N. Bleistein and R. A. Handelsman. Asymptotic Expansions of Integrals. Dover, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... pole and a moving saddlepoint in the integral representation (cf. 17, 2, 29] In general, when algebraic singularity and saddlepoint may coalesce in the integrand, good approximants are parabolic cylinder functions which are certain weighted integrals of the normal distribution function (cf. [1, 29]) Recurrence relation. It is more convenient to work with E nk : Q n0 nk , since E 0k = E 1k = E 2k = E n0 = 0. 7) Using the combinatorial identities j) j 1)H j = n 1)H n 12 (n 1) 31) 8) j) j 1) n 1 11) j) 2 #(3n ....

....singularity in the s plane corresponding to u = 1, and f(s) Note that #(1) 0 for # 0. The restriction that k L (or, equivalently, # 1) can now be removed (with suitable deformation of the integration contour if necessary) Bleistein s method. To evaluate I 4 , write (cf. [1, 29]) f(s) # 0 # 1 (s i#) s(s i#)F (s) where, by l Hopital s rule, # 0 = f(i#) lim s#i# g(u) i# ds = i# 1 g(1)# # (1) # 1 = f(0) g(1) # # # (1) g(#)# #(1 # ## (#) # . Note that both constants are bounded functions for # near unity. It follows ....

N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Dover Publications, Inc., New York, 1985.

.... two results are based on a fundamental (integral) formula , which explicitly isolates the contribution of factors of degree 1, and the singularity analysis of Flajolet and Odlyzko [13] Other analytic tools used are Selberg s method (cf. 35, 39, 26] the uniform asymptotic methods by Bleistein [6, 7] and Temme [38] and some new techniques. The interest of considering convolution approximations to random discrete distributions is threefold. First, for probabilists, such a consideration suggests further discrete approximations (besides Poisson, binomial, etc) for combinatorial distributions, ....

....etc) for combinatorial distributions, the general intuition being that discrete approximations (as opposed to continuous ones) to discrete structures would usually provide better error estimates. Next, this line of study introduces many intriguing problems for uniform asymptotic analysis (cf. [7, 43]) and would be of special interest to analysts. Third, for combinatorialists and number theorists, the quantitative results demand further structural interpretations and characterizations. Problems like why it is Poisson for small m and negative binomial for large m will shed further light on ....

[Article contains additional citation context not shown here]

N. Bleistein and R. A. Handelsman, Asymptotic expansions of integrals, Dover Publications, Inc., New York, 1986.

....of the convolution law of a Poisson and a negative binomial distributions. The analytic context encountered here is more complicated than the previous one (for M n;d ) and consists of a saddlepoint and a pole of order q in the integrand. Thus the appearance of D Gammaq (x) is quite expected; see [5, 31]. An intuitive interpretation of the result is that when Y n is large, most of the irreducible factors are of degree 1. Thus we can write Y n n Z n , where Y n and Z n count the number of irreducible factors of degree 2 and = 1, respectively, and prove that the Poisson behavior comes from ....

N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Second Edition, Dover Publications Inc., New York (1986).

.... pole and a moving saddlepoint in the integral representation (cf. 17, 2, 29] In general, when algebraic singularity and saddlepoint may coalesce in the integrand, good approximants are parabolic cylinder functions which are certain weighted integrals of the normal distribution function (cf. [1, 29]) Recurrence relation. It is more convenient to work with E nk : Q n0 Q nk ; since E 0k = E 1k = E 2k = E n0 = 0: 7) Using the combinatorial identities j(n 1 j) j 1)H j = n 1)H n 12 (n 1) 1 (8) j(n 1 j) j 1) n 1 j(n 1 j) 2 (3n 8) we obtain ....

....in the s plane corresponding to u = 1, and f(s) g(u) 1 s) Note that (1) 0 for 0. The restriction that k L (or, equivalently, 1) can now be removed (with suitable deformation of the integration contour if necessary) Bleistein s method. To evaluate I 4 , write (cf. [1, 29]) f(s) 0 1 (s i ) s(s i )F (s) where, by l H opital s rule, 0 = f(i ) lim s i g(u) i s) ds = i g(1) 1) 1 = 0 f(0) i g(1) 1) g( 1 ) 00 ( Note that both constants are bounded functions for near unity. It follows that ....

N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Dover Publications, Inc., New York, 1985.

.... two results are based on a fundamental (integral) formula , which explicitly isolates the contribution of factors of degree 1, and the singularity analysis of Flajolet and Odlyzko [13] Other analytic tools used are Selberg s method (cf. 35, 39, 26] the uniform asymptotic methods by Bleistein [6, 7] and Temme [38] and some new techniques. The interest of considering convolution approximations to random discrete distributions is threefold. First, for probabilists, such a consideration suggests further discrete approximations (besides Poisson, binomial, etc) for combinatorial distributions, ....

....etc) for combinatorial distributions, the general intuition being that discrete approximations (as opposed to continuous ones) to discrete structures would usually provide better error estimates. Next, this line of study introduces many intriguing problems for uniform asymptotic analysis (cf. [7, 43]) and would be of special interest to analysts. Third, for combinatorialists and number theorists, the quantitative results demand further structural interpretations and characterizations. Problems like why it is Poisson for small m and negative binomial for large m will shed further light on ....

[Article contains additional citation context not shown here]

N. Bleistein and R. A. Handelsman, Asymptotic expansions of integrals, Dover Publications, Inc., New York, 1986.

....terms of the convolution law of a Poisson and a negative binomial distributions. The analytic context encountered here is more complicated than the previous one (for M n,d ) and consists of a saddlepoint and a pole of order q in the integrand. Thus the appearance of D q (x) is quite expected; see [5, 31]. An intuitive interpretation of the result is that when Y n is large, most of the irreducible factors are of degree 1. Thus we can write Y n = Y # n Z n , where Y # n and Z n count the number of irreducible factors of degree 2 and = 1, respectively, and prove that the Poisson behavior comes ....

N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Second Edition, Dover Publications Inc., New York (1986).

....2 gives access to successive lower order terms of the polynomials A k (t) near t = 0. Here, we develop a more sophisticated analysis based on a method of coalescent saddle points whose principles originate with Chester, Friedman, and Ursell [6] and which is exposed in classical treatises like [4, 25, 33]. In particular, we follow closely the treatment o ered by Olver in [25, p. 352 356] Proceeding along these lines, we establish below the existence of an expansion (31) Q(z; q) B k (t; 0 that is valid for t in a closed subinterval of (0; 1] for instance t 2 [ 4 ; 1] It will ....

Bleistein, N., and Handelsman, R. A. Asymptotic expansions of integrals, second ed. Dover Publications Inc., New York, 1986.

....the change of variables u x = 2 1, v y = 2 1 in (2) and find that I u v f u v f u v du dv n n 1 1 4 4 2 2 1 1 1 1 ( log( where f u v u v ( 1 1 . Since 1 2 2 4 = w w O w exp( as w 0, it follows by Laplace s method (see [3], p. 322) that as n (12) I log 4 3 4 which contradicts (iii) if g Q , since then log S d I n n n = 2 for n large. This completes the proof. Corollary 8. If log log S S n n 16 1 infinitely often when d d n n 2 2 2 = then g is irrational. Indeed, if the inequality holds ....

N. Bleistein and R. Handelsman, Asymptotic expansion of integrals, Holt, Rinehart and Winston, 1975.

....that log[F (t) Gamma t = log Z t O(t ) t 0: 3.3) We first consider the limit n; k 1, with n2 j fixed and 0 b. Scaling z = n , 3.1) becomes nf( d (3.4) f( Gamma log 1 : 3.5) We evaluate (3. 4) by the saddle point method (cf. [3, 22]) We can easily show that the equation d f( 0, i.e. b Gamma 1) 1 Gamma ( 0; 3.6) 15 has a unique solution on the real axis. We call it 0 = 0 ( b) It satisfies 0 1 Gamma 1 0; 0 1 b: Using Stirling s formula to simplify ....

....1) 1 Gamma ( 0; 3.6) 15 has a unique solution on the real axis. We call it 0 = 0 ( b) It satisfies 0 1 Gamma 1 0; 0 1 b: Using Stirling s formula to simplify n in (3. 4) and evaluating the integral by the standard saddle point method (cf. [3]) yields n Gamman 1 2 ( 0 ; n nf( 0 ; 3.7) Also, by (3.4) and (3.5) we can show that ( 0 ; Gamma [1 ( Gamma b) 0 Gamma 1) 0 (3.8) with which (3.7) becomes the same as Theorem 2, part (ii) Next we consider the limit n = O(1) ....

N. Bleistein and R. Handelsman, Asymptotic Expansions of Integrals, Dover Publications, New York 1986.

....walks [9] A key advantage of the SchrSdinger approach is that the Fourier integrals for the amplitudes are amenable to analysis in standard ways. There is a well developed theory of the asymptotic expansion of integrals that allows us to determine the behavior of the wave function in the limit [4, 5]. This gives another asymptotic form for the probability distribution. The SchrSdinger approach is also quite general and could be potentially applied to quantum walks on any Cayley graph. Related work Various quantum variants of random walks have previously been studied by a few authors [6, 12, ....

....the Hadamard walk. It is a counterpart of the SchrSdinger approach in quantum mechanics. The basic result is the following lemma. LEMMA T. We have b(n,t) dk ie i 2r x 1 cos 2 k i(wkt kn ) 1 x l cos 2k] sin k where wk sin 1 G [ Using the Method of Stationary Phase [4, 5], it is possi ble to derive the asymptotic form of the amplitudes from their integral representation, and hence also the form of the probability distribution P(n, t) Results for semi infinite and finite Hadamard walks While there are several questions one could ask about about the ....

N. Bleistein and R. Handelsman. Asymptotic Expansions of Integrals. Holt, Rinehart and Winston, New York, 1975.

.... we are able to improve on the complexity of the best known random samplers for multiply connected planar graphs and convex polyhedra from [44] The analysis that we introduce is largely based on a method of coalescing saddle points that was perfected in the 1950 s by applied mathematicians [3, 8, 52] and has found scattered applications in statistical physics and the study of phase transitions [41] However, this method does not appear to have been employed so far in the eld of random combinatorics. We claim some generality for the approach Date: August 31, 2001. Key words and phrases. ....

Bleistein, N., and Handelsman, R. A. Asymptotic Expansions of Integrals. Dover, New York,

....are used in the theory [9] and [10] ca To read a very good book about mathematical methods for wave problems, especially asymptotic ones, the reader is referred to [11] Asymptotic methods can be approached from the mathematical analysis point of view. To do this the reader is referred to [2] [12] and [13] There is a good, but rough, example of how a asymptotic expansion can be interpreted in [14] Of course when Helmholz equation is to be applied to Maxwell s equation there is a need for some knowledge in theoretical electromagnetics. Here [15] and [16] can be useful to introduce the ....

R. A. Handelsman N. Bleistein. Asymptotic expansion of integrals. Dover Publications, Inc., New York, 1975.

....equation (2.2) In practice, finding the optimal exactly is infeasible andsome approximation is required. As in Section 1, let us write F(z) for log G(z) z # D, so that equation (2.5) becomes min E h e 2F(Z) # Z (1 2) 1 D i . 2. 6) The classical Laplace method for integrals (e.g. Bleistein and Handelsman 1975, Chap. 8) suggests that, for any fixed , E h e 2F(Z) Z (1 2) 1 D i # (2#) n 2 Z D e 2F(z) z (1 2) e (1 2)z z dz # constant exp max z#D 2F(z) # z 1 2 # 1 2 z # z . 122 GLASSERMAN, HEIDELBERGER, AND SHAHABUDDIN Substituting this approximation into ....

BLEISTEIN, N., and R. A. HANDELSMAN (1975): Asymptotic Expansions of Integrals. New York: Holt, Rinehart, and Winston.

....walks [9] A key advantage of the Schr odinger approach is that the Fourier integrals for the amplitudes are amenable to analysis in standard ways. There is a well developed theory of the asymptotic expansion of integrals that allows us to determine the behavior of the wave function in the limit [4, 5]. This gives another asymptotic form for the probability distribution. The Schr odinger approach is also quite general and could be potentially applied to quantum walks on any Cayley graph. Related work Various quantum variants of random walks have previously been studied by a few authors [6, ....

....result is the following lemma. Lemma 7. We have L(n; t) Z dk 2 ie ik p 1 cos 2 k e i( k t kn) R (n; t) Z dk 2 1 cos k p 1 cos 2 k e i( k t kn) where k = sin 1 sin k p 2 2 [ 2 ; 2 ] Using the Method of Stationary Phase [4, 5], it is possible to derive the asymptotic form of the amplitudes from their integral representation, and hence also the form of the probability distribution P (n; t) Results for semi infinite and finite Hadamard walks While there are several questions one could ask about about the semi in nite ....

N. Bleistein and R. Handelsman. Asymptotic Expansions of Integrals. Holt, Rinehart and Winston, New York, 1975.

....u) Z Gamma A( e Gammai u e iqr d ; 6.4) where q = p 2 Gamma 1 Gamma , A( o( as Sigma1 and apart from slight deformations, the contour Gamma is of infinite extent, running from Re( Gamma1 to Re( 1. Thus the integral is of the form examined in x9.5 of [21]; the results obtained are as follows. See the given reference for more details. The relevant limit of (6.4) is r 1, u finite. Then one may assume (for u positive; u negative leads to similar results) ffl j u r 2 [0; ff) 6.5) where ff 1. A uniform asymptotic expansion for (6.4) with ffl ....

....ffl) b 1 (ffl)I 1 (r; ffl) 6.6) where b n = O(1) as ffl 0, and I n = Gamma2i(2fle Gammai=2 ) n J n (flr) 6.7) with fl = ffl 2 ffl) 1=2 . The constant will be strictly positive (as a consequence of (6. 3) and for a source consisting of an oscillator at the origin, 1 [21]. Since the result (6.6) is uniform in ffl, one can substitute ffl = u=r and obtain (using asymptotics of Bessel functions [18] I(r; u) O(r Gamma1=4 ) 6.8) This is the behaviour in the limiting case = 0; the more reasonable = 1 yields I(r; u) O(r Gamma3=4 ) 6.9) In any case, for ....

N. Bleistein and R.A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1975).

....hard integrals are of the form f(x) # # # g(t) exp(xh(t) dt , 5.82) 30 and it is necessary to estimate the behavior of f(x) as x ##, with the functions g(t) h(t) and the limits of integration # and # held fixed. There is a substantial theory of such integrals, and good references are [54, 63, 100, 315]. The basic technique is usually referred to as Laplace s method, and consists of approximating the integrand by simpler functions near its maxima. This approach is similar to the one that is discussed at length in Section 5.1 for estimating sums. The contributions of the approximations are then ....

....functions h(t) that behave near t = # like h(#) c(t #) for any 0. When the integral is highly oscillatory, as happens when h(t) iu(t) for a real valued function u(t) still other techniques (such as the stationary phase method) are used. We will not present them here, and refer to [54, 63, 100, 315] for descriptions and applications. In Section 12.1 we will discuss the saddle point method, which is related to both Laplace s method and the stationary phase method. Laplace integrals F (x) # # 0 f(t) exp( xt)dt (5.85) can often be approximated by integration by parts. We have (under ....

[Article contains additional citation context not shown here]

N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, 2nd edition, Holt, Rinehart and Winston, New York, 1975. 163

....The only unusual feature is that the phase is neither real nor purely imaginary. This presents no diculties, but it does necessitate the statement of a result in Section 5 that is a little di erent from the usual results on purely oscillating integrals, found in, for example, Stein (1993) or Bleistein Handelsman (1986). We rst establish that b = 0 is a stationary phase point for the function f when r 2 dir(z) Lemma 4.2. The quantity f(0) always vanishes. If r 2 dir(z) then r f(0) 0 and the real part of f has a strict minimum at 0. Proof. The rst statement is immediate. To prove the ....

Bleistein, N. & Handelsman, R. A. (1986), Asymptotic expansions of integrals, second edn, Dover Publications Inc., New York.

....computing the stationary phase point in the method of stationary phase. Therefore we obtain p Z ( Gammai= Delta Gamma1=2 A (7. 47) Theta e Gamma R z 0 [ 1;zz = 2fl 1 ) cos( Gamma1 ds e Gamma( 2 R z 0 D cos( Gamma1 ds) e i (S Gammat) d as # 0; see [6]. The quantity Delta = Delta(x; z; is the determinant of the Hessian of S with respect to . The above expression is evaluated at the stationary phase point defined as in Appendix F. Modulo the random phase factor and the Gaussian spreading factor, the expression (7.47) is the high ....

N. Bleistein and R. Handelsman. Asymptotic expansions of integrals. Dover Publications, 1986.

....the phase is neither real nor purely imaginary. This presents no diculties, but it does necessitate the statement of a result in Section 5 that is a little di erent from the usual results on purely 14 ROBIN PEMANTLE AND MARK C. WILSON oscillating integrals, found in, for example, Stein (1993) or Bleistein Handelsman (1986). We rst establish that b = 0 is a stationary phase point for the function f when r 2 dir(z) Lemma 4.2. The quantity f(0) always vanishes. If r 2 dir(z) then r f(0) 0 and the real part of f has a strict minimum at 0. Proof. The rst statement is immediate. To prove the ....

Bleistein, N. & Handelsman, R. A. (1986), Asymptotic expansions of integrals, second edn, Dover Publications Inc., New York.

....( 2) 2k (50) for some k 2 Z not depending on . Actually is periodic as we will see below (see Eq. 59) below) The main step in proving Theorem 1 is to estimate these integrals with the aid of a theorem which originates from Airy. We have used the form which appear in [6] and [2]: Theorem 2 (Airy) Let f be a real valued C 1 function near 0 in R 2 such that f = 2 f 2 = 0, but 3 f 3 6= 0 at 0. Then there exists C 1 real valued functions a (g) b (g) near 0 such that a (0) 0, b (0) f (0; 0) and fi fi fi fi Z u ( g) e ivf( g) d ....

N. Bleistein and R. A. Handelsman. Asymptotic Expansions of Integrals. Holt, Rinehart and Winston, New York, 1975.

.... ) det h 2 S x j x k (x ) i : Moreover, if F and S (and x ) depend in addition smoothly on some additional parameter y 2 D 0 R n 0 , then the asymptotic expansion (30) holds uniformly as y ranges over a compact subset of D 0 . See [Fed] Theorems II.4.1 and II.4. 4, or [BH], Section 8.3. As we have already observed in (15) owing to the strict plurisubharmonicity of log the function x 7 (x) y) j (x; y)j 2 has a strict local maximum at x = y. Diminishing U if necessary, we may thus assume that the function S(x) log (x) y) j (x; y)j 2 (31) peaks ....

N. Bleistein, R.A. Handelsman, Asymptotic expansions of integrals, Dover Publ., New York, 1986.

....In Appendix G of [38] we show that we thus can ignore when computing the stationary phase point in the method of stationary phase. Therefore p Z ( i= 1=2 A (7. 46) e R z 0 [ 1;zz = 2 1 ) cos( 1 ds e ( 2 R z 0 D cos( 1 ds) e i (S t) d as # 0; see [6]. The quantity = x; z; is the determinant of the Hessian of S with respect to . The above expression is evaluated at the stationary phase point de ned as in Appendix F of [38] Modulo the random phase factor and the Gaussian spreading factor, the expression (7.46) is the high ....

N. Bleistein and R. Handelsman. Asymptotic expansions of integrals. Dover Publications, 1986.

....1 to a Gamma 1; for example, if a = 0, a Gamma 1 = n Gamma 1. We note that Delta is invertible only for cycles which are not marginal, j c j 6= 1. The j c j = 1 case we would require going beyond the Gaussian saddlepoints studied here, and typically to the Airy function type stationary points [25]. 4 Weak noise perturbation expansion The saddlepoint approximation (8) is a discrete path integral on periodic chain of n points which we shall evaluate by standard field theoretic methods. Separating the quadratic terms we obtain e Wc = 1 j c Gamma 1j Z [d ] e GammaS 0 ( GammaS I ....

N. Bleistein and R. A. Handelsman. Asymptotic Expansions of Integrals. Dover, New York, 1986.

....fY n g: Y n is o p (a n ) if Y n =a n converges in probability to 0 as n 1, and is O p (a n ) if jY n =a n j is bounded in probability as n 1. Asymptotic expansions are used in many areas of mathematical analysis. Three helpful textbooks on asymptotic expansions are Bleistein and Handelsman [9], Jeffreys [25] and DeBruijn [18] An important feature of asymptotic expansions is that they are not in general convergent series and taking successively more terms from the right hand side of (1) is not guaranteed to improve the approximation to the left hand side. In the study of asymptotic 1 ....

Bleistein, N. and Handelsman, R.A. (1975). Asymptotic Expansions of Integrals. Holt, Rinehart and Winston, New York.

.... or surveys of general methods for computing special functions include [Bre78b,DKK81,Gau75,HCL 68,Luk69b,Luk77b,PT84,PTVF92,Tem78,vdLT84] Other books and articles that provide indepth coverage of pertinent topics include: Ask89, survey of compendia] BG81a,BG81b, Pad e approximations] BH75, asymptotic approximations] Bre91, continued fractions, Pad e approximations] BvI93, Pad e approximations] Cod70, polynomial and rational approximations] Fik68, polynomial and rational approximations] FP68, Chebyshev polynomials] JT80, continued fractions] Kar91, power series] ....

N. Bleistein and R. A. Handelsman, Asymptotic expansions of integrals, Holt, Rinehart and Winston, New York, 1975.

....OF RANDOMLY SCATTERED SIGNALS 21 and hence from (3.15) hp refl (t; x; 0)i Gamma 1 2(2 ) 3 3=2 Z Z Z Gamma I ( GammaL) f( 2 (3. 20) Deltae i ( Deltax Gammat Gamma2 ( GammaL) e Gamma 2 fi 1 (L) d d The integral can be approximated by the stationary phase method [68]. To simplify the formulas and to see more clearly the physical meaning of the results let us assume that c 1 is a constant, which means that there are no large scale variations in the sound speed in the randomly layered region. Then from (2.24) GammaL) GammaL(1 Gamma c 2 1 2 ) 1=2 c ....

N. BLEISTEIN and R. HANDELSMAN, Asymptotic Expansion of Integrals, Dover, New York, 1986.

....n2 Gammak j fixed and 0 b. Scaling z = n , 3.1) becomes h k n = n n n 1 2 i I e nf( d (3.4) where f( Gamma log 1 log 1 2 2 2 Delta Delta Delta b b b : 3.5) We evaluate (3. 4) by the saddle point method (cf. [3, 22]) We can easily show that the equation d d f( 0, i.e. Gamma 1 1 Delta Delta Delta ( b Gamma1 = b Gamma 1) 1 Delta Delta Delta ( b =b = 1 Gamma 1 Gamma ( b b 1 1 Delta Delta Delta ( b =b = 0; 3.6) has a ....

....b =b = 0; 3.6) has a unique solution on the real axis. We call it 0 = 0 ( b) It satisfies 0 1 Gamma b b 1 0; 0 1 b Gamma 1 b: Using Stirling s formula to simplify n in (3. 4) and evaluating the integral by the standard saddle point method (cf. [3]) yields h k n p 2 ne Gamman 1 2 1 0 s 2 f 00 ( 0 ; n e nf( 0 ; 3.7) Also, by (3.4) and (3.5) we can show that f 00 ( 0 ; 1 2 0 1 Gamma 1 0 Gamma b 0 Gamma 1 Gamma 1 0 2 = 1 2 0 [1 ( Gamma b) 0 Gamma ....

N. Bleistein and R. Handelsman, Asymptotic Expansions of Integrals, Dover Publications, New York 1986.

.... or surveys of general methods for computing special functions include [Bre78b,DKK81,Gau75,HCL 68,Luk69b,Luk77b,PT84,PTVF92,Tem78,vdLT84] Other books and articles that provide indepth coverage of pertinent topics include: Ask89, survey of compendia] BG81a,BG81b, Pad e approximations] BH75, asymptotic approximations] Bre91, continued fractions, Pad e approximations] BvI93, Pad e approximations] Cod70, polynomial and rational approximations] Fik68, polynomial and rational approximations] FP68, Chebyshev polynomials] JT80, continued fractions] Kar91, power series] ....

N. Bleistein and R. A. Handelsman, Asymptotic expansions of integrals, Holt, Rinehart and Winston, New York, 1975.

....) x 2 R(x s , x) ##(x, x s , x r ) ## d 2 S x , 3.1) where #(x, x r , x s ) 2 x x (x r x s ) 3.2) We assume that the surface B is parametrized as x = x(# 1 , # 2 ) 3.3) 4 In (3. 1) we hold x r and x s constant, and apply the two dimensional stationary phase formula [BH] # g(z)e ik#(z) d 2 z # # 2# k # # j g(z j ) # det H e ik#(z j ) e (i# 4)sigH , 3.4) where H denotes the Hessian of # evaluated at the stationary point z j , and sigH denotes the signature of H. Eq. 3.4) gives the first term in an expansion in decreasing powers of ....

N. Bleistein and R.A. Handelsman, Asymptotic Expansions of Integrals, Dover, New York, 1986.

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N. Bleistein and R. A. Handelsman. Asymptotic Expansions of Integrals. Dover, 1986.

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N. Bleistein and R. A. Handelsman. Asymptotic expansions of integrals. Dover, 1986.

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N. Bleistein and R. A. Handelsman. Asymptotic expansions of integrals. Dover, 1986.

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N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Holt, Rinehart and Winston, 1975.

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N. Bleistein and R. A. Handelsman. Asymptotic expansions of integrals. Dover, 1986.

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N. Bleistein and R. A. Handelsman. Asymptotic expansions of integrals. Dover, 1986.

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N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Holt, Rinehart and Winston, (1975).

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N. Bleistein and R. A. Handelsman. Asymptotic expansions of integrals. Dover Publications Inc., New York, second edition, 1986.

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Bleistein,N. and Handlesman, R.A. (1986). Asymptotic expansions of integrals. Dover Publications, Inc. New York.

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Norman Bleistein and Richard Handelsman. Asymptotic expansions of integrals. Holt, Rinehart and Winston, 1975.

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Norman Bleistein and Richard A. Handelsman. Asymptotic Expansions of Integrals. Holt, Rinehart and Winston, New York, 1975.

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Bleistein, N. & R.A. Handelsman (1975), Asymptotic expansions of integrals, Holt, Rinehart and Winston, New York.

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N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Dover Publications Inc., Mineola, New York, 1986, pp. 187-199.

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N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Dover Publications Inc., Mineola, New York, 1986, pp. 187-199.

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N. Bleistein and R.A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York 1986).

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N. Bleistein and R.A. Handelsman, Asymptotic Expansions of Integrals (Dover Publications Inc., New York, 1986).

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N. Bleistein and R. A. Handelsman. Asymptotic Expansions of Integrals. Holt, Rinehart and Winston, New York, 1975.

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Bleistein, N. & Handelsman, R.A. (1986) Asymptotic Expansions of Integrals. New York: Dover.

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Bleistein, N., and R.A. Handelsman, Asymptotic Expansions of Integrals, Holt, Rinehart, and Winston, New York, 1975.

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Bleistein, N. & Handelsman, R.A. (1986) Asymptotic Expansions of Integrals. New York: Dover.

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N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals. New York: Dover, 1986.

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