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1: 2.9 Difference Equations

… ►If $|{\rho }_2|>|{\rho }_1|$, or if $|{\rho }_2|=|{\rho }_1|$ and $\mathrm{\Re }{\alpha }_2>\mathrm{\Re }{\alpha }_1$, then $w_1(n)$ is recessive and $w_2(n)$ is dominant as $n\to \mathrm{\infty }$. 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. … ►But there is an independent solution … ►

2: 36.5 Stokes Sets

… ►The Stokes sets are defined by the exponential dominance condition: … ►

36.5.7 $$X=\frac{9}{20}+20u^4-\frac{Y^2}{20u^2}+6u^2\mathrm{sign}\left(z\right),$$

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Symbols:

$\mathrm{sign}x$: sign of, $z$: real parameter, $X$: scaled coordinate and $Y$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E7

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Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

… ►For , there are two solutions $u$, provided that $|Y|>{(\frac{2}{5})}^{1/2}$. … ►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 …

3: 2.7 Differential Equations

… ►has twice-continuously differentiable solutions … ► $w_1(x)$ is a recessive (or subdominant) solution as $x\to a_1+$, and $w_4(x)$ is a dominant solution as $x\to a_1+$. … ►as $x\to +\mathrm{\infty }$, $w_2(x)$ being recessive and $w_3(x)$ dominant. … ►

§2.7(iv) Numerically Satisfactory Solutions

… ►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 $\mathrm{\Re }z\to +\mathrm{\infty }$. …

4: 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. …

5: 36.11 Leading-Order Asymptotics

… ►

36.11.1

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Symbols:

$t_j(𝐱)$: solutions

Permalink:

http://dlmf.nist.gov/36.11.E1

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Annotations for §36.11 and Ch.36

… ►

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

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Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Phi }}_K(t;𝐱)$: cuspoid catastrophe of codimension $K$ , $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $t$: variable, $K$: codimension and $t_j(𝐱)$: solutions

Permalink:

http://dlmf.nist.gov/36.11.E2

Encodings:

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Annotations for §36.11 and Ch.36

…

6: 27.15 Chinese Remainder Theorem

… ►The Chinese remainder theorem states that a system of congruences $x\equiv a_1(mod{m}_1),\mathrm{\dots },x\equiv a_k(mod{m}_k)$, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod $m$), where $m$ is the product of the moduli. … ►Their product $m$ has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $(modm)$, which is correct to 20 digits. …

7: 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 $\mathrm{ph}z=\pi$, 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 $\frac{3}{2}\pi$. … ►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 $$w_j(z)={\mathrm{e}}^{{\lambda }_{j}z}{z}^{{\mu }_j}\sum _{s=0}^{n-1}\frac{a_{s,j}}{z^s}+R_n^{(j)}(z),$$ $j=1,2$,

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Symbols:

$\mathrm{e}$: base of natural logarithm, $R_n^{(j)}(z)$: remainder, $a_{s,j}$: coefficients, ${\lambda }_j$: roots, ${\mu }_j$: quantities and $w_j(z)$: solutions

Permalink:

http://dlmf.nist.gov/2.11.E19

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Annotations for §2.11(v), §2.11 and Ch.2

… ►For example, using double precision $d_{20}$ is found to agree with (2.11.31) to 13D. …

8: Bibliography H

… ►

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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External Links:

Document, MathReview (E. L. Koh), ZentralBlatt

Referenced by:

§2.5(ii).

Encodings:

BibTeX

… ►

High frequency solutions of the delta wing equations. Proc. Roy. Soc. Edinburgh Sect. A 81 (3-4), pp. 299–316.

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External Links:

ISSN 0308-2105, MathReview (V. M. Babich), ZentralBlatt

Referenced by:

§29.19(ii).

Encodings:

BibTeX

… ►

Numerical Tools for the Study of Finite Gap Solutions of Integrable Systems. Ph.D. Thesis, Technischen Universität Berlin.

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Notes:

(All relevant material, plus corrections, are included in Deconinck et al. (2004).)

External Links:

ZentralBlatt

Referenced by:

§21.5(i).

Encodings:

BibTeX

… ►

Solutions of Poisson’s equation in channel-like geometries. Comput. Phys. Comm. 115 (1), pp. 45–68.

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External Links:

ISSN 0010-4655, Document, Code, MathReview Entry, ZentralBlatt

Referenced by:

§14.31(i).

Encodings:

BibTeX

…

9: 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 …

10: 3.4 Differentiation

… ►

$$B_{-2}^5=\frac{1}{120}(6-10t-15t^2+20t^3-5t^4),$$

… ►

$$B_{-3}^6=-\frac{1}{720}(12-8t-45t^2+20t^3+15t^4-6t^5),$$

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$$B_{-2}^6=\frac{1}{60}(9-9t-30t^2+20t^3+5t^4-3t^5),$$

… ►

$$B_2^6=-\frac{1}{60}(9+9t-30t^2-20t^3+5t^4+3t^5),$$

… ►With the choice $r=k$ (which is crucial when $k$ is large because of numerical cancellation) the integrand equals ${\mathrm{e}}^k$ at the dominant points $\theta =0,2\pi$, and in combination with the factor $k^{-k}$ in front of the integral sign this gives a rough approximation to $1/k!$. …