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12 Parabolic Cylinder FunctionsProperties

§12.11 Zeros

  1. §12.11(i) Distribution of Real Zeros
  2. §12.11(ii) Asymptotic Expansions of Large Zeros
  3. §12.11(iii) Asymptotic Expansions for Large Parameter

§12.11(i) Distribution of Real Zeros

If a12, then U(a,x) has no real zeros. If 32<a<12, then U(a,x) has no positive real zeros. If 2n32<a<2n+12, n=1,2,, then U(a,x) has n positive real zeros. Lastly, when a=n12, n=1,2, (Hermite polynomial case) U(a,x) has n zeros and they lie in the interval [2a,2a]. For further information on these cases see Dean (1966).

If a>12, then V(a,x) has no positive real zeros, and if a=322n, n, then V(a,x) has a zero at x=0.

§12.11(ii) Asymptotic Expansions of Large Zeros

When a>12, U(a,z) has a string of complex zeros that approaches the ray phz=34π as z, and a conjugate string. When a>12 the zeros are asymptotically given by za,s and za,s¯, where s is a large positive integer and

12.11.1 za,s=e34πi2τs(1iaλs2τs+2a2λs28a2λs+4a2+316τs2+O(λs3τs3)),


12.11.2 τs=(2s+12a)π+iln(π122a12Γ(12+a)),


12.11.3 λs=lnτs12πi.

When a=12 these zeros are the same as the zeros of the complementary error function erfc(z/2); compare (12.7.5). Numerical calculations in this case show that z12,s corresponds to the sth zero on the string; compare §7.13(ii).

§12.11(iii) Asymptotic Expansions for Large Parameter

For large negative values of a the real zeros of U(a,x), U(a,x), V(a,x), and V(a,x) can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). For example, let the sth real zeros of U(a,x) and U(a,x), counted in descending order away from the point z=2a, be denoted by ua,s and ua,s, respectively. Then

12.11.4 ua,s212μ(p0(α)+p1(α)μ4+p2(α)μ8+),

as μ (=2a) , s fixed. Here α=μ43as, as denoting the sth negative zero of the function Ai (see §9.9(i)). The first two coefficients are given by

12.11.5 p0(ζ)=t(ζ),

where t(ζ) is the function inverse to ζ(t), defined by (12.10.39) (see also (12.10.41)), and

12.11.6 p1(ζ)=t36t24(t21)2+548((t21)ζ3)12.

Similarly, for the zeros of U(a,x) we have

12.11.7 ua,s212μ(q0(β)+q1(β)μ4+q2(β)μ8+),

where β=μ43as, as denoting the sth negative zero of the function Ai and

12.11.8 q0(ζ)=t(ζ).

For the first zero of U(a,x) we also have

12.11.9 ua,1212μ(11.85575 708μ4/30.34438 34μ8/30.16871 5μ40.11414μ16/30.0808μ20/3),

where the numerical coefficients have been rounded off.

For further information, including associated functions, see Olver (1959).