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12 Parabolic Cylinder FunctionsProperties

§12.9 Asymptotic Expansions for Large Variable


§12.9(i) Poincaré-Type Expansions

Throughout this subsection δ is an arbitrary small positive constant.

As z

12.9.1 U(a,z)e-14z2z-a-12s=0(-1)s(12+a)2ss!(2z2)s,
|phz|34π-δ(<34π) ,
12.9.2 V(a,z)2πe14z2za-12s=0(12-a)2ss!(2z2)s,
|phz|14π-δ(<14π) .
12.9.3 U(a,z)e-14z2z-a-12s=0(-1)s(12+a)2ss!(2z2)s±i2πΓ(12+a)eiπae14z2za-12s=0(12-a)2ss!(2z2)s,
14π+δ±phz54π-δ ,
12.9.4 V(a,z)2πe14z2za-12s=0(12-a)2ss!(2z2)s±iΓ(12-a)e-14z2z-a-12s=0(-1)s(12+a)2ss!(2z2)s,

To obtain approximations for U(a,-z) and V(a,-z) as z combine the results above with (12.2.15) and (12.2.16). See also Temme (2015, Chapter 11).

§12.9(ii) Bounds and Re-Expansions for the Remainder Terms

Bounds and re-expansions for the error term in (12.9.1) can be obtained by use of (12.7.14) and §§13.7(ii), 13.7(iii). Corresponding results for (12.9.2) can be obtained via (12.2.20).