Parabolic cylinder functions - Read
Parabolic cylinder functions - Read
Parabolic cylinder functions - Read
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III. SOLUTIONS OF EQUATION 2<br />
A. Standard solution<br />
The standard solution W (a, x) of Eq. (2) is<br />
where<br />
3<br />
−<br />
W (a, ±x) = 2 4 (<br />
G1<br />
y1 ∓<br />
G3<br />
4<br />
<br />
2G3<br />
y2), (19)<br />
G1 = |Γ(ia + 1<br />
4 )|, G3 = |Γ(ia + 3<br />
)|,<br />
4<br />
(20)<br />
y1 = 1 + a x2<br />
2! + (a2 − 1<br />
2 )x4<br />
4! + (a3 − 7a<br />
2 )x6<br />
6! + (a4 − 11a 2 + 15<br />
4 )x8 + . . . ,<br />
8!<br />
(21)<br />
y2 = x + a x3<br />
3! + (a2 − 3<br />
2 )x5<br />
5! + (a3 − 13a<br />
2 )x7<br />
7! + (a4 − 17a 2 + 63<br />
4 )x9 + . . . ,<br />
9!<br />
(22)<br />
in which the coefficients An of xn<br />
n!<br />
G1<br />
obey the recurrence relation<br />
An+2 = aAn − 1<br />
4 n(n − 1)An−2. (23)<br />
Relations for gamma function of complex argument are given in Appendix A. At x = 0,<br />
<br />
3<br />
− G1<br />
W (a, 0) = 2 4<br />
G3<br />
W ′ 1<br />
−<br />
(a, 0) = −2 4<br />
G3<br />
B. Relations at large values of argument x<br />
,<br />
. (24)<br />
G1<br />
At large values of the argument x, when x ≫ |a|, there are relations<br />
<br />
2k<br />
W (a, x) =<br />
x [s1(a, x) cos γ − s2(a, x) sin γ],<br />
<br />
2<br />
W (a, −x) =<br />
kx [s1(a, x) sin γ + s2(a, x) cos γ], (25)<br />
where<br />
with<br />
k = √ 1 + e 2πa − e πa , k −1 = √ 1 + e 2πa + e πa , (26)<br />
γ = x2<br />
4<br />
− a ln x + π<br />
4<br />
φ<br />
+ , (27)<br />
2<br />
φ = arg Γ(ia + 1<br />
), (28)<br />
2
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