parabolic cylinder functions
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1: 12.14 The Function
§12.14 The Function
… ►This equation is important when and 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
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►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: , , , and .
…An older notation, due to Whittaker (1902), for is .
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3: 12.2 Differential Equations
§12.2 Differential Equations
►§12.2(i) Introduction
… ►Standard solutions are , , (not complex conjugate), for (12.2.2); for (12.2.3); for (12.2.4), where … ►§12.2(iii) Wronskians
… ►When is real the solution is defined by …4: 14.15 Uniform Asymptotic Approximations
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►Here we introduce the envelopes of the parabolic cylinder functions
, , which are defined in §12.2.
For or , with and nonnegative,
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14.15.24
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14.15.25
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14.15.30
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