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relation to parabolic cylinder functions

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1: 12.7 Relations to Other Functions
§12.7(i) Hermite Polynomials
§12.7(ii) Error Functions, Dawson’s Integral, and Probability Function
§12.7(iii) Modified Bessel Functions
§12.7(iv) Confluent Hypergeometric Functions
2: 10.39 Relations to Other Functions
Parabolic Cylinder Functions
3: 10.16 Relations to Other Functions
Parabolic Cylinder Functions
4: 13.18 Relations to Other Functions
§13.18(iv) Parabolic Cylinder Functions
5: 12.14 The Function W ( a , x )
Bessel Functions
Confluent Hypergeometric Functions
6: 13.6 Relations to Other Functions
§13.6(iv) Parabolic Cylinder Functions
7: 7.18 Repeated Integrals of the Complementary Error Function
Parabolic Cylinder Functions
Probability Functions
8: 12.17 Physical Applications
Dean (1966) describes the role of PCFs in quantum mechanical systems closely related to the one-dimensional harmonic oscillator. …
9: 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 ) . These notations are due to Miller (1952, 1955). An older notation, due to Whittaker (1902), for U ( a , z ) is D ν ( z ) . The notations are related by U ( a , z ) = D - a - 1 2 ( z ) . …
10: 18.15 Asymptotic Approximations
With μ = 2 n + 1 the expansions in Chapter 12 are for the parabolic cylinder function U ( - 1 2 μ 2 , μ t 2 ) , which is related to the Hermite polynomials via …