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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,K1)(p+1)xp+1anp(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(31/3iz)ncn(x)cn(y)n!+Ai(x)Ai(y)n=2(31/3iz)ncn(x)dn(y)n!+Ai(x)Ai(y)n=2(31/3iz)ndn(x)cn(y)n!+Ai(x)Ai(y)n=1(31/3iz)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,xiy121/3)),

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¯),

and

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