# PCFs

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6 matching pages ♦

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## 6 matching pages

##### 1: 12.16 Mathematical Applications

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►PCFs are used as basic approximating functions in the theory of contour integrals with a coalescing saddle point and an algebraic singularity, and in the theory of differential equations with two coalescing turning points; see §§2.4(vi) and 2.8(vi).
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►Sleeman (1968b) considers certain orthogonality properties of the PCFs and corresponding eigenvalues.
In Brazel et al. (1992) exponential asymptotics are considered in connection with an eigenvalue problem involving PCFs.
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►PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs.
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##### 2: 12.18 Methods of Computation

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►Because PCFs are special cases of confluent hypergeometric functions, the methods of computation described in §13.29 are applicable to PCFs.
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##### 3: 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: $U(a,z)$, $V(a,z)$, $\overline{U}(a,z)$, and $W(a,z)$.
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##### 4: 12.17 Physical Applications

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►The main applications of PCFs in mathematical physics arise when solving the Helmholtz equation
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►Buchholz (1969) collects many results on boundary-value problems involving PCFs.
…Dean (1966) describes the role of PCFs in quantum mechanical systems closely related to the one-dimensional harmonic oscillator.
►Problems on high-frequency scattering in homogeneous media by parabolic cylinders lead to asymptotic methods for integrals involving PCFs.
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##### 5: 12.2 Differential Equations

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►PCFs are solutions of the differential equation
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##### 6: 12.10 Uniform Asymptotic Expansions for Large Parameter

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►In this section we give asymptotic expansions of PCFs for large values of the parameter $a$ that are uniform with respect to the variable $z$, when both $a$ and $z$
$(=x)$ are real.
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