§22.2 Definitions
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- Keywords:
- Jacobian elliptic functions, Neville’s theta functions, analytic properties, definitions, modulus, nome, periods, relations to Jacobian elliptic functions, relations to other functions, theta functions
- Notes:
- See Lawden (1989, §2.1), Walker (1996, §§5.1, 6.2), Walker (2003), Whittaker and Watson (1927, §22.1).
- Referenced by:
- §19.25(v), §20.1, §20.11(iii), §20.7(ii), §20.9(ii), §22.11, §22.20(i), §22.21, §29.2(i), §29.3(v), §29.6(i), §29.8, §31.2(iv), Erratum (V1.0.7) for Other
- Permalink:
- http://dlmf.nist.gov/22.2
- Clarification (effective with 1.0.7):
-
The part of the final sentence of this section that follows (22.2.12)
has been clarified.
Suggested 2013-11-15 by Howard Cohl
- See also:
- Annotations for Ch.22
The nome is given in terms of the modulus by
| 22.2.1 | |||
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where , are defined in §19.2(ii). Inversely,
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where and the theta functions are defined in §20.2(i).
With
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| 22.2.6 | |||
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| 22.2.7 | |||
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| 22.2.8 | |||
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| 22.2.9 | |||
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As a function of , with fixed , each of the 12 Jacobian elliptic functions is doubly periodic, having two periods whose ratio is not real. Each is meromorphic in for fixed , with simple poles and simple zeros, and each is meromorphic in for fixed . For , all functions are real for .
Glaisher’s Notation
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- Keywords:
- Neville’s, theta functions
- See also:
- Annotations for §22.2 and Ch.22
The Jacobian functions are related in the following way. Let , , be any three of the letters , , , . Then
| 22.2.10 | |||
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with the convention that functions with the same two letters are replaced by unity; e.g. .
The six functions containing the letter in their two-letter name are odd in ; the other six are even in .
In terms of Neville’s theta functions (§20.1)
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where
| 22.2.12 | |||
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and on the left-hand side of (22.2.11) , are any pair of the letters , , , , and on the right-hand side they correspond to the integers .
