Parabolic cylinder functions - Read

APPENDIX A: RELATIONS FOR GAMMA FUNCTION OF COMPLEX
ARGUMENT
If |z| ≫ 1 and | arg z| ≤ π − ɛ with ɛ > 0, there is relation
ln Γ(z) ∼ (z − 1
1
) ln z − z + ln 2π +
2 2
where B2n are the Bernoulli’s numbers,
Specific values,
Other useful relations,
k−1 2 · (2k)!
B2k = (−1)
(2π) 2k
∞
n=1
n=1
B2 = 1/6 B12 = −691/2730
B4 = −1/30 B14 = 7/6
6
∞ B2n 1
, (A1)
2n(2n − 1) z2n−1 1
, k = 1, 2, . . . (A2)
n2k B6 = 1/42 B16 = −3617/510 (A3)
B8 = −1/30 B18 = 43867/798
B10 = 5/66 B20 = −174611/330
Γ(z + n) = z(z + 1) · · · (z + n − 1)Γ(z),
Γ(z)Γ(−z) = −π
z sin πz
(A4)
[1] H. F. Weber “Ueber die Integration der partiellen Differential-gleichung: ∂ 2 u/∂x 2 + ∂ 2 u/∂y 2 +
k 2 u = 0,” Math. Ann. 1, 1–36 (1869).
[2] J. C. P. Miller “On the choice of standard solutions to Weber’s equation,” Proc. Cambridge
Philos. Soc. 48, 428–435 (1952).
[3] M. Abramowitz and I. Stegun Handbook of Mathematical Functions (New York, 1964).