Asymptotics of integrals

Transcription

1 December 9, Asymptotics o integrals Paul Garrett garrett@math.umn.e garrett/ [This document is garrett/m/v/basic asymptotics.pd]. Heuristic or the main term in asymptotics or Γs. Watson s lemma 3. Watson s lemma and Γs/Γs + a 4. Main term in asymptotics by Laplace s method 5. Stirling s ormula or main term in asymptotics or Γs 6. Laplace s method illustrated on Bessel unctions Standard methods in asymptotic expansions [] o integrals are illustrated: Watson s lemma and Laplace s method. Watson s lemma dates rom at latest [Watson 98a], and Laplace s method at latest rom [Laplace 774]. The exposition here is revisionist, in that Laplace s method is reced to Watson s lemma. For example, a simple argument gives the main term [] in the asymptotics or Γs: Γs π e s s s as s, with Res δ > and obtain a useul result about ratios o gamma unctions, Γs + a Γs s a as s, or ixed a, or Res δ > The latter is awkward to obtain as a corollary rom Stirling s ormula. Laplace s method is urther illustrated by an application to asymptotics o unctions closely related to Bessel unctions, namely, or any ixed spectral parameter ν R, y e u+ u y u iν u π e y as y + One point is avoidance o standard but immemorable arguments special to the gamma unction. O course, these special arguments do bear more orceully upon gamma itsel: see [Whittaker-Watson 97] or [Lebedev 963]. However, to the extent possible, we want to understand the asymptotics o gamma and other important special unctions on general principles. [] The simplest notion o asymptotic F s or s as s goes to + on R, or in a sector in C, is a simpler unction F s such that lim s s/f s, written F. One might require an error estimate, or example, F s F s + O s A more precise orm is to say that c s s s α + c s α+ + c s α with any auxiliary unction, is an asymptotic expansion or when c s s α + c s α cn s α+n + O s α+n+ [] The main term in the asymptotics or Γs is e to Stirling. Higher terms are e to Binet, and perhaps Laplace.

2 Paul Garrett: Asymptotics o integrals December 9,. Heuristic or the main term in asymptotics or Γs A memorable heuristic or Stirling s ormula or the main term in the asymptotics o Γs, namely Γs e s s s π Using Euler s integral, s Γs Γs + e u u s+ u e u u s e u+s log u The trick is to replace the exponent u+s log u by the quadratic polynomial in u best approximating it near its maximum, and evaluate the resulting integral. This replacement can be justiied via Watson s lemma and Laplace s method, below, but the heuristic is simpler than the justiication. More precisely, the exponent takes its maximum where its derivative vanishes, at the unique solution u o s o + s u The second derivative in u o the exponent is s/u, which takes value /s at u o s. Thus, near u o s, the quadratic Taylor-Maclaurin polynomial in t approximating the exponent is Thus, we imagine that s + s log s u s! s s Γs e s+s log s s u s e s s s e s u s Note that the latter integral is taken over the whole real line. [3] u by su and cancel a actor o s rom both sides, To simpliy the remaining integral, replace Γs e s s s e su / Then replace u by u +, and then u by u π/s, obtaining e su / e su / π s e πu π s We obtain Γs e s s s π Several aspects o this heuristic are bious, so it is striking that it can be made rigorous, as below. [3] Evaluation o the integral over the whole line, and simple estimates on the integral over, ], show that the integral over, ] is o a lower order o magnitude than the whole. Thus, the leading term o the asymptotics o the integral over the whole line is the same than the integral rom to +.

3 Paul Garrett: Asymptotics o integrals December 9,. Watson s lemma The oten-rediscovered Watson s lemma [4] gives asymptotic expansions valid in hal-planes in C or Laplace transorm integrals. For example, or smooth h on, + with all derivatives o polynomial growth, and expressible or small x > as hx x α gx or x >, some α C where gx is dierentiable [5] on R near. Thus, hx has an asymptotic expansion at + hx x α c n x n Taylor-Maclaurin asymptotic expansion or x + n Watson s Lemma gives an asymptotic expansion o the Laplace transorm o h: e sx hx dx x Γα c s α + Γα + c s α+ + Γα + c s α or Res > The error estimates below give e sx hx dx x e sx x α gx dx x Γα g s α + O s Re α+ Similar conclusions hold or errors ater inite sum o terms. The idea is straightorward: the expansion is obtained rom e sx hx dx x e sx x α c c n x n dx x + e sx x α gx c c n x n dx x The irst integral gives the asymptotic expansion, and or Res > the second integral can be integrated by parts and trivially bounded to give the error term. To understand the error, let ε be the smallest such that N Reα + n ε Z The subtraction o the initial polynomial and re-allocation o the /x rom the measure makes x α gx c c n x n vanish to order N at. This, with the exponential e sx and the presumed polynomial growth o h and its derivatives, allows integration by parts N times without boundary terms, giving e sx hx dx Γα c s α + Γα + c s α+ + s N Γα + n c n s α+n e sx N x α gx c c n x n dx x [4] This lemma appeared in the treatise [Watson 9] on page 36, citing [Watson 98a], page 33. Curiously, the aggregate bibliography o [Watson 9] omitted [Watson 98a], and the ootnote mentioning it gave no title. Happily, [Watson 98a] is mentioned by title in [Blaustein-Handelsman 975]. In the bibliography at the end, we note [Watson 97], [Watson 98a], [Watson 98b]. [5] g need not be real-analytic near, only smooth to the right o, so it and its derivatives have inite Taylor- Maclaurin expansions approximating it as x +. 3

4 Paul Garrett: Asymptotics o integrals December 9, Although the indicated letover term is typically larger than the last term in the asymptotic expansion, it is smaller than the next-to-last term, so the desired conclusion holds: or hx with asymptotic expansion at + hx x α c n x n Taylor-Maclaurin asymptotic expansion or x + n and it and its derivatives o polynomial growth as h +, the Laplace transorm has asymptotic expansion e sx hx dx Γα c s α + Γα + c s α Γα + n c n s α+n +O s Re α+n+ or n,, 3, Watson s lemma and Γs/Γs + a A useul asymptotic awkward to derive rom Stirling s ormula or Γs, but easy to obtain rom Watson s lemma, is an asymptotic or Euler s beta integral [6] Bs, a x s x a dx Γs Γa Γs + a Fix a with Rea >, and consider this integral as a unction o s. Setting x e u gives an integrand itting Watson s lemma, Bs, a e su e u a e su u u! +...a e su u a u! +...a u Γa s a or ixed a taking just the irst term in an asymptotic expansion, using Watson s lemma. Thus, Γs Γa Γs + a Γa s a or ixed a giving That is, Γs Γs + a s a or ixed a Γs Γs + a s a + O s Re a+ or ixed a [6] We recall how to obtain the expression or beta in terms o gamma. With x u/u + in the beta integral, Bs, a u s u + s a u s u + s a Γs + a u s e vu+ v s+a dv v using e vy v b dv/v Γb/y b. Replacing u by u/v gives Bs, a ΓsΓa/Γs + a. u 4

5 Paul Garrett: Asymptotics o integrals December 9, 4. Main term in asymptotics by Laplace s method Laplace s method [7] obtains asymptotics in s or integrals e s u or real-valued, Res > Inormation attached to u minimizing u dominate. For a unique minimum, at u o, with u o >, the main term o the asymptotic expansion is π e s u e suo u o or s, with Res δ > s This reces to a variant [8] o Watson s lemma, breaking the integral at points where the derivative changes sign, and changing variables to convert each ragment to a Watson-lemma integral. The unction must be smooth, with all derivatives o at most polynomial growth and at most polynomial decay, as u +. [4..] Example: An integral e sy hy dy y is not quite in the orm to apply the simplest version o Watson s lemma. Replacing y by x corrects the exponential e sy hy dy y e sx h x dx x but the asymptotic expansion o h x at + will be in powers o x. This is harmless, by a variety o possible adaptations. [4..] Remark: In act, in the discussion below, the odd powers o x will cancel. For simplicity assume exactly one point u o at which u o, and that u o >, and that u goes to + at + and at +. [9] Since u > or u > u o and u < or < u < u o, there are unctions F, G smooth near such that F u u o u or u o < u < + G u u o u or < u < u o Let y u u o in both integrals, noting that F y u gives dy F y, obtaining integrals almost as in Watson s lemma: e s u e e suo uo suo uo e sy + e sy + u o u o e sy e sy e suo e suo e sy F y + G y dy dy e sy y F y + G y y [7] [8] Perhaps the irst appearance o this is in [Laplace 774]. See [Miller 6] or a thorough discussion o variants o Watson s lemma. [9] The hypothesis o exactly one point u o at which u o, that u o >, and that u goes to + at + and at +, holds in two important examples, namely, u u log u or Euler s integral or Γs, and u u+ u or the Bessel unction. 5

6 Paul Garrett: Asymptotics o integrals December 9, Since F, G are smooth near y, they do have Taylor-Maclaurin asymptotics in y near. To convert the integrals to integrals o the orm in Watson s lemma, replace y by x. This would seem to require extending Watson s lemma to tolerate asymptotic expansion o F x + G x in powers o x, but, in act, the odd powers o x cancel. Derivatives o must increase or decrease only polynomially as u +. An asymptotic near x o the orm F x + G x c + c x + c x + c 3 x as x + ollows rom a Taylor-Maclaurin expansion o F y + G y. Watson s lemma gives asymptotic expansion e s u e suo Γ c s dy e sy y F y+g y y + Γ 3 c s 3 + Γ 5 c s 5 e suo e sx x +... or Res > F x+g dx x x To determine only the leading coeicient F, F y u gives F y dy, so F y / dy. From the derivative is and u u o y u u o u o At y, also u u o, so u o u u o + O u u o 3 /! / u o + Ou u o u u o + Ou u o dy F y dy u o + Ou u o F u o + Ou u o u o The same argument applied to G gives G F, and Watson s lemma gives e s u e suo Γ u o s e suo π u o s Last, this outcome would be obtained by replacing u by its quadratic approximation Integrating over the whole line, e suo u u o + u o! e s u o+ u ou u o e suo e s u ou e suo 6 u u o π u o e s u ou u o s e suo π u o s

7 Paul Garrett: Asymptotics o integrals December 9, This does indeed agree. Last, veriy that the integral o the exponentiated quadratic approximation over, ] is o a lower order o magnitude. Indeed, or u and u o > we have u u o u + u o, and u o > by assumption, so e suo e suo e s u o u o s e u ou u o e suo e s u o u o e s u ou e suo e s u o u o e s u ou π u o Thus, the integral over, ] has an additional exponential decay by comparison to the integral over the whole line. 5. Stirling s ormula or main term in asymptotics or Γs Stirling s ormula or main term in asymptotics or Γs can be obtained in this context. For real s >, replacing u by su expresses Euler s integral or Γs as a proct o an exponential and a Watson s-lemma integral: so s Γs Γs + e u u s e u+s log u e su+s log u+s log s s s e s log s e su log u Γs e s log s e su log u For complex s with Res >, both s Γs and the integral s e s log s e su+log u are holomorphic in s, and they agree or real s. The identity principle gives equality or Res >. With u u log u, the derivative u u has unique zero at u o, and +. Thus, s Γs e s log s e suo π u o s π e s log s e s π e s log s e s s [5..] Remark: As noted earlier, the odd powers o x cancel, so F x + G x has an expansion c + c x + c x +..., and the error estimate in the asymptotic expansion is Γs π e s log s e s + O s 7

8 Paul Garrett: Asymptotics o integrals December 9, 6. Laplace s method illustrated on Bessel unctions The above discussion extends to treat the standard integral [] y y e u+ u y u iν u The exponent u + u y is o the desired orm, with the earlier s replaced by y, but the uiν in the integrand does not it into the simpler Laplace method. Thus, consider integrals e su gu where is real-valued, but g may be complex-valued. The minimum values o u should still dominate, and the leading term o the asymptotics should be assuming a unique minimum at u o e s u gu e suo π guo u o As in the simpler case, rece to Watson s lemma by breaking the integral where changes sign, and change variables to convert each ragment to a Watson-lemma integral. The course o the argument uncovers conditions on and g. For simplicity assume that there is exactly one point u o at which u o, that u o >, and that u goes to + at + and at +. Thus, on these intervals there are smooth square roots u u o and smooth unctions F, G such that F u u o u or u o < u < + With y u u o in both integrals, s G u u o u or < u < u o uo e su gu e suo e s u uo gu + e suo e s u uo gu e suo e s y gf y F y + ggy G y dy The assumptions u o and u o > assure that F y has a Taylor series expansion near, which gives an asymptotic expansion at + gf y F y gf F u o +... small y > [] This integral is almost the Bessel unction K νy e u+ u y/ u ν u The unction K ν is variously called a Bessel unction o imaginary argument or MacDonald s unction or modiied Bessel unction o third kind. Being interested mainly in the case that the parameter ν here is purely imaginary, in the text we replace ν by iν. The leading actor o y arises in applications, where such an integral appears as a Whittaker unction. 8

9 Paul Garrett: Asymptotics o integrals December 9, The exponential e sy is not quite right to apply Watson s lemma, so replace y by x, obtaining an asymptotic expansion in powers o x. The main term o the Watson s lemma asymptotics or the integral involving F would be e sx As beore, F rom F y u and F y dy give F y dy and F u o and G F. Thus, x gf x F x dx x Γ gf F u o + Ou u o s e s u gu e suo Γ gu o u o s e suo π guo u o s Returning to y e u+ u y u iν u we have critical point u o, and u o and u o. Applying the just-derived asymptotic, y e u+ u y u iν u π y e y iν π e y as y + y Even though the exponent is plausible, it would be easy to lose track o the power o y, which might matter. Note that the leading constant does not depend upon the index ν. [Bleistein-Handelsman 975] N. Bleistein, R.A. Handelsman, Asmptotic expansions o integrals, Holt, Rinehart, Winston, 975, reprinted 986, Dover. [Laplace 774] P.S. Laplace, Memoir on the probability o causes o events, Mémoires de Mathématique et de Physique, Tome Sixi eme. English trans. S.M. Stigler, 986. Statist. Sci., [Lebedev 963] N. Lebedev, Special unctions and their applications, translated by R. Silverman, Prentice- Hall 965, reprinted Dover, 97. [Miller 6] P.D. Miller, Applied Asymptotic Analysis, AMS, 6. [Watson 97] G.N. Watson, Bessel unctions and Kapteyn series, Proc. London Math. Soc. xvi 97, , 98. [Watson 98a] G.N. Watson, Harmonic unctions associated with the parabolic cylinder, Proc. Math. Soc. 7 98, London [Watson 98b] G.N. Watson, Asymptotic expansions o hypergeometric unctions, Trans. Cambridge Phil. Soc., 77-38, 98. [Watson 9] G.N. Watson, Treatise on the Theory o Bessel Functions, Cambridge Univ. Press, 9. [Whittaker-Watson 97] E.T. Whittaker, G.N. Watson, A Course o Modern Analysis, Cambridge University Press, 97, 4th edition, 95. 9