Jacobi%20nome
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11: 20.3 Graphics
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§20.3(i) -Functions: Real Variable and Real Nome
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… ► …12: 20.1 Special Notation
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►The main functions treated in this chapter are the theta functions where and .
When is fixed the notation is often abbreviated in the literature as , or even as simply , it being then understood that the argument is the primary variable.
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►Primes on the symbols indicate derivatives with respect to the argument of the function.
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►Jacobi’s original notation: , , , , respectively, for , , , , where .
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►Neville’s notation: , , , , respectively, for , , , , where again .
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13: 20.15 Tables
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►This reference gives , , and their logarithmic -derivatives to 4D for , , where is the modular angle given by
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20.15.1
►Spenceley and Spenceley (1947) tabulates , , , to 12D for , , where and is defined by (20.15.1), together with the corresponding values of and .
►Lawden (1989, pp. 270–279) tabulates , , to 5D for , , and also to 5D for .
►Tables of Neville’s theta functions , , , (see §20.1) and their logarithmic -derivatives are given in Abramowitz and Stegun (1964, pp. 582–585) to 9D for , where (in radian measure) , and is defined by (20.15.1).
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14: 20.2 Definitions and Periodic Properties
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§20.2(i) Fourier Series
… ►Corresponding expansions for , , can be found by differentiating (20.2.1)–(20.2.4) with respect to . … ►For fixed , each is an entire function of with period ; is odd in and the others are even. For fixed , each of , , , and is an analytic function of for , with a natural boundary , and correspondingly, an analytic function of for with a natural boundary . … ►For , the -zeros of , , are , , , respectively.15: 20.5 Infinite Products and Related Results
16: 20.9 Relations to Other Functions
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20.9.1
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20.9.3
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20.9.4
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►The relations (20.9.1) and (20.9.2) between and (or ) are solutions of Jacobi’s inversion problem; see Baker (1995) and Whittaker and Watson (1927, pp. 480–485).
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17: 22.3 Graphics
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►Line graphs of the functions , , , , , , , , , , , and for representative values of real and real illustrating the near trigonometric (), and near hyperbolic () limits.
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, , and as functions of real arguments and .
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18: 23.15 Definitions
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►In §§23.15–23.19, and
denote the Jacobi modulus and complementary modulus, respectively, and () denotes the nome; compare §§20.1 and 22.1.
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23.15.6
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23.15.7
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23.15.8
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23.15.9
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19: 22.20 Methods of Computation
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►To compute , , to 10D when , .
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►If either or is given, then we use , , , and , obtaining the values of the theta functions as in §20.14.
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