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.

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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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