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11: 14.21 Definitions and Basic Properties
14.21.1 ( 1 z 2 ) d 2 w d z 2 2 z d w d z + ( ν ( ν + 1 ) μ 2 1 z 2 ) w = 0 .
§14.21(iii) Properties
This 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). …
12: 14.2 Differential Equations
14.2.1 ( 1 x 2 ) d 2 w d x 2 2 x d w d x + ν ( ν + 1 ) w = 0 .
14.2.2 ( 1 x 2 ) d 2 w d x 2 2 x d w d x + ( ν ( ν + 1 ) μ 2 1 x 2 ) w = 0 .
14.2.3 𝒲 { 𝖯 ν μ ( x ) , 𝖯 ν μ ( x ) } = 2 Γ ( μ ν ) Γ ( ν + μ + 1 ) ( 1 x 2 ) ,
14.2.4 𝒲 { 𝖯 ν μ ( x ) , 𝖰 ν μ ( x ) } = Γ ( ν + μ + 1 ) Γ ( ν μ + 1 ) ( 1 x 2 ) ,
13: 34.11 Higher-Order 3 n j Symbols
§34.11 Higher-Order 3 n j Symbols
14: 30.1 Special Notation
x real variable. Except in §§30.7(iv), 30.11(ii), 30.13, and 30.14, 1 < x < 1 .
m order, a nonnegative integer.
15: 14.6 Integer Order
§14.6 Integer Order
§14.6(i) Nonnegative Integer Orders
§14.6(ii) Negative Integer Orders
For connections between positive and negative integer orders see (14.9.3), (14.9.4), and (14.9.13). …
16: 14.5 Special Values
14.5.1 𝖯 ν μ ( 0 ) = 2 μ π 1 / 2 Γ ( 1 2 ν 1 2 μ + 1 ) Γ ( 1 2 1 2 ν 1 2 μ ) ,
14.5.2 d 𝖯 ν μ ( x ) d x | x = 0 = 2 μ + 1 π 1 / 2 Γ ( 1 2 ν 1 2 μ + 1 2 ) Γ ( 1 2 ν 1 2 μ ) ,
14.5.3 𝖰 ν μ ( 0 ) = 2 μ 1 π 1 / 2 sin ( 1 2 ( ν + μ ) π ) Γ ( 1 2 ν + 1 2 μ + 1 2 ) Γ ( 1 2 ν 1 2 μ + 1 ) , ν + μ 1 , 2 , 3 , ,
14.5.4 d 𝖰 ν μ ( x ) d x | x = 0 = 2 μ π 1 / 2 cos ( 1 2 ( ν + μ ) π ) Γ ( 1 2 ν + 1 2 μ + 1 ) Γ ( 1 2 ν 1 2 μ + 1 2 ) , ν + μ 1 , 2 , 3 , .
17: 10.24 Functions of Imaginary Order
§10.24 Functions of Imaginary Order
and J ~ ν ( x ) , Y ~ ν ( x ) are linearly independent solutions of (10.24.1): … In consequence of (10.24.6), when x is large J ~ ν ( x ) and Y ~ ν ( x ) comprise a numerically satisfactory pair of solutions of (10.24.1); compare §2.7(iv). … …
18: 10.45 Functions of Imaginary Order
§10.45 Functions of Imaginary Order
and I ~ ν ( x ) , K ~ ν ( x ) are real and linearly independent solutions of (10.45.1): … The corresponding result for K ~ ν ( x ) is given by …
19: 10.26 Graphics
§10.26(i) Real Order and Variable
§10.26(ii) Real Order, Complex Variable
§10.26(iii) Imaginary Order, Real Variable
See accompanying text
Figure 10.26.7: I ~ 1 / 2 ( x ) , K ~ 1 / 2 ( x ) , 0.01 x 3 . Magnify
See accompanying text
Figure 10.26.8: I ~ 1 ( x ) , K ~ 1 ( x ) , 0.01 x 3 . Magnify
20: 10.41 Asymptotic Expansions for Large Order
§10.41 Asymptotic Expansions for Large Order
§10.41(i) Asymptotic Forms
§10.41(ii) Uniform Expansions for Real Variable