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12 Parabolic Cylinder FunctionsProperties

§12.4 Power-Series Expansions

12.4.1 U(a,z)=U(a,0)u1(a,z)+U(a,0)u2(a,z),
12.4.2 V(a,z)=V(a,0)u1(a,z)+V(a,0)u2(a,z),

where the initial values are given by (12.2.6)–(12.2.9), and u1(a,z) and u2(a,z) are the even and odd solutions of (12.2.2) given by

12.4.3 u1(a,z)=e-14z2(1+(a+12)z22!+(a+12)(a+52)z44!+),
12.4.4 u2(a,z)=e-14z2(z+(a+32)z33!+(a+32)(a+72)z55!+).


12.4.5 u1(a,z)=e14z2(1+(a-12)z22!+(a-12)(a-52)z44!+),
12.4.6 u2(a,z)=e14z2(z+(a-32)z33!+(a-32)(a-72)z55!+).

These series converge for all values of z.