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11: 12.15 Generalized Parabolic Cylinder Functions
§12.15 Generalized Parabolic Cylinder Functions
12: 12.20 Approximations
§12.20 Approximations
13: 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. …
14: 10.66 Expansions in Series of Bessel Functions
§10.66 Expansions in Series of Bessel Functions
10.66.1 ber ν x + i bei ν x = k = 0 e ( 3 ν + k ) π i / 4 x k J ν + k ( x ) 2 k / 2 k ! = k = 0 e ( 3 ν + 3 k ) π i / 4 x k I ν + k ( x ) 2 k / 2 k ! .
15: 10.23 Sums
§10.23(i) Multiplication Theorem
If 𝒞 = J and the upper signs are taken, then the restriction on λ is unnecessary.
§10.23(ii) Addition Theorems
The restriction | v | < | u | is unnecessary when 𝒞 = J and ν is an integer. … The restriction | v e ± i α | < | u | is unnecessary in (10.23.7) when 𝒞 = J and ν is an integer, and in (10.23.8) when 𝒞 = J . …
16: 12.17 Physical Applications
§12.17 Physical Applications
By using instead coordinates of the parabolic cylinder ξ , η , ζ , defined by … Dean (1966) describes the role of PCFs in quantum mechanical systems closely related to the one-dimensional harmonic oscillator. … Lastly, parabolic cylinder functions arise in the description of ultra cold atoms in harmonic trapping potentials; see Busch et al. (1998) and Edwards et al. (1999).
17: 12.7 Relations to Other Functions
§12.7(i) Hermite Polynomials
12.7.1 U ( - 1 2 , z ) = D 0 ( z ) = e - 1 4 z 2 ,
§12.7(ii) Error Functions, Dawson’s Integral, and Probability Function
§12.7(iii) Modified Bessel Functions
§12.7(iv) Confluent Hypergeometric Functions
18: 10.25 Definitions
Symbol 𝒵 ν ( z )
Corresponding to the symbol 𝒞 ν introduced in §10.2(ii), we sometimes use 𝒵 ν ( z ) to denote I ν ( z ) , e ν π i K ν ( z ) , or any nontrivial linear combination of these functions, the coefficients in which are independent of z and ν . …
19: 12.2 Differential Equations
§12.2 Differential Equations
§12.2(i) Introduction
§12.2(iii) Wronskians
§12.2(iv) Reflection Formulas
§12.2(v) Connection Formulas
20: 12.3 Graphics
§12.3(i) Real Variables
See accompanying text
Figure 12.3.8: V ( a , x ) , - 2.5 a 2.5 , - 2.5 x 2.5 . Magnify 3D Help
§12.3(ii) Complex Variables