F. W. J. Olver, 1974. Asymptotics and Special Functions. Computer Science and Applied Mathematics. Academic Press, Inc.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....3.3. Example. Suppose u(x) x) Then (5) for this potential is # xx (# 2 sech (x) # = 0 Using the substitution y : R #] 1, 1[ defined by y = tanh(x) we get d# 2 # = 0. This is the associated Legendre equation. It is possible to verify (see [DJ89] or [Olv74] ) that if # 0 the only bounded (and normalized) solution is #(x) # 2 sech(x) for # = and as x #, #(x) # 2e x . Moreover, if # = k 0 any solution with asymptotic behavior #(x, k) e ikx as x behaves as x # as #(x, k) a(k)e ikx and therefore b(k) 0, i.e. the potencial u ....

F. Olver. Asymptotics and Special Functions. Academic Press, 1974.

....(z) 1 # . Similar remarks hold for (2.30) 2. 7 Asymptotics near the branch points Near the branch points, the asymptotic formulas involve the Airy function Ai, which is the solution of the di#erential equation y ## (z) zy(z) Ai(z) 1 2 # # arg z #, see, e.g. [28] for more details. We only deal with the asymptotic behavior of P n , Q n and E n near z 1 . Similar results can be given for the behavior near the other branch points. Here we will take the branch of the function # P (z) which is 0 at z = z 1 . So it behaves like # P (z) c(z 5 2 ....

F. Olver, Asymptotics and Special Functions, Academic Press, San Diego, 1974.

....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 ....

Olver, F. W. J. Asymptotics and Special Functions. Academic Press, 1974.

....for jzj 1. Then (f n (z) 2 S(X) and the asymptotic expansion for (f n (z) holds uniformly for jzj r 1. 6. xn ) 2 S(C) k=1 xk 2 ) Q n k=1 (1 2 ) 2 S(C) Proof: 1:a) and (2: are straightforward extensions of properties of asymptotic expansions of functions (cf. e.g. [3]) to the case of sequences in a Banach space. 4: is also a classical result. 1:b) The proof of this statement is based on the expansion 1 (n Sigma 1) 1 Sigma Upsilon k 1 (3: It is sufficient to prove this when limxn = e, the unit. Let yn = e Gamma xn . Then for ....

.... a k Gamma1 1 Gamma z Phi(z; 1; n)a 1 Delta Delta Delta Phi(z; k Gamma 1; n)a k Gamma1 ; 3) where (cf. 2] Phi(z; s; x) l=0 (x l) l 1 Gamma(s) t s Gamma1 e Gammaxt Gammat dt (jzj 1; s 0; x 0) Applying Watson s lemma (cf. e.g. [3]) to this integral, we have the asymptotic expansion Phi(z; s; x) c 0 (s; z)x c 1 (s; z)x Delta Delta Delta as x 1; 4) which holds uniformly for jzj r 1. Hence, as k M k Phi(jzj; k; n) the last term in (3) is uniformly of order n k for jzj r and by (4) the other ....

F.W.J. Olver, Asymptotics and special functions. Academic Press, 1974.

....same lines as in [5] an integral expression for the error. Then, we give in Lemma 2 an asymptotic expansion for the error kernel, valid near the singularity. This expansion is obtained from asymptotic expansions for the Legendre polynomials and the functions of the second kind derived in [3] and [9]. Bringing the results of the lemmas together we obtain the proof of the above theorem. Finally, the result is extended to logarithmic singularities and numerical examples are given. 2 The error representation in case of an analytic integrand For the Cauchy kernel function c z (x) z Gamma ....

.... curves Gamma 1 and Gamma 3 can be written as an integral on the interval [1; 2 ] so that En (f) Gamma r r Gamma1 (z Gamma 1) g(z)e n (z)dz O( Substituting r = e ae and z = cosh x in this equation, we obtain (5) 2 4 An asymptotic expansion for the error kernel In [3] and [9], asymptotic expansions are derived for classes of functions including Pn (cosh x) and Qn (cosh x) These expansions involve the modified Bessel functions I (x) and K (x) The proof is based on regarding the (suitably transformed) Legendre differential equation satisfied by Pn and Qn for large ....

F.W.J. Olver, Asymptotics and special functions. Academic Press, 1974.

....in our abstract. Throughout this paper, A B means that A and B share the same leading term ( when expanded in terms of a certain asymptotic parameter ) The symbol a 1 means that a is small enough ( this usually can be characterized by some asymptotic parameter ) We refer to [1, 5, 7, 11] for a full theory of the asymptotic analysis of integrals. 2. ACCURACY OF APPROXIMATIONS We define the following approximations to f p ( t ) f O p (v)e dv (frequency limited to v p) 6) iarg ( f O p ) dv (magnitude taken as 1) 7) F p ( t ) dv (leading term of ....

....iii ) FisC around this critical point and F 9 (c) x 0 . Then the leading asymptotic magnitude is proportional to 1 l: f(c) lF9(c) e 0i[lF(c) sign ( F 0 ( c) p 4] o(l 01 2 ) 25) The proofs of these two statements can be found in many asymptotic analysis textbooks ( for instance [1, 5, 7, 11] ) with a little modification on the regularity of F . Next, let s consider the doubly parameterized Fourier integral (DPFI) e 0ilF(v,t) dv, for real t. 26) At any fixed time t, Statements 1 and 2 can be applied to DPFI. As long as the regularity conditions for v are satisfied uniformly ....

F. W. J. Olver, "Asymptotics and Special Functions," Academic Press, New York, 1974.

....1 7 6 #(2 3) Gi # (0) Hi # (0) Bi # (0) Ai # (0) 1 5 6 #(1 3) 1. 2) Because Scorer functions and Airy functions solve the inhomogeneous equation w ## zw = K,with K constant, Scorer functions appear in asymptotic expansions for inhomogeneous equations around a turning point ([9], Pg. 429) Scorer functions appear in a number of applications in physics and chemistry (see, for example, 7, 8, 11] for real variables and [10] for complex variables) Properties of the Scorer functions are given in Chapter 10 of [1] In [3] stable integral representations of the Scorer ....

F.W.J. Olver. Asymptotics and Special Functions. Reprinted by A.K. Peters Ltd., 1997.

....the initial values. We use the asymptotic expansions Gi(z) ph z # #, 1.8) Hi(z) ph( z) # #, 1.9) # being an arbitrary positive constant. These expansions follow from (1.1) and (1. 2) and by using standard methods from asymptotics (Watson s lemma; see [4], page 112 and page 431) 2. Qualitative properties of the real zeros of Gi(z) and Gi # (z) From (1.2) we see that Hi(z) 0andHi # (z) 0 for real finite z. However, Gi(z)andGi # (z)have real zeros. First we show that Gi(z) does not have positive zeros. Later, we study properties of the ....

F.W.J. Olver. Asymptotics and Special Functions. Academic Press, New York. Reprinted in 1997 by A.K. Peters.

....applicability than in the case of real , the reason being that the values of (ia, m) increase more rapidly than in the case of real orders. Sequence transformations [21] may be used to improve the performance of asymptotic series. An error bound of the remainder in this expansion follows from [13]. We have for n = 0,1,2, Kia(X) J [k=0 (2x)m q Rn(a where the remainder satisfies the simple bound (2.15) IRn,x)l 2e I(i,n)l (2.16) 2x)n For a given pair a, x and precision e, it is easy to check if a number n can be found such that la,x)l 2.3 Continued fraction The Bessel ....

Olver, F.W.J., Asymptotics and special functions, Academic Press, New York, 1974, Reprinted in 1997 by A.K. Peters.

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F. W. J. Olver, 1974. Asymptotics and Special Functions. Computer Science and Applied Mathematics. Academic Press, Inc.

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Olver, F. W. J. Asymptotics and Special Functions. Academic Press, 1974.

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F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974.

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F. W. J. Olver. Asymptotics and Special Functions. Academic Press, 1974.

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Olver, F. W. J. Asymptotics and Special Functions. Academic Press, 1974.

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F. W. Olver. Asymptotics and Special Functions. Academic Press, New York-London, 1974.

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F. W. J. Olver, Asymptotics and Special Functions, A. K. Peters Classics, Massachusetts, 1997.

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F.W.J. Olver. Asymptotics and Special Functions. Academic Press, New York, 1974.

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F.W.Olver, Asymptotic and Special Functions (Academic Press, New York and London 1974).

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F. W. J. Olver, Asymptotics and special functions, Academic Press, New York, 1974.

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F. W. J. Olver, Asymptotics and special functions, Academic Press, Inc., New York, 1974.

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F. W. J. Olver, Asymptotics and special functions, Academic Press, Inc., New York, 1974.

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F. W. J. Olver, Asymptotics and special functions, Academic Press, New York, 1974.

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Olver, F. W. J. Asymptotics and Special Functions. Academic Press, 1974.

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F. W. J. Olver, Asymptotics and special functions, Academic Press, New York and London, 1974.

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OLVER, F.W.J. (1974), Asymptotics and Special Functions, Academic Press, New York. Reprinted in 1997 by A.K. Peters.

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