Legendre functions on the cut
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11—19 of 19 matching pages
11: 30.16 Methods of Computation
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30.16.9
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12: 14.21 Definitions and Basic Properties
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►When is complex , , and are defined by (14.3.6)–(14.3.10) with replaced by : the principal branches are obtained by taking the principal values of all the multivalued functions appearing in these representations when , and by continuity elsewhere in the -plane with a cut along the interval ; compare §4.2(i).
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13: 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 .14: 10.43 Integrals
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§10.43(iii) Fractional Integrals
… ►For the second equation there is a cut in the -plane along the interval , and all quantities assume their principal values (§4.2(i)). For the Ferrers function and the associated Legendre function , see §§14.3(i) and 14.21(i). … … ►The Kontorovich–Lebedev transform of a function is defined as …15: 19.2 Definitions
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§19.2(ii) Legendre’s Integrals
… ►Legendre’s complementary complete elliptic integrals are defined via … ►Bulirsch’s integrals are linear combinations of Legendre’s integrals that are chosen to facilitate computational application of Bartky’s transformation (Bartky (1938)). … ►Lastly, corresponding to Legendre’s incomplete integral of the third kind we have … ►§19.2(iv) A Related Function:
…16: 30.1 Special Notation
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►The main functions treated in this chapter are the eigenvalues and the spheroidal wave functions
, , , , and , .
…Meixner and Schäfke (1954) use , , , for , , , , respectively.
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Other Notations
►Flammer (1957) and Abramowitz and Stegun (1964) use for , for , and …17: 19.3 Graphics
§19.3 Graphics
… ►See Figures 19.3.1–19.3.6 for complete and incomplete Legendre’s elliptic integrals. … ►In Figures 19.3.7 and 19.3.8 for complete Legendre’s elliptic integrals with complex arguments, height corresponds to the absolute value of the function and color to the phase. … ► … ►18: 30.6 Functions of Complex Argument
§30.6 Functions of Complex Argument
►The solutions … ►Relations to Associated Legendre Functions
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30.6.5
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