C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....from x , and the amplitude of e grows exponentially. If Im . 0, then Imk becomes positive as x increases and the amplitude decays exponentially. The latter is the condition for a localized wave packet to be possible. This argument can be made systematic by the method of WKBJ analysis [5], as follows. Let a vector v be de ned by v j = x j )e ; 8) where is continuous and is twice continuously di erentiable. Then v is an eigenfunction of A with eigenvalue if c p (x j ) x j p )e iN (x j p ) x j )e for all j. Expanding this equation for large N we nd ....

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978.

....P0 (the interface width is illustrated in Fig. 8 below) Indeed, it is possible to derive an e#ective interface description systematically by performing an expansion of the equations in powers of (technically, this is done using singular perturbation theory or matched asymptotic expansions [5,6,21 23]) In such an analysis, the wall or front is treated as a sharp interface when viewed on the outer pattern forming length scale l # This will be illustrated, e.g. by the Landau free energy f in Fig. 4b below. Here the states # 0 and # # s both correspond to minima of the free energy density ....

....Eq. 53) it ensures that the interface width scales as # and that the time scale # for the order parameter relaxation is also of order #. It thus allows us to derive the e#ective interface equations mathematically using the methods of matched asymptotic expansions or singular perturbation theory [5,21,22] by taking the limit #P0. Since both and # scale as # the response of the interface stays finite as # goes to zero. Although the mathematical analysis by which e#ective interface equations can be obtained is certainly more sophisticated and systematical than what will transpire from the brief ....

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C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978.

....variable fi 2 = Gamma k The expression in (3.13) becomes (x; z) Gamma v j (x; v j (x 0 ; ae 0 fi 2 ) kdfi 2 : 3.14) The weight dae ( ae ( d is given in Theorem 2.1. Since the integrand is bounded uniformly in fi 2 , we can apply Laplace s method [1] and obtain for jzj 1 that (x; z) Gamma 0 )v j (x 0 ; k 0 ) ae 0 )k; where v j (x; have been given in (2.3) and ae 0 ) OE j (h; k j (h; k Since OE (h; k 0 ) O(k) and v(x; k 0 ) O(1) we see that the factor multiplying 1=kjzj behaves ....

....a similar analysis for the radiation component. We use the variable fi = 0 Gamma =k and rewrite (3.12) as ikfijzj Theta )v j (x 0 ; ae 0 Gammafi kdfi: 3.15) The term multiplying the exponent in the integrand is analytic in fi. We use the Method of Steepest Descent [1]. Deform the contour of integration in the complex fi plane as shown in Figure 3.1. If we let R 1 in Figure 3.1, the contribution from the vertical parts of the contour are 0 k kdfi 2 ik(n cl ifi2 )jzj Theta 0 Gammak (n cl ifi2 ) 2 kdfi 2 : We note ....

C. Bender and S. Orszag, Advanced Mathematical Methods for Scientists and Engineers McGraw Hill, New York, 1978.

....components v 0 ; vN Gamma1 as v n = i Z n and require Z 0 = ZN = 0. The last component is then vN = i 1 GammaN ZN Gamma1 = 2N) We obtain the recurrence relation Z n Gamma1 Z n 1 = Gamma2inZ n (28) whose solutions for nonzero are linear combinations of Bessel functions [1] Z n = ffJ n ( fiY n ( 29) where we have defined = Gammai= We can obtain eigenvectors v provided that is a root of the equation det J 0 ( Y 0 ( JN ( YN ( 0 (30) or equivalently arg A = 0; 31) where Hankel functions H n ( J n ( iY n ( are ....

Carl M. Bender and Steven A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill Book Company, New York, 1978.

....(5) being the dispersion relation. Given (x; 0) the integral can be well approximated by a wave train solution via the stationary phase method. Since this technique will be used several times in this paper, we quote the theorem below. For its detailed proof and various applications, see [4, 14]. Theorem 1. Let a(x) 2 C c (R ) be an C function with a compact support. Suppose that function (x) is smooth, the set fy : D (y) 0g supp(a) has nitely many points fy 1 ; y 2 ; yN g and the matrix D (y j ) is nonsingular. Then, for 1, I = y) a(y)dy = ....

C. Bender, S. Orszag, Advanced mathematical methods for scientists and engineers, McGrawHill, 1978.

...., I.1) where #M : #(z M ) and z M is the location such that y(z M ) y M and z z M z #Z. Let us rewrite #N s as exp[# M h(z) dz, where #M : # #M , h(z) 1 #M #. Since #M 1, we can approximate the integral using Laplace s method for integrals (e.g. [11, 54]) exp[# M h(z) dz # 1 2 2h zz (z M ) Since # z (z M ) 0, h zz (z M ) 1 # zz (z M ) Similarly, since y z (z M ) 0, from (I.1) we have # zz (z M ) y zz (z M ) Di#erentiating (4.11) gives y zz (z M ) Therefore, and #N s is given by ....

C. Bender and S. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978.

.... under limiting conditions, are asymptotic expansions, or asymptotic power series in particular: x 0 ) x =n x 0 , but some asymptotic expansions can be divergent and still be useful if a finite number of terms are used, such as the expansion of the famous Stieltjes integral [10] (1 xt) 1) n x 0, which clearly diverges. For asymptotic applications, we are usually only interested in a few terms, whether the expansion is convergent or divergent. Limits play a different role in asymptotic expansions then they do for Taylor series, in that limits of the ....

.... say OE(x) ffl, has a maximum at x within the interior of the range of integration (a# b) such that OE ) 0 and OE ) 0, i.e. OE(x) 0:5OE ) x while f(x) is continuous and subdominant, as x and 0 ffl 1, Laplace s method for asymptotic evaluation of integrals [10] leads to the asymptotic approximation, OE(x) ffl 2ffl ;OE ) ffl )# (1.26) as ffl . If x = a or x = b, i.e. an end point maximum, then the integral is asymptotic to one half the above approximation. For example, the factorial function or Gamma function for real x with ....

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C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill Book Co., New York, NY, 1978.

....The work of this author was partially supported by National Science Council of ROC under the Grants NSC 90 2118 M 001 034 and NSC 90 2118 M 001 028. are also referred to in the literature as Euler equations, Euler Cauchy equations, or equidimensional equations; see Bender and Orszag [10], Ince [65] Equations of this type often appear in analysis of algorithms, notably in analysis of quicksort and search trees. We propose in this paper an asymptotic theory for the coecients of z in the Taylor expansion of f when the c j s are given and asymptotics of the coecients of is ....

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Springer, NY, 1999.

....The work of this author was partially supported by National Science Council of ROC under the Grants NSC 90 2118 M 001 034 and NSC 90 2118 M 001 028. are also referred to in the literature as Euler equations, Euler Cauchy equations, or equidimensional equations; see Bender and Orszag [10], Ince [65] Equations of this type often appear in analysis of algorithms, notably in analysis of quicksort and search trees. We propose in this paper an asymptotic theory for the coe#cients of z in the Taylor expansion of f when the c j s are given and asymptotics of the coe#cients of # is ....

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Springer, NY, 1999.

.... Then, using equation (3) we get fi fi jCl d;D 1 (w d;D )j Gamma jCl d;D (w d;D Gamma (d Gamma 1) j = Gamma cos(x(d Gamma 1) d sin(x) dx : 4) The asymptotic evaluation of this ratio is done using Laplace s method, a comprehensive description of which can be found in [2]. Proposition 5. fi fi Proof. Laplace s theorem states that if two real continuous functions f and g on [a; b] are such that g(t) has its maximum at t = a and f(a) 6= 0, then we may approximate the integral R b dt by R a ffl dt for any 0 ffl b Gamma a and x AE 1, as the ....

C. M. Bender and S. A. Orszag. Advanced mathematical methods for scientists and engineers, chapter 6.4 - Laplace's method and Watson's lemma, pages 261--

....we may truncate this series at k = N and use the data C n , C n 1 , C n N to determine the a k . As n approaches infinity, C n approaches a 0 ; therefore, a 0 serves as the best estimate of C. This technique is known as Richardson extrapolation, and for a given N , it turns out that (see [1]) a 0 = C n k (n k) N ( 1) k N k (N k) For all constraints, the best estimates of the capacity, along with the number of digits of precision in the estimate, have been listed in Table 1. If A n is real and symmetric (conditions met by the square and diamond constraints) we may ....

C. M. Bender and S. A. Orszag. Advanced Mathematical Methods For Scientists and Engineers. McGraw-Hill, Inc., St. Louis, 1978.

....which may be coded by a colour formalism [Ric88] The solutions resulting from integrations of a complex ODE are elements of C , and need four dimensions for their visualization. In this particular case, we can nevertheless use a visualization technique derived from the extended phase portraits [BO78]. This visualization technique provides a way to visualize a group of solutions, specified by different initial conditions or by different values of a parameter. If we want to integrate the differential equation (1) where x is taken on a complex path P, and y 0 2 fl ae C (the initial conditions ....

.... = y 1 (t) iy 2 (t) and (t)f( Phi(t) y(t) f 1 (t; y 1 ; y 2 ) if 2 (t; y 1 ; y 2 ) we obtain a 2 Theta 2 real dynamical system : f 1 (t; y 1 ; y 2 ) f 2 (t; y 1 ; y 2 ) 4) 4 The solutions of this dynamical system can be visualized on a phase portrait [BO78]. But this dynamical system is generally not autonomous, even if equation (1) was autonomous, and as a consequence different solution curves (issued from different initial conditions) can intersect, and the interpretation of the phase portrait can be obscured. Lifted orbits Omega (x 0 ; y 0 )can ....

C. M. Bender and S. A. Orszag. Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill, 1978.

....with ff i 6= ff j 8i 6= j; 20) u n ) 21) For the general case L(u n ) 1( Omega Delta Delta Delta( Omega Gamma ff l ) s l (u n ) the results are trivially obtained from the ones for these two types of operators. From the theory of linear difference operators (see for instance [2]) we obtain: L 1 (u n ) 0 , u n = 1 i ; A i constants 11 ; A i constants For the corresponding transformations given by (19) the structure of the kernels follows immediately: Theorem 6 Let us consider the sequence transformations L i (S n =D n ) L i (1=D n ) i = 1; 2; 22) ....

....of the extrapolation method T . But, if we can t find a basis of solutions, we can obtain, using different techniques than above, the asymptotic behavior of the solution when (n 1) and, for some classes of operators, also the asymptotic expansion for a linearly independent set of solutions [2]. From the knowledge of this asymptotic behavior, we are going to: a) give the acceleration properties for the sequence transformation T corresponding to a given operator L; b) propose a method of constructing an operator L and the corresponding sequence transformation T to accelerate a class of ....

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C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engeneers. McGraw-Hill International Editions, Mathematics Series (1987).

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C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978.

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C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers. Springer, 1999.

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C. Bender and S. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Mc-Graw Hill, 1978. 3

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C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Springer, New York, 1999.

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C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978.

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C. M. Bender and S. A. Orszag. Advanced Mathematical Methods for Scientists and Engineers. McGrawHill, Inc., 1978.

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C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).

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C. M. Bender and S. A. Orszag, Advanced mathematical methods for scientists and engineers (McGraw-Hill, 1978).

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C. M. Bender and S. A. Orszag. Advanced Mathematical Methods for Scientists and Engineers. McGrawHill, Inc., 1978.

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C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, 1978.

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C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill Book Company, 1978.

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C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw{Hill, New York, 1978.

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