PCFs

SearchAdvancedHelp

(0.001 seconds)

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: $U(a,z)$, $V(a,z)$, $\overline{U}(a,z)$, and $W(a,z)$. …

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. …