unstable pairs
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1: 28.17 Stability as
§28.17 Stability as
►If all solutions of (28.2.1) are bounded when along the real axis, then the corresponding pair of parameters is called stable. All other pairs are unstable. ►For example, positive real values of with comprise stable pairs, as do values of and that correspond to real, but noninteger, values of . ►However, if , then always comprises an unstable pair. …2: 29.9 Stability
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►The Lamé equation (29.2.1) with specified values of is called stable if all of its solutions are bounded on ; otherwise the equation is called unstable.
If is not an integer, then (29.2.1) is unstable iff or lies in one of the closed intervals with endpoints and , .
If is a nonnegative integer, then (29.2.1) is unstable iff or for some .
3: 16.25 Methods of Computation
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►In these cases integration, or recurrence, in either a forward or a backward direction is unstable.
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4: 17.12 Bailey Pairs
§17.12 Bailey Pairs
… ►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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►Hence from §28.17 the corresponding Mathieu equation is stable or unstable according as is in the intersection of with the colored or the uncolored open regions depicted in Figure 28.17.1.
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6: 24.19 Methods of Computation
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►For number-theoretic applications it is important to compute for ; in particular to find the irregular pairs
for which .
We list here three methods, arranged in increasing order of efficiency.
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7: 11.13 Methods of Computation
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►For both forward and backward integration are unstable, and boundary-value methods are required (§3.7(iii)).
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8: 32.4 Isomonodromy Problems
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– can be expressed as the compatibility condition of a linear system, called an isomonodromy problem or Lax pair.
Suppose
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9: 10.25 Definitions
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