Digital Library of Mathematical Functions
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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((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(14π)+o(1)),y+,π/|y|exp(-14π)(1+2exp(-14y2)+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)5/4-18π)+o(1)),x+,cos(4(|x|/5)5/4-14π)+o(1),x-.
36.11.5 Ψ3(0,y,0) =Ψ3*(0,-y,0)
=exp(14π)π/y(1-(/3)exp(32(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)5/2-14π)+o(1)),z-.
36.11.7 Ψ(E)(0,0,z) =πz(+3exp(427z3)+o(1)),
z±,
36.11.8 Ψ(H)(0,0,z) =2πz(1-3exp(127z3)+o(1)),
z±.