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36 Integrals with Coalescing SaddlesProperties

§36.9 Integral Identities

36.9.1 |Ψ1(x)|2 =25/30Ψ1(22/3(3u2+x))du;
equivalently,
36.9.2 (Ai(x))2 =22/3π0Ai(22/3(u2+x))du.
36.9.3 |Ψ1(x)|2 =8π30u-1/2cos(2u(x+u2)+14π)du.
36.9.4 |Ψ2(x,y)|2 =0(Ψ1(4u3+2uy+xu1/3)+Ψ1(4u3+2uy-xu1/3))duu1/3.
36.9.5 |Ψ2(x,y)|2 =20cos(2xu)Ψ1(2u2/3(y+2u2))duu1/3.
36.9.6 |Ψ3(x,y,z)|2 =24/5-Ψ3(24/5(x+2uy+3u2z+5u4),0,22/5(z+10u2))du.
36.9.7 |Ψ3(x,y,z)|2 =27/451/40(e2iu(u4+zu2+x)Ψ2(27/451/4yu3/4,2u5(3z+10u2)))duu1/4.
36.9.8 |Ψ(H)(x,y,z)|2=8π2(29)1/3--Ai((43)1/3(x+zv+3u2))Ai((43)1/3(y+zu+3v2))dudv.
36.9.9 |Ψ(E)(x,y,z)|2=8π232/3002π(Ai(131/3(x+iy+2zuexp(iθ)+3u2exp(-2iθ)))×Bi(131/3(x-iy+2zuexp(-iθ)+3u2exp(2iθ))))×ududθ.

For these results and also integrals over doubly-infinite intervals see Berry and Wright (1980). This reference also provides a physical interpretation in terms of Lagrangian manifolds and Wigner functions in phase space.