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12 Parabolic Cylinder FunctionsProperties

§12.11 Zeros


§12.11(i) Distribution of Real Zeros

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

If a>-12, then V(a,x) has no positive real zeros, and if a=32-2n, 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(1-iaλs2τs+2a2λs2-8a2λs+4a2+316τs2+O(λs3τs-3)),


12.11.2 τs=(2s+12-a)π+iln(π-122-a-12Γ(12+a)),


12.11.3 λs=lnτs-12πi.

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=2-a, 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=μ-43as, 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)=t3-6t24(t2-1)2+548((t2-1)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=μ-43as, 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(1-1.85575 708μ-4/3-0.34438 34μ-8/3-0.16871 5μ-4-0.11414μ-16/3-0.0808μ-20/3-),

where the numerical coefficients have been rounded off.

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