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§10.69 Uniform Asymptotic Expansions for Large Order

Let Uk(p) and Vk(p) be the polynomials defined in §10.41(ii), and

10.69.1 ξ=(1+ix2)1/2.

Then as ν+,

10.69.2 berν(νx)+ibeiν(νx) eνξ(2πνξ)1/2(xe3πi/41+ξ)νk=0Uk(ξ-1)νk,
10.69.3 kerν(νx)+ikeiν(νx) e-νξ(π2νξ)1/2(xe3πi/41+ξ)-νk=0(-1)kUk(ξ-1)νk,
10.69.4 berν(νx)+ibeiν(νx) eνξx(ξ2πν)1/2(xe3πi/41+ξ)νk=0Vk(ξ-1)νk,
10.69.5 kerν(νx)+ikeiν(νx) -e-νξx(πξ2ν)1/2(xe3πi/41+ξ)-νk=0(-1)kVk(ξ-1)νk,

uniformly for x (0,). All fractional powers take their principal values.

All four expansions also enjoy the same kind of double asymptotic property described in §10.41(iv).

Accuracy in (10.69.2) and (10.69.4) can be increased by including exponentially-small contributions as in (10.67.3), (10.67.4), (10.67.7), and (10.67.8) with x replaced by νx.