unstable pairs

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1: 28.17 Stability as $x\to \pm \mathrm{\infty }$

§28.17 Stability as $x\to \pm \mathrm{\infty }$

►If all solutions of (28.2.1) are bounded when $x\to \pm \mathrm{\infty }$ along the real axis, then the corresponding pair of parameters $(a,q)$ is called stable. All other pairs are unstable. ►For example, positive real values of $a$ with $q=0$ comprise stable pairs, as do values of $a$ and $q$ that correspond to real, but noninteger, values of $\nu$. ►However, if $\mathrm{\Im }\nu \ne 0$, then $(a,q)$ always comprises an unstable pair. …

2: 29.9 Stability

… ►The Lamé equation (29.2.1) with specified values of $k,h,\nu$ is called stable if all of its solutions are bounded on $ℝ$; otherwise the equation is called unstable. If $\nu$ is not an integer, then (29.2.1) is unstable iff $h\le a_{\nu }^0\left(k^2\right)$ or $h$ lies in one of the closed intervals with endpoints $a_{\nu }^m\left(k^2\right)$ and $b_{\nu }^m\left(k^2\right)$, $m=1,2,\mathrm{\dots }$. If $\nu$ is a nonnegative integer, then (29.2.1) is unstable iff $h\le a_{\nu }^0\left(k^2\right)$ or $h\in [b_{\nu }^m\left(k^2\right),a_{\nu }^m\left(k^2\right)]$ for some $m=1,2,\mathrm{\dots },\nu$.

3: 16.25 Methods of Computation

… ►In these cases integration, or recurrence, in either a forward or a backward direction is unstable. …

4: 17.12 Bailey Pairs

§17.12 Bailey Pairs

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Bailey Pairs

… ►When (17.12.5) is iterated the resulting infinite sequence of Bailey pairs is called a Bailey Chain. … ►The Bailey pair that implies the Rogers–Ramanujan identities §17.2(vi) is: … ►The Bailey pair and Bailey chain concepts have been extended considerably. …

5: 28.33 Physical Applications

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Boundary-values problems arising from solution of the two-dimensional wave equation in elliptical coordinates. This yields a pair of equations of the form (28.2.1) and (28.20.1), and the appropriate solution of (28.2.1) is usually a periodic solution of integer order. See §28.33(ii).

… ►Hence from §28.17 the corresponding Mathieu equation is stable or unstable according as $(q,a)$ is in the intersection of $ℒ$ with the colored or the uncolored open regions depicted in Figure 28.17.1. …

6: 24.19 Methods of Computation

… ►For number-theoretic applications it is important to compute $B_{2n}(modp)$ for $2n\le p-3$; in particular to find the irregular pairs $(2n,p)$ for which $B_{2n}\equiv 0(modp)$. We list here three methods, arranged in increasing order of efficiency. … ►

A method related to “Stickelberger codes” is applied in Buhler et al. (2001); in particular, it allows for an efficient search for the irregular pairs $(2n,p)$. Discrete Fourier transforms are used in the computations. See also Crandall (1996, pp. 120–124).

7: 11.13 Methods of Computation

… ►For ${𝐌}_{\nu }\left(x\right)$ both forward and backward integration are unstable, and boundary-value methods are required (§3.7(iii)). …

8: 32.4 Isomonodromy Problems

… ► ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$ can be expressed as the compatibility condition of a linear system, called an isomonodromy problem or Lax pair. Suppose …

9: 10.25 Definitions

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§10.25(iii) Numerically Satisfactory Pairs of Solutions

►Table 10.25.1 lists numerically satisfactory pairs of solutions (§2.7(iv)) of (10.25.1). … ►

Table 10.25.1: Numerically satisfactory pairs of solutions of the modified Bessel’s equation.

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Pair	Region
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10: 27.5 Inversion Formulas

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27.5.3 $$g(n)=\sum _{d|n}f(d)⟺f(n)=\sum _{d|n}g(d)\mu \left(\frac{n}{d}\right).$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $d$: positive integer, $n$: positive integer, $f$: function and $g$: function

A&S Ref:

24.3.1 II.C

Permalink:

http://dlmf.nist.gov/27.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §27.5 and Ch.27

►Special cases of Möbius inversion pairs are: …