exponent pairs
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7 matching pages
1: 15.11 Riemann’s Differential Equation
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►Here , , are the exponent pairs at the points , , , respectively.
Cases in which there are fewer than three singularities are included automatically by allowing the choice for exponent pairs.
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2: 14.2 Differential Equations
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§14.2(iii) Numerically Satisfactory Solutions
►Equation (14.2.2) has regular singularities at , , and , with exponent pairs , , and , respectively; compare §2.7(i). …3: 15.10 Hypergeometric Differential Equation
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►It has regular singularities at , with corresponding exponent pairs
, , , respectively.
When none of the exponent pairs differ by an integer, that is, when none of , , is an integer, we have the following pairs
, of fundamental solutions.
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4: 28.29 Definitions and Basic Properties
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§28.29(ii) Floquet’s Theorem and the Characteristic Exponent
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28.29.9
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►Given together with the condition (28.29.6), the solutions of (28.29.9) are the characteristic
exponents of (28.29.1).
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►If
is a solution of (28.29.9), then , comprise a fundamental pair of solutions of Hill’s equation.
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5: 36.5 Stokes Sets
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►They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4).
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6: 31.11 Expansions in Series of Hypergeometric Functions
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►Then the Fuchs–Frobenius solution at belonging to the exponent
has the expansion (31.11.1) with
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►For example, consider the Heun function which is analytic at and has exponent
at .
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►Here one of the following four pairs of conditions is satisfied:
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7: 28.2 Definitions and Basic Properties
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►This equation has regular singularities at 0 and 1, both with exponents 0 and , and an irregular singular point at .
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►(28.2.1) possesses a fundamental pair of solutions called basic solutions with
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§28.2(iii) Floquet’s Theorem and the Characteristic Exponents
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28.2.16
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►Either or is called a characteristic exponent of (28.2.1).
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