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1: 35.11 Tables
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►Tables of zonal polynomials are given in James (1964) for , Parkhurst and James (1974) for , and Muirhead (1982, p. 238) for .
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2: 10.37 Inequalities; Monotonicity
3: 20.1 Special Notation
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►The main functions treated in this chapter are the theta functions where and .
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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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►McKean and Moll’s notation: , .
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, | integers. |
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the nome, , . Since is not a single-valued function of , it is assumed that is known, even when is specified. Most applications concern the rectangular case , , so that and and are uniquely related. | |
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4: Sidebar 7.SB1: Diffraction from a Straightedge
5: 4.18 Inequalities
6: 27.9 Quadratic Characters
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►For an odd prime , the Legendre symbol
is defined as follows.
If divides , then the value of is .
…The Legendre symbol , as a function of , is a Dirichlet character (mod ).
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►If an odd integer has prime factorization , then the Jacobi symbol
is defined by , with .
The Jacobi symbol is a Dirichlet character (mod ).
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7: 20.2 Definitions and Periodic Properties
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►Corresponding expansions for , , can be found by differentiating (20.2.1)–(20.2.4) with respect to .
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►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 .
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►For , the -zeros of , , are , , , respectively.
8: 27.6 Divisor Sums
9: 27.7 Lambert Series as Generating Functions
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►If , then the quotient is the sum of a geometric series, and when the series (27.7.1) converges absolutely it can be rearranged as a power series:
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27.7.2
►Again with , special cases of (27.7.2) include:
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27.7.3
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27.7.4
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