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limiting form as a Bessel function

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1: 18.11 Relations to Other Functions
Laguerre
2: 18.34 Bessel Polynomials
18.34.8 lim α P n ( α , a - α - 2 ) ( 1 + α x ) P n ( α , a - α - 2 ) ( 1 ) = y n ( x ; a ) .
3: 10.24 Functions of Imaginary Order
§10.24 Functions of Imaginary Order
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). Also, in consequence of (10.24.7)–(10.24.9), when x is small either J ~ ν ( x ) and tanh ( 1 2 π ν ) Y ~ ν ( x ) or J ~ ν ( x ) and Y ~ ν ( x ) comprise a numerically satisfactory pair depending whether ν 0 or ν = 0 . …
4: 10.28 Wronskians and Cross-Products
§10.28 Wronskians and Cross-Products
10.28.1 𝒲 { I ν ( z ) , I - ν ( z ) } = I ν ( z ) I - ν - 1 ( z ) - I ν + 1 ( z ) I - ν ( z ) = - 2 sin ( ν π ) / ( π z ) ,
10.28.2 𝒲 { K ν ( z ) , I ν ( z ) } = I ν ( z ) K ν + 1 ( z ) + I ν + 1 ( z ) K ν ( z ) = 1 / z .
5: 10.45 Functions of Imaginary Order
§10.45 Functions of Imaginary Order
The corresponding result for K ~ ν ( x ) is given by … In consequence of (10.45.5)–(10.45.7), I ~ ν ( x ) and K ~ ν ( x ) comprise a numerically satisfactory pair of solutions of (10.45.1) when x is large, and either I ~ ν ( x ) and ( 1 / π ) sinh ( π ν ) K ~ ν ( x ) , or I ~ ν ( x ) and K ~ ν ( x ) , comprise a numerically satisfactory pair when x is small, depending whether ν 0 or ν = 0 . …
6: 10.5 Wronskians and Cross-Products
§10.5 Wronskians and Cross-Products
10.5.2 𝒲 { J ν ( z ) , Y ν ( z ) } = J ν + 1 ( z ) Y ν ( z ) - J ν ( z ) Y ν + 1 ( z ) = 2 / ( π z ) ,
10.5.3 𝒲 { J ν ( z ) , H ν ( 1 ) ( z ) } = J ν + 1 ( z ) H ν ( 1 ) ( z ) - J ν ( z ) H ν + 1 ( 1 ) ( z ) = 2 i / ( π z ) ,
10.5.4 𝒲 { J ν ( z ) , H ν ( 2 ) ( z ) } = J ν + 1 ( z ) H ν ( 2 ) ( z ) - J ν ( z ) H ν + 1 ( 2 ) ( z ) = - 2 i / ( π z ) ,
10.5.5 𝒲 { H ν ( 1 ) ( z ) , H ν ( 2 ) ( z ) } = H ν + 1 ( 1 ) ( z ) H ν ( 2 ) ( z ) - H ν ( 1 ) ( z ) H ν + 1 ( 2 ) ( z ) = - 4 i / ( π z ) .
7: 10.2 Definitions
§10.2 Definitions
§10.2(i) Bessel’s Equation
Bessel Function of the First Kind
Bessel Function of the Second Kind (Weber’s Function)
Bessel Functions of the Third Kind (Hankel Functions)
8: 33.10 Limiting Forms for Large ρ or Large | η |
§33.10 Limiting Forms for Large ρ or Large | η |
§33.10(i) Large ρ
§33.10(ii) Large Positive η
F 0 ( η , ρ ) e - π η ( π ρ ) 1 / 2 I 1 ( ( 8 η ρ ) 1 / 2 ) ,
§33.10(iii) Large Negative η
9: 10.52 Limiting Forms
§10.52 Limiting Forms
10.52.1 j n ( z ) , i n ( 1 ) ( z ) z n / ( 2 n + 1 ) !! ,
10: 10.7 Limiting Forms
§10.7 Limiting Forms
For H - ν ( 1 ) ( z ) and H - ν ( 2 ) ( z ) when ν > 0 combine (10.4.6) and (10.7.7). For H i ν ( 1 ) ( z ) and H i ν ( 2 ) ( z ) when ν and ν 0 combine (10.4.3), (10.7.3), and (10.7.6). … For the corresponding results for H ν ( 1 ) ( z ) and H ν ( 2 ) ( z ) see (10.2.5) and (10.2.6).