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1: 27.14 Unrestricted Partitions
§27.14(vi) Ramanujan’s Tau Function
The discriminant function Δ ( τ ) is defined by
27.14.16 Δ ( τ ) = ( 2 π ) 12 ( η ( τ ) ) 24 , τ > 0 ,
27.14.17 Δ ( a τ + b c τ + d ) = ( c τ + d ) 12 Δ ( τ ) ,
2: 23.3 Differential Equations
The discriminant1.11(ii)) is given by …
3: Bibliography M
  • H. R. McFarland and D. St. P. Richards (2001) Exact misclassification probabilities for plug-in normal quadratic discriminant functions. I. The equal-means case. J. Multivariate Anal. 77 (1), pp. 21–53.
  • H. R. McFarland and D. St. P. Richards (2002) Exact misclassification probabilities for plug-in normal quadratic discriminant functions. II. The heterogeneous case. J. Multivariate Anal. 82 (2), pp. 299–330.
  • 4: 23.1 Special Notation
    (For other notation see Notation for the Special Functions.)
    𝕃 lattice in .
    = e i π τ nome.
    Δ discriminant g 2 3 - 27 g 3 2 .
    The main functions treated in this chapter are the Weierstrass -function ( z ) = ( z | 𝕃 ) = ( z ; g 2 , g 3 ) ; the Weierstrass zeta function ζ ( z ) = ζ ( z | 𝕃 ) = ζ ( z ; g 2 , g 3 ) ; the Weierstrass sigma function σ ( z ) = σ ( z | 𝕃 ) = σ ( z ; g 2 , g 3 ) ; the elliptic modular function λ ( τ ) ; Klein’s complete invariant J ( τ ) ; Dedekind’s eta function η ( τ ) . …
    5: 1.11 Zeros of Polynomials
    The discriminant of f ( z ) is defined by …The elementary symmetric functions of the zeros are (with a n 0 ) … The discriminant of g ( w ) is
    1.11.12 D = - 4 p 3 - 27 q 2 .
    The discriminant of g ( w ) is …
    6: 28.29 Definitions and Basic Properties
    §28.29(iii) Discriminant and Eigenvalues in the Real Case
    The function …is called the discriminant of (28.29.1). It is an entire function of λ . … For this purpose the discriminant can be expressed as an infinite determinant involving the Fourier coefficients of Q ( x ) ; see Magnus and Winkler (1966, §2.3, pp. 28–36). …
    7: 23.19 Interrelations