ( 1 ) , M Ξ½ ( 3 ) } = 𝒲 ⁑ { M Ξ½ ( 1 ) , M Ξ½ ( 4 ) } = 1 2 ⁒ 𝒲 ⁑ { M Ξ½ ( 3 ) , M Ξ½ ( 4 ) } = 2 ⁒ i / Ο€ .
20: 32.8 Rational Solutions
β–Ίwhere Ο„ n ⁑ ( z ) is the n × n Wronskian determinant β–Ί
32.8.10 Ο„ n ⁑ ( z ) = 𝒲 ⁑ { p 1 ⁑ ( z ) , p 3 ⁑ ( z ) , , p 2 ⁒ n 1 ⁑ ( z ) } .