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1: 12.14 The Function W ( a , x )
§12.14 The Function W ( a , x )
This equation is important when a and z ( = x ) are real, and we shall assume this to be the case. …
Bessel Functions
Confluent Hypergeometric Functions
§12.14(x) Modulus and Phase Functions
2: 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 ) . …
3: 12.2 Differential Equations
§12.2 Differential Equations
§12.2(i) Introduction
Standard solutions are U ( a , ± z ) , V ( a , ± z ) , U ¯ ( a , ± x ) (not complex conjugate), U ( - a , ± i z ) for (12.2.2); W ( a , ± x ) for (12.2.3); D ν ( ± z ) for (12.2.4), where …
§12.2(iii) Wronskians
When z ( = x ) is real the solution U ¯ ( a , x ) is defined by …
4: 14.15 Uniform Asymptotic Approximations
Here we introduce the envelopes of the parabolic cylinder functions U ( - c , x ) , U ¯ ( - c , x ) , which are defined in §12.2. For U ( - c , x ) or U ¯ ( - c , x ) , with c and x nonnegative, …
14.15.24 P ν - μ ( x ) = 1 ( ν + 1 2 ) 1 / 4 2 ( ν + μ ) / 2 Γ ( 1 2 ν + 1 2 μ + 3 4 ) ( ζ 2 - α 2 x 2 - a 2 ) 1 / 4 ( U ( μ - ν - 1 2 , ( 2 ν + 1 ) 1 / 2 ζ ) + O ( ν - 2 / 3 ) env U ( μ - ν - 1 2 , ( 2 ν + 1 ) 1 / 2 ζ ) ) ,
14.15.25 Q ν - μ ( x ) = π ( ν + 1 2 ) 1 / 4 2 ( ν + μ + 2 ) / 2 Γ ( 1 2 ν + 1 2 μ + 3 4 ) ( ζ 2 - α 2 x 2 - a 2 ) 1 / 4 ( U ¯ ( μ - ν - 1 2 , ( 2 ν + 1 ) 1 / 2 ζ ) + O ( ν - 2 / 3 ) env U ¯ ( μ - ν - 1 2 , ( 2 ν + 1 ) 1 / 2 ζ ) ) ,
14.15.30 P ν - μ ( x ) = 1 ( ν + 1 2 ) 1 / 4 2 ( ν + μ ) / 2 Γ ( 1 2 ν + 1 2 μ + 3 4 ) ( ζ 2 + α 2 x 2 + a 2 ) 1 / 4 U ( μ - ν - 1 2 , ( 2 ν + 1 ) 1 / 2 ζ ) ( 1 + O ( ν - 1 ln ν ) ) ,
5: 12.16 Mathematical Applications
§12.16 Mathematical Applications
For examples see §§13.20(iii), 13.20(iv), 14.15(v), and 14.26. …
6: 12.15 Generalized Parabolic Cylinder Functions
§12.15 Generalized Parabolic Cylinder Functions
7: 12 Parabolic Cylinder Functions
Chapter 12 Parabolic Cylinder Functions
8: 12.6 Continued Fraction
§12.6 Continued Fraction
9: 12.20 Approximations
§12.20 Approximations
10: 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