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11—20 of 97 matching pages
11: 30.17 Tables
12: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►Assume that is dense in , i.
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, corresponding to distinct eigenvalues, are orthogonal: i.
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►This insures the vanishing of the boundary terms in (1.18.26), and also is a choice which indicates that , as and satisfy the same boundary conditions and thus define the same domains.
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►In what follows will be taken to be a self adjoint extension of following the discussion ending the prior sub-section.
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►, and for .
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13: 14.33 Tables
14: 10.29 Recurrence Relations and Derivatives
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►With defined as in §10.25(ii),
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►For results on modified quotients of the form see Onoe (1955) and Onoe (1956).
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15: 10.13 Other Differential Equations
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10.13.1
,
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►In (10.13.9)–(10.13.11) , are any cylinder functions of orders , respectively, and .
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10.13.9
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10.13.10
,
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10.13.11
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16: 10.36 Other Differential Equations
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►The quantity in (10.13.1)–(10.13.6) and (10.13.8) can be replaced by if at the same time the symbol in the given solutions is replaced by .
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10.36.1
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10.36.2
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17: 28.35 Tables
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►For other tables prior to 1961 see Fletcher et al. (1962, §2.2) and Lebedev and Fedorova (1960, Chapter 11).
18: 23.18 Modular Transformations
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equals
…according as the elements of in (23.15.3) have the respective forms
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23.18.3
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23.18.5
►where the square root has its principal value and
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19: 31.10 Integral Equations and Representations
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►and the kernel is a solution of the partial differential equation
…where is Heun’s operator in the variable
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►The kernel must satisfy
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►where is given by (31.10.4).
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►The kernel must satisfy
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