G. Darboux, Memoire sur l'approximation des fonctions de tres grands nombres, Journal de Mathematiques 4 (1878) 5---56, 377---416.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....example, if f(t) t, t # 0, then f(s) s 1 and Q(z) 2(1 z) 1 , so that Q has a simple pole at z = 1 and q n = 2( 1) n . However, in many applications Q will have an algebraic singularity on the unit circle. With an algebraic singularity, we can apply Darboux s theorem; see Darboux [18], p. 179 of Wilf [52] and pp. 445 450 of Henrici [24] We will actually apply only a simple form of Darboux s theorem. Suppose that Q has one singularity on the unit circle, which is an algebraic singularity at z = 1 (corresponding to a singularity of f at s = #) Then Q(z) 1 z) # ....

G. Darboux, 1878. Memoire sur l'approximation des fonctions de tres grands nombres, J. de Mathematiques 4, 5--56 and 377--416.

.... A singularity of f(z) at z = w is called algebraic if f(z) can be written as the sum of a function analytic in a neighborhood of w and a finite number of terms of the form (1 z w) # g(z) 11.33) where g(z) is analytic near w, g(w) #= 0, and # ## 0, 1, 2, Darboux s theorem [87] gives asymptotic expansions for functions with algebraic singularities on the circle of convergence. We state one form of Darboux s result, derived from Theorem 8.4 of [354] Theorem 11.3. Suppose that f(z) is analytic for z r, r 0, and has only algebraic singularities on z = r. Let a ....

....circle. Existence of higher derivatives leads to even better estimates. We do not attempt to state a general theorem, but illustrate an application of this method with an example. The same technique can be used in other situations, for example in obtaining better error terms in Darboux s theorem [87]. Example 11.4. Permutations with distinct cycle lengths. Example 8.5 showed that for the function f(z) defined by Eq. 8.58) z n ]f(z) # exp( #) as n # #. This coe#cient is the probability that a random permutation on n letters has distinct cycle lengths. The more precise 113 estimate ....

G. Darboux, Memoire sur l'approximation des fonctions de tres-grands nombres, et sur une classe etendue de developpements en serie, J. Math. Pures Appl., 4 (1878), 5--56, 377--416.

....at z = r of the kind S(z) c 1 c 2 (r z) 1 2 O( r z ) 1.1) s(z) c 3 c 4 (r z) 3 2 O( r z 2 ) 1.2) where the c i are constants depending on the trees being considered) and no other singularities on z = r. It can then be deduced using the Darboux method [10] that for some constants b 1 and b 2 , S n b 1 n 3 2 r n as n , s n b 2 n 5 2 r n as n . The above sketch should not be taken to imply that the Polya method is trivial. Many of the steps leading to (1.1) and (1.2) which involve obtaining relations involving the ....

....z. Equation (1.5) can be solved explicitly; we find that B(z) 2z 1 1 4z . 1.6) We immediately see that B(z) is analytic in the entire complex domain with the exception of an algebraic singularity (a square root one) at z = 1 4. Hence an application of the Darboux method [10] easily yields the result B n p 1 2 n 3 2 4 n as n . 1.7) In this particular case there is no need to use the Darboux method. If we expand (1 4z) 1 2 by the binomial theorem, we find that B n = n 1 1 ( n 2n ) 1.8) so that B n is one of the famous Catalan ....

G. Darboux, Memoire sur l'approximation des fonctions de tres grands nombres, et sur une classe etendu de developpements en serie, J. Math. Pures et Appliq. (3) 4 (1878), 5-56.

No context found.

G. Darboux, Memoire sur l'approximation des fonctions de tres grands nombres, Journal de Mathematiques 4 (1878) 5---56, 377---416.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC