N. Bleistein, Uniform asymptotic expansions of integrals with stationary point near algebraic singularity, Communications on Pure and Applied Mathematics, 19:353--370 (1966).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....b 0 = b 1 = 12(1 . Note that each b j has a removable singularity at r = 1. It should be mentioned that (3) is also derivable by classical methods for uniform asymptotic expansions of integrals having a saddlepoint and a simple pole (one being allowed to approach the other) see [47, 7, 36, 29, 11, 24] and [50, pp. 356 360] In particular, error bounds for (3) are discussed in [29, 39, 40] 4. By the definition of #m (#) 4) which is itself an asymptotic expansion for m = o(#) First of all, from (4) we have roughly # j m # , 5) and we expect that the last ....

N. Bleistein, Uniform asymptotic expansions of integrals with stationary point near algebraic singularity, Communications on Pure and Applied Mathematics, 19:353--370 (1966).

....; b 1 = 12(1 Gamma r) Note that each b j has a removable singularity at r = 1. It should be mentioned that (3) is also derivable by classical methods for uniform asymptotic expansions of integrals having a saddle point and a simple pole (one being allowed to approach the other) see [47, 7, 36, 29, 11, 24] and [50, pp. 356 360] In particular, error bounds for (3) are discussed in [29, 39, 40] 4. By the definition of Pi m ( 4) which is itself an asymptotic expansion for m = o( First of all, from (4) we have roughly e 1 Gamma m= 5) and we expect that the last expression ....

N. Bleistein, Uniform asymptotic expansions of integrals with stationary point near algebraic singularity, Communications on Pure and Applied Mathematics, 19:353--370 (1966).

.... 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, ....

....(19) By definition, we have # 0 = 1 3 # # q O # 1 = q 1 3 12 # # 2q ) O for m in (II) Next, from (18) it follows that b # m = 6 # # D q ( b # m) D q ( This completes the proof. We now prove (19) using the methods of Temme [38] and Bleistein [6, 7]. Assume for the moment that # 1. Let I q = I q (m 1) 1 z m (1 dz. By Cauchy s theorem we may straighten the integration path so that we have I q = 1 z m e z) q dz (20) m#(s) #s) q ds (0 c 1) 21) where #(s) s log s. The steepest descent curve is ....

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N. Bleistein, Uniform asymptotic expansions of integrals with stationary point near algebraic singularity, Communications on Pure and Applied Mathematics, 19, 353--370 (1966).

.... 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, ....

....we have j 0 = 1 Gamma q O q 1 3 (1 Gamma 2q ) O for m in (II) Next, from (18) it follows that m = Gamma m) D Gammaq ( Gamma) This completes the proof. We now prove (19) using the methods of Temme [38] and Bleistein [6, 7]. Assume for the moment that ae 1. Let I q = I q (m Gamma 1) 1 dz: By Cauchy s theorem we may straighten the integration path so that we have I q = 1 dz (20) mOE(s) 1 Gamma aes) ds (0 c 1) 21) where OE(s) s Gamma 1 Gamma log s. The steepest descent ....

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N. Bleistein, Uniform asymptotic expansions of integrals with stationary point near algebraic singularity, Communications on Pure and Applied Mathematics, 19, 353--370 (1966).

....its distribution under the hypothesis. This statement will be made more precise in the following section. It may be noted that in terms of asymptotic approximation an entirely equivalent result may be obtained by use of a dioeerent type of Laplace approximation to a tail integral, by a method from Bleistein (1966), also known from Lugannani Rice (1980) This gives the right tail probability as 1 Gamma Phi(R) OE(R) R i R= U Gamma 1 j ; 3) instead of 1 Gamma Phi( R) The equivalence of the two expression is proved in Jensen (1992, Lemma 2.1) Numerical examples indicate, however, that ....

Bleistein, N. (1966). Uniform asymptotic expansions of integrals with stationary point near algebraic singularity. Comm. Pure Appl. Math. 19, 353370.

....S X s=1 s d s (z ) ds = C; z 0 : 4.13) Since F 0 (z) is strictly monotonic increasing with z, any simple procedure, such as bisection, quickly yields z . A uniform asymptotic approximation to the contour integral in (4. 9) was derived by means of the technique given by Bleistein [2]. The approximation remains valid when the stationary point at z = z is close to the pole at z = 1, and it is also valid when z is not close to 1. We have [20] Proposition 4.2. Let V = S X s=1 s d 2 s (z ) ds ; 4.14) and M = 1 2 Erfc sgn(1 Gamma z ) q GammaF (z ....

N. Bleistein. Uniform asymptotic expansions of integrals with stationary point near algebraic singularity. Commun. Pure Appl. Math., 19:353--370, 1966.

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Bleistein, N. (1966), Uniform asymptotic expansions of integrals with stationary points and algebraic singularity, Comm. Pure Appl. Math., 19, 353--370.

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N. Bleistein (1966). Uniform asymptotic expansions of integrals with stationary points and algebraic singularity, Comm. Pure Appl. Math., 19 353--370.

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