# PCFs

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## 6 matching pages

##### 1: 12.16 Mathematical Applications
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). … 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. … PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs. …
##### 2: 12.18 Methods of Computation
Because PCFs are special cases of confluent hypergeometric functions, the methods of computation described in §13.29 are applicable to PCFs. …
##### 3: 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: $U\left(a,z\right)$, $V\left(a,z\right)$, $\overline{U}\left(a,z\right)$, and $W\left(a,z\right)$. …
##### 4: 12.17 Physical Applications
The main applications of PCFs in mathematical physics arise when solving the Helmholtz equation … 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. …
##### 5: 12.2 Differential Equations
PCFs are solutions of the differential equation …
##### 6: 12.10 Uniform Asymptotic Expansions for Large Parameter
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. …