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36 Integrals with Coalescing SaddlesProperties

§36.8 Convergent Series Expansions

36.8.1 ΨK(x) =2K+2n=0exp(iπ(2n+1)2(K+2))Γ(2n+1K+2)a2n(x),
K even,
ΨK(x) =2K+2n=0incos(π(n(K+1)-1)2(K+2))Γ(n+1K+2)an(x),
K odd,

where

36.8.2 a0(x) =1,
an+1(x) =in+1p=0min(n,K-1)(p+1)xp+1an-p(x),
n=0,1,2,.

For multinomial power series for ΨK(x), see Connor and Curtis (1982).

36.8.3 32/34π2Ψ(H)(31/3x)=Ai(x)Ai(y)n=0(-3-1/3iz)ncn(x)cn(y)n!+Ai(x)Ai(y)n=2(-3-1/3iz)ncn(x)dn(y)n!+Ai(x)Ai(y)n=2(-3-1/3iz)ndn(x)cn(y)n!+Ai(x)Ai(y)n=1(-3-1/3iz)ndn(x)dn(y)n!,

and

36.8.4 Ψ(E)(x)=2π2(23)2/3n=0(-i(2/3)2/3z)nn!(fn(x+iy121/3,x-iy121/3)),

where

36.8.5 fn(ζ,ζ)=cn(ζ)cn(ζ)Ai(ζ)Bi(ζ)+cn(ζ)dn(ζ)Ai(ζ)Bi(ζ)+dn(ζ)cn(ζ)Ai(ζ)Bi(ζ)+dn(ζ)dn(ζ)Ai(ζ)Bi(ζ),

with asterisks denoting complex conjugates, and

36.8.6 c0(t) =1,
d0(t) =0,
cn+1(t) =cn(t)+tdn(t),
dn+1(t) =cn(t)+dn(t).