G. Doetsch. Handbuch der Laplace Transformation, Vol. 1--3. Birkhauser Verlag, Basel, 1955.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....For the range h n we use the Taylor expansion again to obtain fi fi = O(nq 2h ) 5.8) Inserting (5.6) 5.7) and (5.8) into (5.5) we obtain EM n = 1 Gamma exp( Gammanpq O(1)O e Gamman log n O(n log n) O(n (5. 9) We now apply the Mellin transform (cf. [2, 4]) to the function f(t) h1 (1 Gamma exp( Gammatpq ) This yields the transformed function f (s) Gamma Gamma(s)p ; for Gamma 1 s 0: 5.10) Application of the Mellin inversion formula, shifting the line of integration to the right and collecting residues yields f(t) ....

G. Doetsch, Handbuch der laplace transformation, Birkhauser, Basel, 1955.

....to De Bruijn (see [4, pp. 131 et seq. The Mellin transform of a function f(x) defined for x 0, x real, is by definition the complex function f (s) given by f (s) Mir(x) s] f(x) x dx. 17) We succinctly recall the salient properties of the Mellin transform, referring the reader to [ 1 ] for precise statements. The Mellin transform of a function f is defined in a strip of the complex plane that is determined by the asymptotic behaviours of f at 0 and oo. It satisfies the important functional property M[f(ax) s] a .f (s) 18) Finally there is a complex inversion formula ....

G. DOETSCH, "Handbuch der Laplace-Transformation," Birkhauser, Basel, 1950.

....the same voltage range is used in both cases. 3. 2 300 hz frequency Without precompensation the input signal is hardly recognizable (figure 4) The precompensation still gives 2 One may compare the demonstrations based on Mikusinski operational calculus and Laplace transform (see, for instance, [4]) p. 2 0 0.02 0.04 0.06 0.08 0.1 0.12 0 5 10 15 20 25 Commande v(0,t) u(t) 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.2 0.4 0.6 0.8 1 Sortie v(l,t) y(t) Figure 2: No compensation. 50Hz frequency 0 0.02 0.04 0.06 0.08 0.1 0.12 20 10 0 10 20 30 Commande v(0,t) u(t) 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.2 ....

G. DOETSCH, "Handbuch der LaplaceTransformation ", 3. Bd., Birkhauser, Bale, 1956.

....loc ( 0; 1) X) is given by the set of analytic functions r defined on some post sectorial region S with values in X which are asymptotically equal to the finite Laplace transform of f ; i.e. ffg T : fr 2 A(P ; X ) r( Gamma Z T 0 e Gammat f(t) dt T 0g; 3 See, for example, G. Doetsch [15], Kap. 2.2. A function F is of exponential type if there exist positive constants M and such that kF (t)k Me t for all t 0. 4 The concept of asymptotic Laplace transforms has its roots in the theory of asymptotic power series of analytic functions; see, e.g. R. Remmert [21] 9.6. With a ....

....k ffi ( e Gamma ffi . For the special case ffi = 1=2, the regularizer k ffi is given by k 1=2 (t) 1 2 p t Gamma3=2 e Gamma 1 4t . Also, all functions m( e Gamma fl with fl 1 can not be represented as Laplace transforms of functions in k 2 L 1 loc ; see G. Doetsch [15], p.163, 229, 263. 11 See alse B. Baumer [2] and B. Baumer, G. Lumer and F. Neubrander [3] 10 G. Lumer, F. Neubrander Then f 2 C 0 ( 0; 1) X) sup t 0 ke Gamma t 1 t b f(t)k 1 and q( b f( for Re( see, e.g. B. Baumer and F. Neubrander [5] Thus q( u( f( k( ....

G. Doetsch, Handbuch der Laplace Transformation, Vol.I, Birkhauser Verlag, Basel-Stuttgart, 1950.

....transforms can be used to transform generating functions into other forms. For example, to transform an ordinary generating function F (u) # a n u n into an exponential one, we can use 1 2#i # u =r F (u) exp(w u)du . 14.2) The basic references for asymptotics of integral transforms are [89, 95, 299, 347]. This section will only highlight some of the main properties of Mellin transforms and illustrate how they are used. For a more detailed survey, especially to analysis of algorithms, see [137] Let f(t) be a measurable function defined for real t # 0. The Mellin transform f # (z) of f(t) is a ....

.... General presentations of asymptotic methods, although usually with emphasis on applications to applied mathematics (di#erential equations, special functions, and so on) are available in the books [54, 100, 114, 115, 315, 344, 354, 372, 382, 385] Integral transforms are treated extensively in [89, 95, 116, 299, 365]. Books that deal with asymptotics arising in the analysis of algorithms or probabilistic methods include [11, 55, 108, 209, 223, 240, 241, 270, 338] Nice general introductions to combinatorial identities, generating functions, and related topics are presented in [81, 351, 377] Further material ....

G. Doetsch, Handbuch der Laplace Transformation, Birkhauser, Basel, 1955. 166

.... 0; Re ae 0 with b = 1 if = 2; 2.7) lim x#0 L(aex) L(x) 1 for Re ae 0; ae 6= 0; Remark 2.1. The principal value of ae Gamma1 is defined so that ae Gamma1 0 for ae 0. 2 For examples of distributions B(t) of the type (2.3) and with LST as given by (2. 6) see [1] cf. further [2], vol. I, p. 467, 501. Note, that (2.6) and (2.7) imply, that: for Re ae 0, g(ae) 0 and ae Gamma1 L(ae) 0 for jaej 0: 2.8) The contraction coefficient Delta(a) is defined as that zero of 4 x Gamma1 L(x) 1 Gamma a ac ; x 0; 2.9) which tends to zero for a 1. From ....

.... 1) n Gamma(n( Gamma 1) 2) a.6) Since y(r) is the LST of P Gamma1 (t) we have: for Re r 0, 1 Gamma y(r) r = 1 Z 0 e Gammart f1 Gamma P Gamma1 (t)gdt: a.7) From (a.6) and (a. 7) we derive the asymptotic relation for 1 Gamma P Gamma1 (t) 1, by using Theorem 2 of [2], vol.II. p.159. The function y(r) is regular for Re r 0, continuous for Re r 0 and r = 0 is a branch point of y(r) From (3.12) it is seen that y(r) can be continued analytically into fr : jrj 0; Gamma 1 2 Gamma arg r 1 2 g for a 2 (0; 1 2 ) and that y(r) is absolutely ....

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Doetsch, G., Handbuch der Laplace-Transformation, Vol. I, II, III, Birkhauser Verlag, Basel, 1950-55.

....[26] for detailed validity conditions. There is also a well known correspondence called the mapping property that fares both ways and relates the asymptotic expansions of an original function f(t) at 0 and 1 and poles of the transform f (s) in a left and right half plane respectively; see [7, 10, 26] for details. In what follows, an essential r ole is played by what may be called the root zeta function of the Airy function. This function is de ned by (25) s) 1 X k=1 ( k ) s ( s) 3 2 ) where the sum is a priori de ned and analytic for (s) 3=2, given the growth of the k ....

G. Doetsch. Handbuch der Laplace Transformation, Vol. 1-3. Birkhauser Verlag, Basel, 1955.

.... Gamma( sin( Gamma 1) f1 c 1 c ( Gamma1) k Gamma1 Gamma( sin( Gamma 1) Gamma(k Gamma 1) fiae) k Gamma logfiaeg: iii. The case 1 Gamma B(t) c( fi t ) flog t fi Gamma Gamma (1) 1 Gamma ) Gamma(1 Gamma ) g G 2 (t) for t fi; 3.10) with c 0; 1 2. From [13], Vol. I, p. 469, it is seen that: for Re ae 0, 1 Gamma 1 Gamma fifaeg fiae = g 4 (fiae) c Gamma(1 Gamma ) fiae) Gamma1 logfiae; 3.11) and this agrees again with (2.6) and (2.7) note that c Gamma(1 Gamma ) Gamma c Gamma( sin( Gamma 1) so that for the present case ....

....(3.11) and this agrees again with (2.6) and (2.7) note that c Gamma(1 Gamma ) Gamma c Gamma( sin( Gamma 1) so that for the present case L(fiae) Gamma( sin( Gamma 1) log 1 fiae ; Re ae 0; ae 6= 0: iv) The case 1 Gamma B(t) c( fi t ) 2 G 2 (t) t fi: 3. 12) From [13], Vol. I, p. 467, it is seen that: for Re ae 0, 1 Gamma 1 Gamma fifaeg aefi = g 5 (fiae) cfiaelog 1 fiae ; 3.13) with g 5 (fiae) regular for Re ae Gammaffi and g 5 (0) 0. Obviously, we have here an example with = 2 and L(fiae) log 1 fiae : 3.14) A heavy tailed distribution of ....

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G. Doetsch (1950). Handbuch der Laplace Transformation, Vol. I,II,III (Birkhauser Verlag, Basel).

....2; r # 0, Phi(r=fl) Gamma max(d 1 ; d 2 )r Gamma1 f1 O(r ( Gamma1) g: 8.4) Consequently, from (8.3) and (8.4) we obtain for 1 2; r # 0, 1 Z 0 e Gammart f1 Gamma W Gamma1 (flt)gdt = max(d 1 ; d 2 )r Gamma2 f1 O(r ( Gamma1) g: 8. 5) 17 By applying theorem 2 of [5], vol. II. p. 159, we obtain from (8.5) for 1 2; t 1, 1 Gamma W Gamma1 (flt) max(d 1 ; d 2 ) Gamma(2 Gamma ) 1 t Gamma1 f1 O( 1 t Gamma1 )g; 8.6) with d 1 and d 2 given by (3.13) For the case that the tail of B(t) is heavier than that of A(t) it is seen from (6.8) ....

....= Gamma min(d 1 ; d 2 )r Gamma1 f1 O(r Gamma1 )g: 8.21) Hence 1 Z 0 e Gammart f1 Gamma I Gamma1 (t)gdt = 1 r f1 Gamma e Phi( Gammar=fl) g = min(d 1 ; d 2 )r Gamma2 f1 O(r Gamma1 )g for r # 0: 8. 22) From which it follows by using theorem 2 of [5], vol. II, p. 159 that: for t 1; 1 2, 1 Gamma I Gamma1 (flt) min(d 1 ; d 2 ) Gamma(2 Gamma ) 1 t Gamma1 f1 O( 1 t Gamma1 )g: 8.23) The relations (8.20) and (8.23) lead to an interesting conclusion. Note that they apply for the case that A(t) and B(t) have similar heavy ....

Doetsch, G., Handbuch der Laplace Transformation, Vol. I, II, III, Birkhauser Verlag, Basel, 1950-56.

....= 0 it follows by analytic continuation that (5.3) holds for Re ae 0. The relation (5.3) will be the starting point for the derivation of the expression for the tail probabilities of the stationary distribution U(t) of u. Essential in our analysis is the application of the Theorem of Doetsch, cf. [6], vol. II p. 159 or appendix A of [2] In [2] it has been shown that [1 Gamma Efe Gammaaew g] ae is regular in the domain D Gamma , see Figure 4, for the case that B(t) is given by (4.1) Figure 4 14 Here D Gamma is the domain in the ae plane to the right of the open contour D, which ....

....[ 2 p ffi a 2 fi 1 Gamma a Gamma fi 1 Gamma a 2ffi 1 1 2 a p a a 1 Gamma a (1 Gamma p a) 5 O(jaefij) Cf1 O(jaefijg O(jfiaej) ae f1 Gamma 2 ffi fi affi 1 2 1 Gamma a (jaefij) 1 2 [1 O(jaefij) g: To the relation (5. 22) we apply the theorem of Doetsch [6], vol. II, p. 159, see also appendix A of [2] As in appendix A of [2] it is shown that this theorem can be applied. Its application yields: for t 1, 1 Gamma U(t) Gamma f1 Gamma W (t)g = Gamma a Gamma( 1 2 ) C fi t 1 2 Gamma 2 p ffi a 1 Gamma a 1 Gamma( Gamma1 1 2 ) ....

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Doetsch G., Handbuch der Laplace-Transformation, vol. I, II, III, Verlag Birkhauser Basel, 1950.

....of terms ae n m ; n = 0; 1; 2; m = 0; 1; for jae=sj 1, and hence (5. 5) shows that such a series expansion also exists for 1 Gamma (ae) Once such a series expansion has been obtained the asymptotics of 1 Gamma W (t) for t 1 can be readily obtained by using a theorem in [4], vol. II, see appendix A. In this section we assume that is rational, so we put = M N ; M N; 5.8) with M;N 2 f1; 2; g and g.c.d. M; N) 1: 5.9) Put y = ae s ) 1=N ; s 0; Re ae 0; 5.10) with the principal value of y positive for ae 0, so y is well defined by (5.10) ....

....with Zn : 1 Gamma y N n ) 2 yn Yn ; n = 1; 2N Gamma 2: 5.27) Hence from (5.10) and (5.26) for jae=sj Y N ; j arg ae s j , 1 Gamma (ae) ae = 1 s 2N Gamma2 X n=1 Zn 1 X k=0 y Gamma(N GammaM k) n ( ae s ) k Gammam) N : 5. 28) We next apply theorem 2 of [4], vol. II, p. 159 (see also appendix A) In appendix A it is shown that the conditions of this theorem are fulfilled by (5.28) It follows from (5.3) 5.28) and the quoted theorem that: for s 0; t 1 and every finite H 2 f0; 1; 2; g, 1 Gamma W (t) 2N Gamma2 X n=1 Zn H X k=0 ....

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Doetsch, G., Handbuch der Laplace-Transformation, Birkhauser Verlag, Basel, 1955.

....for detailed validity conditions. There is also a well know correspondence called the mapping property that fares both ways and relates the asymptotic expansions of an original function f(t) at 0 and 1 and poles of the transform f (s) in a left and right half plane respectively; see [7, 10, 26] for details. In what follows, an essential role is played by what may be called the root zeta function of the Airy function. This function is defined by (s) 1 X k=1 (ff k ) Gammas ( s) 3 2 ) 25) where the sum is a priori defined and analytic for (s) 3=2, given the growth of ....

G. Doetsch. Handbuch der Laplace Transformation, Vol. 1--3. Birkhauser Verlag, Basel, 1955.

....= 2 S ) j e G(z)e z j Ae ffjzj : Proof: Let S 0 be the linear cone for which the polynomial bound over g(z) holds. Let al.so g (s) be the Laplace transform of the function g(x) of a real variable x: g (s) Z 1 0 g(x)e Gammasx dx defined for (s) 0. It is well known that (cf. [5]) the inverse Laplace transform of g (s) exists in (s) 0 and one can write g(x) 1 2 i Z i1 Gammai1 g (s)e sx ds with 0. In addition, g (s) is absolutely integrable on the line of the integration. Observe now that the exponential generating function G(z) e G(z)e z of ....

....theory. APPENDIX A: Laplace Transform of a Complex Variable We want to extend the Laplace transform of a function g(x) of real x to a function g(z) of complex z where g(z) is an analytic continuation of g(x) to a cone S 0 for 0 =2. We expected to find such an extension in books (cf. [5] Chap. 11) but since we fail to provide a definite reference, we include here a brief discussion. First of all, if g(z) is an analytic continuation of g(x) in a cone S 0 , then we can write z = xe i . When Gamma 0 0 , then z = xe i sweeps the whole cone as 0 x 1. But, one can ....

G. Doetsch, Handbuch der Laplace Transformation, Vol. 1--3, Birkhauser-Verlag, Basel 1955.

....(s) as F (s) M[f(t) s] Z 1 0 f(t)t s Gamma1 dt : In some of our arguments (e.g. depoissonization of Section 3.2 and singularity analysis of Section 4.1) we could use either Mellin transform of a complex variable function f(z) or an analytical continuation argument. It is known (cf. [5, 16]) that as long as arg(z) belongs to some cone around the real axis, the Mellin transform F (s) of a function f(x) of a real argument and its corresponding function of a complex argument is the same. Therefore, we work most of the time with the Mellin transform of a function of real variable as ....

G. Doetsch, Handbuch der Laplace Transformation, Verlag Birkhauser, Basel (1950).

....19. Show that, given a polynomial p(z) 2 Q[z] it is decidable whether one of its roots is a root of unity or has an argument that is commensurate with . Implement such an algorithm in a computer algebra system. Hint: Use resultants. A complete discussion of this and related issues is given in [4]. Fluctuations. Take the polynomial D(z) 1 Gamma 6 5 z z 2 , whose roots are i = 3 5 i 4 5 ; i = 3 5 Gamma i 4 5 ; both of modulus 1 (the numbers 3; 4; 5 form a Pythagorean triple ) with argument Sigma where = arctan( 4 3 ) 0:9279. The expansion of the function f(z) ....

....Central and local limit theorems applied to asymptotic enumeration. Journal of Combinatorial Theory 15 (1973) 91 111. 2] Dieudonn e, J. Calcul Infinit esimal. Hermann, Paris, 1968. 3] Feller, W. An Introduction to Probability Theory and its Applications, third ed. vol. 1. John Wiley, 1968. [4] Gourdon, X. and Salvy, B. Asymptotics of linear recurrences with rational coefficients. Tech. Rep. 1887, INRIA, Mar. 1993. To appear in Proceedings FPACS 93. 5] Henrici, P. Applied and Computational Complex Analysis. John Wiley, New York, 1977. 3 volumes. 6] Knopp, K. Theory of Functions. ....

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Doetsch, G. Handbuch der Laplace Transformation, Vol. 1--3. Birkhauser Verlag, Basel, 1955.

.... asymptotic analysis of coefficients of functions with nonpolar singularities (the method of singularity analysis of [8] to the application of Mellin Perron formulae to Dirichlet series with algebraic or logarithmic singularities [15] or to the analysis of Mellin transforms in the nonpolar case [5]. Rather than stating general conditions that would be rather heavy, we content ourselves here with presenting in some detail the analysis of sums that generalize the S n (m) when m is no longer an integer. Figure 3 then summarizes the general correspondence between the nature of singularities and ....

Doetsch, G. Handbuch der Laplace Transformation, Vol. 1--3. Birkhauser Verlag, Basel, 1955.

....For completeness, we give a brief outline of the Perron formula by relating it to the Mellin transform. The resulting summation formulae are classical, so that we content ourselves with a sketchy description of the analysis involved. The major reference for Mellin transforms is Doetsch s book [Do50]. Mellin summation is briefly surveyed in [FRS84] which is directed towards applications in the average case analysis of algorithms, while in the context of integrals (rather than sums) a useful reference is [Wo89, Chap. III] The classical Perron formula is discussed at length in Apostol s book ....

....k f (s) The condition is for s to belong to a fundamental strip defined by the property that the integral giving f (s) and the sum P k k Gammas k are both absolutely convergent. 8 P. FLAJOLET et al. Similar to the Laplace transform there is an inversion theorem (cf. [Do50]) When applied to (2.2) it provides (2.3) X k k f( k x) 1 2 i c i1 Z c Gammai1 X k k Gammas k f (s)x Gammas ds; with c in the fundamental strip. Formula (2.3) could be called Mellin s summation formula. It is especially useful when the integral can be computed by ....

G. Doetsch, Handbuch der Laplace Transformation, Birkhauser Verlag, Basel, 1950.

.... Gamma ( Gamma1) k k (s ) k 1 O(x ffi ) Meromorphicity to the right, till (s) Gammaffi Figure 7.7: The correspondence between asymptotics of f(x) and poles of f (s) which is meromorphic in all C and provides the singular expansion of f (s) in the extended strip. See also [11]. 2 The notation of singular expansions gives a transparent form to the correspondence between asymptotic expansions of original functions and poles of transforms. For instance, in the simpler case of an asymptotic expansion in increasing powers of x that will frequently arise from a Taylor ....

.... digits theorem [10] given here is typical. Mellin transforms are close relatives of the integral transforms of Laplace and Fourier. As such, they play an important role in applied mathematics. Good general references that include a treatment of Mellin transforms are the books of Doetsch [11], Titchmarsh [43] and Widder [45] The book by Wong [46] is in spirit especially close to us as it focuses on asymptotic analysis, in particular as applied to harmonic integrals , a continuous analogue of our harmonic sums. Mellin himself formalized his transform for the purpose of analyzing both ....

Doetsch, G. Handbuch der Laplace Transformation, Vol. 1--3. Birkhauser Verlag, Basel, 1955.

.... denoted as f (s) and defined by F (s) Z 1 0 F (x)x s Gamma1 dx: We consider the real function: F (x) X 2 k [1 Gamma e Gammax=2 k e m Gamma1 (x=2 k ) Its Mellin transform F (s) exists for Gamma2 (s) Gamma1 and is easily determined using basic principles [2], as we now explain. 1. The Mellin transform of e Gammax is the Gamma function and more generally, one has Z 1 0 [1 Gamma e Gammax e m Gamma1 (x) x s Gamma1 dx = Gamma(s) s m Gamma 1 m Gamma 1 ; an equation valid for Gammam (s) 0. 2. The transform of f(ax) is a ....

G. Doetsch. Handbuch der Laplace Transformation, Vol. 1--3. Birkhauser Verlag, Basel, 1955.

....in the books by Hofri [44, p. 48ff ] Kemp [51, p. 141] Mahmoud [62] and in the handbook chapter [84] We follow here the architecture of the informal survey [31] General properties of the Mellin transform are usually treated in detail in books on integral transforms, like those of Doetsch [18], Widder [86] or Titchmarsh [82] Asymptotic methods in connection with Mellin transforms are discussed within the context of applied mathematics in treatises by Davies [13] Dingle [17] and Wong [88] In particular, our work is close in spirit to Wong s who discusses extensively an analogue of ....

....integrable with fundamental strip hff; fii and be of bounded variation in a neighbourhood of x 0 . Then, for any c in the interval (ff; fi) lim T 1 1 2i Z c iT c GammaiT f (s)x Gammas ds = f(x 0 ) f(x Gamma 0 ) 2 : 12 P. FLAJOLET, X. GOURDON, AND P. DUMAS Proof. See [18, 82, 86]. Figure 2 presents a few classical original transform pairs (f; f ) More can be found in standard tables of integral transforms [19, 66, 67] and in [61, 77] 2. The fundamental correspondence There is a very precise correspondence between the asymptotic expansion of a function at 0 (and 1) ....

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Doetsch, G. Handbuch der Laplace Transformation, Vol. 1--3. Birkhauser Verlag, Basel, 1955.

....is based on integral splittings; from transformed to original functions, it is called the Converse Mapping Property and it is based on a residue evaluation of inverse Mellin integrals, so that it holds under conditions of smallness of the transform at Sigmai1. These mappings are described in [1], as well as in [3] Theorems 3 and 4, pp. 16 22) and [14] Theorem 5, p. 153) 6 PHILIPPE FLAJOLET Lemma 2. The functions Li ff;r (z) satisfy the singular expansions (9) and (10) as z 1 in the domain (ffl an arbitrary positive real) Phi jzj 1 Psi [ n z fi fi Gamma 2 ffl ....

G. Doetsch, Handbuch der Laplace Transformation, vol. 1--3, Birkhauser Verlag, Basel, 1955.

.... (s) satisfies certain smallness conditions towards i1 (cf. 12] for Gammafi (s) M , M 0) and f (s) K X k=0 d k (s Gamma b) k 1 ; 37) then as x 1 f(x) K X k=0 d k k x Gammab ( Gamma log x) k O(x GammaM ) 38) P5) Mellin Transform in Complex Plane (cf. [5, 6, 13]) If F (z) is analytic in a cone 1 arg(z) 2 with 1 0 2 , then the Mellin transform F (s) can be defined by replacing the path of integration [0; 1[ by any curve starting at z = 0 and going to 1 inside the cone, and it is identical with the real transform f (s) of f(z) F (z) ....

G. Doetsch, Handbuch der Laplace Transformation, Vol. 1--3, Birkhauser-Verlag, Basel 1955.

....point z = 1. 2.2 Mellin transforms The idea is to estimate G(z) as z tends to 1 directly, for z in a neighbourhood of 1. This asymptotic expansion provides via the mapping z 7 z= 1 Gamma z) the behaviour of F (z) near its singularity z = 1. The analysis relies on Mellin transforms (see [4] or [23] in the context of analysis of algorithms) The convenient form is to set z = 1=t and we need to consider t 0. Define first Q(u) 1 Y j=0 (1 u 2 j ) Then, from Lemma 1, we have G(t Gamma1 ) Q(t=2) b = 1 X k=0 2 k (2 k t) b P ( 1 2 k t ) Q(2 k t) b ....

....z tends to 1 along the real ray [0; 1] Some of the difficulty of our problems comes from the fact that we need a continuation into the complex plane of these asymptotic expansions. As is usual in a Mellin analysis, we apply the inversion theorem in order to recover H(t) from H (s) We have [4] G(t Gamma1 ) Q b (t=2) 1 2i Z d i1 d Gammai1 H (s)t Gammas ds; d 1: 11) Apart from the explicit form of H (s) we also require in passing some growth estimates for the Mellin transform h (s) h (c ix) O i jxj b Gamma1 e Gamma jxj j as x Sigma1, for c ....

Doetsch, G. Handbuch der Laplace Transformation, Vol. 1--3. Birkhauser Verlag, Basel, 1955.

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G. Doetsch. Handbuch der Laplace Transformation, Vol. 1--3. Birkhauser Verlag, Basel, 1955.

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G. DOETSCH, "Handbuch der LaplaceTransformat ion", 3. Bd., Birkhauser, Bale, 1956.

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G. DOETSCH, "Handbuch der LaplaceTransformation ", 3. Bd., Birkhauser, Bale, 1956.

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Doetsch, G. Handbuch der Laplace Transformation, vol. 1-3. Birkha#serVerlag. Basel, 1955.

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G. Doetsch. Handbuch der Laplace Transformation, Band I. Birkhauser Verlag, Basel, 1950.

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G. Doetsch, Handbuch der Laplace Transformation I, Birkh/user Verlag 1971.

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G. Doetsch. Handbuch der Laplace-Transformation, volume 2. Birkhauser Verlag, Basel, Stuttgart, 1955.

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G. Doetsch, Handbuch der Laplace-Transformation, Band I, Verlag Birkhauser, Basel, 1950.

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G. Doetsch (1950). Handbuch der Laplace-Transformation I. Verlag Birkhauser, Basel 1950.

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G. Doetsch (1950). Handbuch der Laplace-Transformation I. Verlag Birkhauser, Basel.

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DOETSCH, G. Handbuch der Laplace-Transformation. Band 1, Theorie der Laplace-Transformation, Birkhauser, Basel, 1950.

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G. Doetsch, Handbuch der Laplace--Transformation, Band III (Birkhauser Verlag, Basel, 1956).

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