Legendre
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11: 14.4 Graphics
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§14.4(iii) Associated Legendre Functions: 2D Graphs
… ► … ► ►§14.4(iv) Associated Legendre Functions: 3D Surfaces
… ►12: 14.2 Differential Equations
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§14.2(i) Legendre’s Equation
… ►Standard solutions: , , , , , . … ►§14.2(ii) Associated Legendre Equation
… ►Ferrers functions and the associated Legendre functions are related to the Legendre functions by the equations , , , , . … ►§14.2(iii) Numerically Satisfactory Solutions
…13: 14.9 Connection Formulas
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§14.9(i) Connections Between , , ,
… ►§14.9(ii) Connections Between , ,
… ►§14.9(iii) Connections Between , , ,
… ►§14.9(iv) Whipple’s Formula
…14: 14.10 Recurrence Relations and Derivatives
§14.10 Recurrence Relations and Derivatives
… ► also satisfies (14.10.1)–(14.10.5). ►
14.10.6
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also satisfies (14.10.6) and (14.10.7).
In addition, and satisfy (14.10.3)–(14.10.5).
15: 14.22 Graphics
16: 14.28 Sums
17: 14.33 Tables
§14.33 Tables
►Abramowitz and Stegun (1964, Chapter 8) tabulates for , , 5–8D; for , , 5–7D; and for , , 6–8D; and for , , 6S; and for , , 6S. (Here primes denote derivatives with respect to .)
Zhang and Jin (1996, Chapter 4) tabulates for , , 7D; for , , 8D; for , , 8S; for , , 8D; for , , , , 8S; for , , 8S; for , , , 5D; for , , 7S; for , , 8S. Corresponding values of the derivative of each function are also included, as are 6D values of the first 5 -zeros of and of its derivative for , .
Žurina and Karmazina (1963) tabulates the conical functions for , , 7S; for , , 7S. Auxiliary tables are included to assist computation for larger values of when .
18: 14.24 Analytic Continuation
§14.24 Analytic Continuation
►Let be an arbitrary integer, and and denote the branches obtained from the principal branches by making circuits, in the positive sense, of the ellipse having as foci and passing through . … ►Next, let and denote the branches obtained from the principal branches by encircling the branch point (but not the branch point ) times in the positive sense. … ►For fixed , other than or , each branch of and is an entire function of each parameter and . ►The behavior of and as from the left on the upper or lower side of the cut from to can be deduced from (14.8.7)–(14.8.11), combined with (14.24.1) and (14.24.2) with .19: 14.5 Special Values
20: 14.17 Integrals
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►In (14.17.1)–(14.17.4), may be replaced by , and in (14.17.3) and (14.17.4), may be replaced by .
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