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parabolic cylinder functions

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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
Bessel Functions
Confluent Hypergeometric Functions
§12.14(x) Modulus and Phase Functions
3: 12.16 Mathematical Applications
§12.16 Mathematical Applications
For examples see §§13.20(iii), 13.20(iv), 14.15(v), and 14.26. …
4: 12.15 Generalized Parabolic Cylinder Functions
§12.15 Generalized Parabolic Cylinder Functions
5: 12 Parabolic Cylinder Functions
Chapter 12 Parabolic Cylinder Functions
6: 12.6 Continued Fraction
§12.6 Continued Fraction
7: 12.20 Approximations
§12.20 Approximations
8: 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
9: 12.18 Methods of Computation
§12.18 Methods of Computation
10: 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).