Physics 4 Spring 16 Method of sttionry phse Lecture notes by M. G. Rozmn Lst modified: April 13, 16 There is n immedite generliztion of the Lplce integrls f t)e φt) dt 1) which we obtin by llowing the function φt) in Eq. 1) to be comple. We my ssume tht f t) is rel; if it were comple, f t) could be decomposed into sum of its rel nd imginry prts. However, llowing φt) to be comple poses nontrivil problems. We consider the specil cse in which φt) is pure imginry: φt) = iψt) where ψt) is rel. The resulting integrl I) = f t)e ψt) dt ) with f t), ψt),, b, ll rel is clled generlized Fourier integrl. When ψt) = t, I) is n ordinry Fourier integrl. The method of sttionry phse gives the leding symptotic behvior of generlized Fourier integrls hving sttionry points, ψ =. This method is similr to Lplce s method in tht the leding contribution to I) comes from smll intervl surrounding the sttionry points of ψ. Recll tht e ±iu du = π e ±i π 4, e ±iu du = π e±i π 4. 3) Emple 1. Find the leding term of the symptotics of the following integrl for : 4 I) = cos sinh ) ) 1 + d. 4) 3 Since only smll, such tht 1 1 re importnt, sinh, 5) Pge 1 of 6 1641316
Physics 4 Method of sttionry phse Spring 16 1..8.6.4 Figure 1: The grphs of the oscillting fctor, cos sinh ) ) in Eq. 4), for = 1, 4. ) ) sinh ) cos 4 sinh ) cos.. -. -.4 -.6 -.8-1. 1..8.6.4.. -. -.4 -.6 -.8-1. -. -1.. 1.. -. -1.. 1.. New integrtion vrible, cos sinh ) ) cos ) = Re e i 6) 1 + 1. 7) 4 I) Re 3 e i d Re e i d. 8) u = = u = u d = 1 du. 9) I) Re 1 e iu du = } {{ } πe i π 4 π Re ) e i π 4 } {{ } 1 = π 1) Pge of 6 1641316
Physics 4 Method of sttionry phse Spring 16 1. I) = 4 3 cos sinh )) 1 + d.9 Figure : Asymptotics Eq. 1) solid line) compred to numericlly evluted Eq. 4) dshed line) for 1. I).8.7.6.5.4.3. 4. 6. 8. 1. 1. Emple. Find the leding term of the symptotics of the Bessel function J ) for : J ) = 1 π cos cos θ) dθ 11) Bessel function J ) is solution of the following second order liner differentil eqution: y + y + y =. 1) Let s show first tht Eq. 11) is indeed solution of Eq. 1). d d J ) = 1 π sin cos θ) cos θ dθ, 13) d d J ) = 1 π cos cos θ) cos θ dθ. 14) Pge 3 of 6 1641316
Physics 4 Method of sttionry phse Spring 16 ) d d J ) + J ) = π = π 1 cos θ ) cos cos θ) dθ sin θ cos cos θ) dθ = 1 π sin θ cos cos θ) d cos θ) = 1 π sin θ d sin cos θ)) = 1 π sin θ sin cos θ) π π + 1 π sin cos θ) cos θ dθ = d d J ), 15) which is indeed in greement with Eq. 1)..8 J ) = 1 π π/ π/ cos cos)) d.6 Figure 3: Asymptotics Eq. 18) solid line) compred to numericlly evluted Eq. 11) dshed line) for 1. J).4.. -. -.4 Let s rewrite integrl Eq. 11) in the eponentil form: -.6 4 6 8 1 1 14 16 18 J ) = 1 π Re e cos θ dθ. 16) The sttionry point of the phse fctor is t θ =. Only smll θ contribute to the integrl. Therefore. cos θ 1 θ. 17) Pge 4 of 6 1641316
Physics 4 Method of sttionry phse Spring 16 J ) 1 π = 1 π Re e θ i 1 e dθ π Re e e i θ d Re e ) πe i π 4 = π cos π ) 4 ) θ 18) Integrtion by prts If ψt) in the integrl Eq. ) hs no sttionry point, ψ t) =, in the integrtion rnge [, b], the method of sttionry phse is not pplicble. In this cse simple integrtion by prts gives the leding symptotic behviour. I) = = 1 f t)e ψt) dt = 1 f t) b ψ t) eψt) 1 f t) ψ t) d e ψt)) ) d f t) dt ψ t) e ψt) dt. 19) The integrl on the right vnishes more rpidly thn 1/ Riemnn Lebesgue lemm). Therefore, s. I) 1 f t) b ψ t) eψt) ) Emple 3. cost) I) = 1 + t Integrting the lst integrl by prts, we obtin e t 1 + t dt = 1 dt = Re 1 1 + t d e t) = 1 ) e 1 e t dt. 1) 1 + t + 1 e t dt. ) 1 + t) The lst term on the right is see below), therefore the leding term in the pproimtion of Eq. 1) when is { )} 1 e I) Re 1 = sin). 3) We cn continue the integrtion by prts of the integrl in the right hnd side of Eq. ): e t 1 + t) dt = 1 1 1 + t) d e t) = 1 ) e 4 1 + e t dt. 4) 1 + t) 3 Thus, e t 1 + t dt = 1 ) e 1 1 ) e 4 1 e t dt. 5) 1 + t) 3 Pge 5 of 6 1641316
Physics 4 Method of sttionry phse Spring 16 The lst term in the right hnd side of Eq. 5) is of order 3 nd cn be neglected, therefore { ) 1 e I) Re 1 1 )} e 4 1 = sin) 1 ) cos) 1 6) 4.8 I) = 1 cost) 1+t dt Figure 4: Asymptotics Eq. 3) dshed line) nd Eq. 6) dotted line) compred to numericlly evluted Eq. 1) solid line) for 8. I).6.4.. -. -.4 -.6 8. 1. 1. 14. 16. 18.. References [1] Lorell M. Jones. An introduction to mthemticl methods of physics. Benjmin Cummings, 1979. [] Crl M. Bender nd Steven A. Orszg. Advnced Mthemticl Methods for Scientists nd Engineers. Springer Verlg, 1999. Pge 6 of 6 1641316