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1: 12.1 Special Notation
Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. The main functions treated in this chapter are the parabolic cylinder functions (PCFs), also known as Weber parabolic cylinder functions: U ( a , z ) , V ( a , z ) , U ¯ ( a , z ) , and W ( a , z ) . …An older notation, due to Whittaker (1902), for U ( a , z ) is D ν ( z ) . …
2: 12.14 The Function W ( a , x )
§12.14 The Function W ( a , x )
§12.14(iii) Graphs
§12.14(iv) Connection Formula
Bessel Functions
3: 10.29 Recurrence Relations and Derivatives
With 𝒵 ν ( z ) defined as in §10.25(ii),
𝒵 ν 1 ( z ) 𝒵 ν + 1 ( z ) = ( 2 ν / z ) 𝒵 ν ( z ) ,
𝒵 ν 1 ( z ) + 𝒵 ν + 1 ( z ) = 2 𝒵 ν ( z ) .
𝒵 ν ( z ) = 𝒵 ν 1 ( z ) ( ν / z ) 𝒵 ν ( z ) ,
For results on modified quotients of the form z 𝒵 ν ± 1 ( z ) / 𝒵 ν ( z ) see Onoe (1955) and Onoe (1956). …
4: 10.13 Other Differential Equations
§10.13 Other Differential Equations
10.13.1 w ′′ + ( λ 2 ν 2 1 4 z 2 ) w = 0 , w = z 1 2 𝒞 ν ( λ z ) ,
In (10.13.9)–(10.13.11) 𝒞 ν ( z ) , 𝒟 μ ( z ) are any cylinder functions of orders ν , μ , respectively, and ϑ = z ( d / d z ) .
10.13.9 z 2 w ′′′ + 3 z w ′′ + ( 4 z 2 + 1 4 ν 2 ) w + 4 z w = 0 , w = 𝒞 ν ( z ) 𝒟 ν ( z ) ,
10.13.10 z 3 w ′′′ + z ( 4 z 2 + 1 4 ν 2 ) w + ( 4 ν 2 1 ) w = 0 , w = z 𝒞 ν ( z ) 𝒟 ν ( z ) ,
5: 10.36 Other Differential Equations
The quantity λ 2 in (10.13.1)–(10.13.6) and (10.13.8) can be replaced by λ 2 if at the same time the symbol 𝒞 in the given solutions is replaced by 𝒵 . …
10.36.1 z 2 ( z 2 + ν 2 ) w ′′ + z ( z 2 + 3 ν 2 ) w ( ( z 2 + ν 2 ) 2 + z 2 ν 2 ) w = 0 , w = 𝒵 ν ( z ) ,
10.36.2 z 2 w ′′ + z ( 1 ± 2 z ) w + ( ± z ν 2 ) w = 0 , w = e z 𝒵 ν ( z ) .
6: 12.16 Mathematical Applications
§12.16 Mathematical Applications
For examples see §§13.20(iii), 13.20(iv), 14.15(v), and 14.26. …
7: 10.6 Recurrence Relations and Derivatives
With 𝒞 ν ( z ) defined as in §10.2(ii),
𝒞 ν 1 ( z ) + 𝒞 ν + 1 ( z ) = ( 2 ν / z ) 𝒞 ν ( z ) ,
𝒞 ν 1 ( z ) 𝒞 ν + 1 ( z ) = 2 𝒞 ν ( z ) .
𝒞 ν ( z ) = 𝒞 ν 1 ( z ) ( ν / z ) 𝒞 ν ( z ) ,
For results on modified quotients of the form z 𝒞 ν ± 1 ( z ) / 𝒞 ν ( z ) see Onoe (1955) and Onoe (1956). …
8: 12.6 Continued Fraction
§12.6 Continued Fraction
9: 10.44 Sums
10.44.1 𝒵 ν ( λ z ) = λ ± ν k = 0 ( λ 2 1 ) k ( 1 2 z ) k k ! 𝒵 ν ± k ( z ) , | λ 2 1 | < 1 .
If 𝒵 = I and the upper signs are taken, then the restriction on λ is unnecessary. …
10.44.3 𝒵 ν ( u ± v ) = k = ( ± 1 ) k 𝒵 ν + k ( u ) I k ( v ) , | v | < | u | .
The restriction | v | < | u | is unnecessary when 𝒵 = I and ν is an integer. …
10: 12 Parabolic Cylinder Functions
Chapter 12 Parabolic Cylinder Functions