Jacobi function
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11—20 of 121 matching pages
11: 22.21 Tables
§22.21 Tables
… ► …12: 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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13: 22.17 Moduli Outside the Interval [0,1]
14: 22.7 Landen Transformations
15: 22.2 Definitions
16: 22.13 Derivatives and Differential Equations
17: 20.11 Generalizations and Analogs
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►
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►For , , and , define twelve combined theta functions
by
►
20.11.6
,
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20.11.7
,
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20.11.8
.
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18: 22.18 Mathematical Applications
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Ellipse
… ►where is Jacobi’s epsilon function (§22.16(ii)). … ►By use of the functions and , parametrizations of algebraic equations, such as … ►§22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem
… ►19: 25.1 Special Notation
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►The main related functions are the Hurwitz zeta function
, the dilogarithm , the polylogarithm (also known as Jonquière’s function
), Lerch’s transcendent , and the Dirichlet -functions
.