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1: 2.9 Difference Equations
As in the case of differential equations (§§2.7(iii), 2.7(iv)) recessive solutions are unique and dominant solutions are not; furthermore, one member of a numerically satisfactory pair has to be recessive. When | ρ 2 | = | ρ 1 | and α 2 = α 1 neither solution is dominant and both are unique. …
2: 2.7 Differential Equations
2.7.30 w 1 ( x ) / w 4 ( x ) 0 , x a 1 + ,
w 1 ( x ) is a recessive (or subdominant) solution as x a 1 + , and w 4 ( x ) is a dominant solution as x a 1 + . … The solutions w 1 ( z ) and w 2 ( z ) are respectively recessive and dominant as z - , and vice versa as z + . …
3: 36.5 Stokes Sets
The Stokes sets are defined by the exponential dominance condition: … For z < 0 , there are two solutions u , provided that | Y | > ( 2 5 ) 1 / 2 . … The first sheet corresponds to x < 0 and is generated as a solution of Equations (36.5.6)–(36.5.9). …For | Y | > Y 1 the second sheet is generated by a second solution of (36.5.6)–(36.5.9), and for | Y | < Y 1 it is generated by the roots of the polynomial equation …
4: 36.11 Leading-Order Asymptotics
36.11.1 t 1 ( x ) < t 2 ( x ) < < t j max ( x ) ,
36.11.2 Ψ K ( x ) = 2 π j = 1 j max ( x ) exp ( i ( Φ K ( t j ( x ) ; x ) + 1 4 π ( - 1 ) j + K + 1 ) ) | 2 Φ K ( t j ( x ) ; x ) t 2 | - 1 / 2 ( 1 + o ( 1 ) ) .
5: 2.11 Remainder Terms; Stokes Phenomenon
In effect, (2.11.7) “corrects” (2.11.6) by introducing a term that is relatively exponentially small in the neighborhood of ph z = π , is increasingly significant as ph z passes from π to 3 2 π , and becomes the dominant contribution after ph z passes 3 2 π . … Rays (or curves) on which one contribution in a compound asymptotic expansion achieves maximum dominance over another are called Stokes lines ( θ = π in the present example). …
§2.11(v) Exponentially-Improved Expansions (continued)
2.11.19 w j ( z ) = e λ j z z μ j s = 0 n - 1 a s , j z s + R n ( j ) ( z ) , j = 1 , 2 ,
6: 3.6 Linear Difference Equations
§3.6 Linear Difference Equations
§3.6(ii) Homogeneous Equations
Thus Y n ( 1 ) is dominant and can be computed by forward recursion, whereas J n ( 1 ) is recessive and has to be computed by backward recursion. …
7: Bibliography H
  • R. A. Handelsman and J. S. Lew (1971) Asymptotic expansion of a class of integral transforms with algebraically dominated kernels. J. Math. Anal. Appl. 35 (2), pp. 405–433.
  • B. A. Hargrave (1978) High frequency solutions of the delta wing equations. Proc. Roy. Soc. Edinburgh Sect. A 81 (3-4), pp. 299–316.
  • M. Heil (1995) Numerical Tools for the Study of Finite Gap Solutions of Integrable Systems. Ph.D. Thesis, Technischen Universität Berlin.
  • M. Hoyles, S. Kuyucak, and S. Chung (1998) Solutions of Poisson’s equation in channel-like geometries. Comput. Phys. Comm. 115 (1), pp. 45–68.