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36 Integrals with Coalescing SaddlesProperties

§36.11 Leading-Order Asymptotics

With real critical points (36.4.1) ordered so that

36.11.1 t1(𝐱)<t2(𝐱)<<tjmax(𝐱),

and far from the bifurcation set, the cuspoid canonical integrals are approximated by

36.11.2 ΨK(𝐱)=2πj=1jmax(𝐱)exp(i(ΦK(tj(𝐱);𝐱)+14π(1)j+K+1))×|2ΦK(tj(𝐱);𝐱)t2|1/2(1+o(1)).

Asymptotics along Symmetry Lines

36.11.3 Ψ2(0,y)={π/y(exp(14iπ)+o(1)),y+,π/|y|exp(14iπ)(1+i2exp(14iy2)+o(1)),y.
36.11.4 Ψ3(x,0,0) =2π(5|x|3)1/8{exp(22(x/5)5/4)(cos(22(x/5)5/418π)+o(1)),x+,cos(4(|x|/5)5/414π)+o(1),x.
36.11.5 Ψ3(0,y,0) =Ψ3(0,y,0)¯=exp(14iπ)π/y(1(i/3)exp(32i(2y/5)5/3)+o(1)),
y+.
36.11.6 Ψ3(0,0,z) =Γ(13)|z|1/33+{o(1),z+,2π51/4(3|z|)3/4(cos(23(3|z|5)5/214π)+o(1)),z.
36.11.7 Ψ(E)(0,0,z) =πz(i+3exp(427iz3)+o(1)),
z±,
36.11.8 Ψ(H)(0,0,z) =2πz(1i3exp(127iz3)+o(1)),
z±.