Jacobi%E2%80%99s
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11: 19.2 Definitions
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►Because is a polynomial, we have
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§19.2(ii) Legendre’s Integrals
… ►Legendre’s complementary complete elliptic integrals are defined via … ►§19.2(iii) Bulirsch’s Integrals
►Bulirsch’s integrals are linear combinations of Legendre’s integrals that are chosen to facilitate computational application of Bartky’s transformation (Bartky (1938)). …12: 22.21 Tables
§22.21 Tables
►Spenceley and Spenceley (1947) tabulates , , , , for and to 12D, or 12 decimals of a radian in the case of . … ►Lawden (1989, pp. 280–284 and 293–297) tabulates , , , , to 5D for , , where ranges from 1. … ►Zhang and Jin (1996, p. 678) tabulates , , for and to 7D. …13: 22.7 Landen Transformations
14: 22.4 Periods, Poles, and Zeros
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►For example, the poles of , abbreviated as in the following tables, are at .
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►Then: (a) In any lattice unit cell has a simple zero at and a simple pole at .
(b) The difference between p and the nearest q is a half-period of .
This half-period will be plus or minus a member of the triple ; the other two members of this triple are quarter periods of .
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►For example, .
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15: 20.4 Values at = 0
16: 20.7 Identities
17: 22.1 Special Notation
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►The functions treated in this chapter are the three principal Jacobian elliptic functions , , ; the nine subsidiary Jacobian elliptic functions , , , , , , , , ; the amplitude function ; Jacobi’s epsilon and zeta functions and .
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►The notation , , is due to Gudermann (1838), following Jacobi (1827); that for the subsidiary functions is due to Glaisher (1882).
Other notations for are and with ; see Abramowitz and Stegun (1964) and Walker (1996).
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18: 18.7 Interrelations and Limit Relations
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