# dominant solutions

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## 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 $|\rho_{2}|=|\rho_{1}|$ and $\Re\alpha_{2}=\Re\alpha_{1}$ neither solution is dominant and both are unique. …
##### 2: 2.7 Differential Equations
2.7.30 $w_{1}(x)/w_{4}(x)\to 0,$ $x\to a_{1}+$,
$w_{1}(x)$ is a recessive (or subdominant) solution as $x\to a_{1}+$, and $w_{4}(x)$ is a dominant solution as $x\to a_{1}+$. … The solutions $w_{1}(z)$ and $w_{2}(z)$ are respectively recessive and dominant as $\Re z\to-\infty$, and vice versa as $\Re z\to+\infty$. …
##### 3: 18.39 Applications in the Physical Sciences
The solutions of (18.39.8) are subject to boundary conditions at $a$ and $b$. … 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 $l$, but $n$ still dominates. … The radial Coulomb wave functions $R_{n,l}(r)$, solutions of …
##### 4: 36.5 Stokes Sets
The Stokes sets are defined by the exponential dominance condition: … For $z<0$, there are two solutions $u$, provided that $|Y|>(\frac{2}{5})^{1/2}$. … The first sheet corresponds to $x<0$ and is generated as a solution of Equations (36.5.6)–(36.5.9). …For $\left|Y\right|>Y_{1}$ the second sheet is generated by a second solution of (36.5.6)–(36.5.9), and for $\left|Y\right| it is generated by the roots of the polynomial equation …
##### 5: 36.11 Leading-Order Asymptotics
36.11.1 $t_{1}(\mathbf{x})
36.11.2 $\Psi_{K}\left(\mathbf{x}\right)=\sqrt{2\pi}\sum\limits_{j=1}^{j_{\max}(\mathbf% {x})}\exp\left(i\left(\Phi_{K}\left(t_{j}(\mathbf{x});\mathbf{x}\right)+\tfrac% {1}{4}\pi(-1)^{j+K+1}\right)\right)\left|\frac{{\partial}^{2}\Phi_{K}\left(t_{% j}(\mathbf{x});\mathbf{x}\right)}{{\partial t}^{2}}\right|^{-1/2}(1+o\left(1% \right)).$
##### 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 $\operatorname{ph}z=\pi$, is increasingly significant as $\operatorname{ph}z$ passes from $\pi$ to $\frac{3}{2}\pi$, and becomes the dominant contribution after $\operatorname{ph}z$ passes $\frac{3}{2}\pi$. … Rays (or curves) on which one contribution in a compound asymptotic expansion achieves maximum dominance over another are called Stokes lines ($\theta=\pi$ in the present example). …
###### §2.11(v) Exponentially-Improved Expansions (continued)
2.11.19 $w_{j}(z)=e^{\lambda_{j}z}z^{\mu_{j}}\sum_{s=0}^{n-1}\frac{a_{s,j}}{z^{s}}+R_{n% }^{(j)}(z),$ $j=1,2$,
##### 7: 3.6 Linear Difference Equations
###### §3.6(ii) Homogeneous Equations
Thus $Y_{n}\left(1\right)$ is dominant and can be computed by forward recursion, whereas $J_{n}\left(1\right)$ is recessive and has to be computed by backward recursion. …
##### 8: 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.
• ##### 9: 1.2 Elementary Algebra
Square $n\times n$ matrices (said to be of order $n$ ) dominate the use of matrices in the DLMF, and they have many special properties. … has a unique solution, $\mathbf{b}={\mathbf{A}}^{-1}\mathbf{c}$. If $\det(\mathbf{A})=0$ then, depending on $\mathbf{c}$, 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 $\mathbf{A}\mathbf{b}=\boldsymbol{{0}}$. 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). …