dominant solutions
♦
9 matching pages ♦
(0.001 seconds)
9 matching pages
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 and neither solution is dominant and both are unique.
…
2: 2.7 Differential Equations
…
►
2.7.30
,
►
is a recessive (or subdominant) solution as , and is a dominant solution as .
…
►The solutions
and are respectively recessive and dominant as , and vice versa as .
…
3: 18.39 Applications in the Physical Sciences
…
►The solutions of (18.39.8) are subject to boundary conditions at and .
…
►The solutions (18.39.8) are called the stationary states as separation of variables in (18.39.9) yields solutions of form
…
►Brief mention of non-unit normalized solutions in the case of mixed spectra appear, but as these solutions are not OP’s details appear elsewhere, as referenced.
…
►Interactions between electrons, in many electron atoms, breaks this degeneracy as a function of , but still dominates.
…
►The radial Coulomb wave functions
, solutions of
…
4: 36.5 Stokes Sets
…
►The Stokes sets are defined by the exponential dominance condition:
…
►For , there are two solutions
, provided that .
…
►The first sheet corresponds to and is generated as a solution of Equations (36.5.6)–(36.5.9).
…For the second sheet is generated by a second solution of (36.5.6)–(36.5.9), and for it is generated by the roots of the polynomial equation
…
5: 36.11 Leading-Order Asymptotics
6: 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 , is increasingly significant as passes from to , and becomes the dominant contribution after passes .
…
►Rays (or curves) on which one contribution in a compound asymptotic expansion achieves maximum dominance over another are called Stokes lines ( in the present example).
…
►
§2.11(v) Exponentially-Improved Expansions (continued)
… ►
2.11.19
,
…
7: 3.6 Linear Difference Equations
§3.6 Linear Difference Equations
… ►§3.6(ii) Homogeneous Equations
… ► … ► … ►Thus is dominant and can be computed by forward recursion, whereas is recessive and has to be computed by backward recursion. …8: Bibliography H
…
►
Asymptotic expansion of a class of integral transforms with algebraically dominated kernels.
J. Math. Anal. Appl. 35 (2), pp. 405–433.
…
►
High frequency solutions of the delta wing equations.
Proc. Roy. Soc. Edinburgh Sect. A 81 (3-4), pp. 299–316.
…
►
Numerical Tools for the Study of Finite Gap Solutions of Integrable Systems.
Ph.D. Thesis, Technischen Universität Berlin.
…
►
Solutions of Poisson’s equation in channel-like geometries.
Comput. Phys. Comm. 115 (1), pp. 45–68.
…
9: 1.2 Elementary Algebra
…
►Square matrices (said to be of order
) dominate the use of matrices in the DLMF, and they have many special properties.
…
►has a unique solution, .
If then, depending on , there is either no solution or there are infinitely many solutions, being the sum of a particular solution of (1.2.61) and any solution of .
Numerical methods and issues for solution of (1.2.61) appear in §§3.2(i) to 3.2(iii).
…
►Numerical methods and issues for solution of (1.2.72) appear in §§3.2(iv) to 3.2(vii).
…