exponent
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11: 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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12: 30.2 Differential Equations
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►This equation has regular singularities at with exponents
and an irregular singularity of rank 1 at (if ).
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13: 31.2 Differential Equations
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►This equation has regular singularities at , with corresponding exponents
, , , , respectively (§2.7(i)).
All other homogeneous linear differential equations of the second order having four regular singularities in the extended complex plane, , can be transformed into (31.2.1).
►The parameters play different roles: is the singularity parameter; are exponent parameters; is the accessory parameter.
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14: 3.5 Quadrature
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►In the case of Chebyshev weight functions
on , with , the nodes , weights , and error constant , are as follows:
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15: 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(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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16: 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). …17: 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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18: 36.8 Convergent Series Expansions
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19: 36.10 Differential Equations
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