# dominant solutions

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## 7 matching pages

##### 1: 2.9 Difference Equations

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►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 $\mathrm{\Re}{\alpha}_{2}=\mathrm{\Re}{\alpha}_{1}$ neither solution is dominant and both are unique.
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##### 2: 2.7 Differential Equations

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2.7.30
$${w}_{1}(x)/{w}_{4}(x)\to 0,$$
$x\to {a}_{1}+$,

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${w}_{1}(x)$ 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 $\mathrm{\Re}z\to -\mathrm{\infty}$, and*vice versa*as $\mathrm{\Re}z\to +\mathrm{\infty}$. …##### 3: 36.5 Stokes Sets

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►The Stokes sets are defined by the exponential dominance condition:
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►For $$, there are two solutions
$u$, provided that $|Y|>{(\frac{2}{5})}^{1/2}$.
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►The first sheet corresponds to $$ 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 $$ it is generated by the roots of the polynomial equation
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##### 4: 36.11 Leading-Order Asymptotics

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36.11.1
$$

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36.11.2
$${\mathrm{\Psi}}_{K}\left(\mathbf{x}\right)=\sqrt{2\pi}\sum _{j=1}^{{j}_{\mathrm{max}}(\mathbf{x})}\mathrm{exp}\left(\mathrm{i}\left({\mathrm{\Phi}}_{K}({t}_{j}(\mathbf{x});\mathbf{x})+\frac{1}{4}\pi {(-1)}^{j+K+1}\right)\right){\left|\frac{{\partial}^{2}{\mathrm{\Phi}}_{K}({t}_{j}(\mathbf{x});\mathbf{x})}{{\partial t}^{2}}\right|}^{-1/2}(1+o\left(1\right)).$$

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##### 5: 2.11 Remainder Terms; Stokes Phenomenon

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►In effect, (2.11.7) “corrects” (2.11.6) by introducing a term that is relatively exponentially small in the neighborhood of $\mathrm{ph}z=\pi $, is increasingly significant as $\mathrm{ph}z$ passes from $\pi $ to $\frac{3}{2}\pi $, and becomes the dominant contribution after $\mathrm{ph}z$ passes $\frac{3}{2}\pi $.
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►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)

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2.11.19
$${w}_{j}(z)={\mathrm{e}}^{{\lambda}_{j}z}{z}^{{\mu}_{j}}\sum _{s=0}^{n-1}\frac{{a}_{s,j}}{{z}^{s}}+{R}_{n}^{(j)}(z),$$
$j=1,2$,

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##### 6: 3.6 Linear Difference Equations

###### §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. …##### 7: Bibliography H

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Asymptotic expansion of a class of integral transforms with algebraically dominated kernels.
J. Math. Anal. Appl. 35 (2), pp. 405–433.
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High frequency solutions of the delta wing equations.
Proc. Roy. Soc. Edinburgh Sect. A 81 (3-4), pp. 299–316.
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Numerical Tools for the Study of Finite Gap Solutions of Integrable Systems.
Ph.D. Thesis, Technischen Universität Berlin.
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Solutions of Poisson’s equation in channel-like geometries.
Comput. Phys. Comm. 115 (1), pp. 45–68.
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