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limiting forms as trigonometric functions

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1: 18.11 Relations to Other Functions
Hermite
2: 22.5 Special Values
§22.5 Special Values
For the other nine functions ratios can be taken; compare (22.2.10). …
§22.5(ii) Limiting Values of k
In these cases the elliptic functions degenerate into elementary trigonometric and hyperbolic functions, respectively. …
3: 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
F ( η , ρ ) = sin ( θ ( η , ρ ) ) + o ( 1 ) ,
§33.10(ii) Large Positive η
§33.10(iii) Large Negative η
4: 10.45 Functions of Imaginary Order
§10.45 Functions of Imaginary Order
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 . … In this reference I ~ ν ( x ) is denoted by ( 1 / π ) sinh ( π ν ) L i ν ( x ) . …
5: 10.24 Functions of Imaginary Order
Y ~ ν ( x ) = 2 / ( π x ) sin ( x - 1 4 π ) + O ( x - 3 2 ) .
6: 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 .
7: 11.9 Lommel Functions
§11.9 Lommel Functions
The inhomogeneous Bessel differential equation … the right-hand side being replaced by its limiting form when μ ± ν is an odd negative integer. …
8: 10.5 Wronskians and Cross-Products
§10.5 Wronskians and Cross-Products
10.5.1 𝒲 { J ν ( z ) , J - ν ( z ) } = J ν + 1 ( z ) J - ν ( z ) + J ν ( z ) J - ν - 1 ( z ) = - 2 sin ( ν π ) / ( π z ) ,
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 ) ,
10.5.5 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 ) .
9: 6.2 Definitions and Interrelations
This is also true of the functions Ci ( z ) and Chi ( z ) defined in §6.2(ii). … The logarithmic integral is defined by … Si ( z ) is an odd entire function. … Cin ( z ) is an even entire function. …
§6.2(iii) Auxiliary Functions
10: 33.5 Limiting Forms for Small ρ , Small | η | , or Large
§33.5 Limiting Forms for Small ρ , Small | η | , or Large
§33.5(i) Small ρ
§33.5(ii) η = 0
§33.5(iii) Small | η |
§33.5(iv) Large