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21: 1.2 Elementary Algebra
§1.2(vi) Square Matrices
Special Forms of Square Matrices
Norms of Square Matrices
Non-Defective Square Matrices
22: Bibliography
  • T. M. Apostol and H. S. Zuckerman (1951) On magic squares constructed by the uniform step method. Proc. Amer. Math. Soc. 2 (4), pp. 557–565.
  • T. M. Apostol (1952) Theorems on generalized Dedekind sums. Pacific J. Math. 2 (1), pp. 1–9.
  • T. M. Apostol (2006) Bernoulli’s power-sum formulas revisited. Math. Gaz. 90 (518), pp. 276–279.
  • R. Askey and G. Gasper (1976) Positive Jacobi polynomial sums. II. Amer. J. Math. 98 (3), pp. 709–737.
  • R. Askey (1974) Jacobi polynomials. I. New proofs of Koornwinder’s Laplace type integral representation and Bateman’s bilinear sum. SIAM J. Math. Anal. 5, pp. 119–124.
  • 23: 3.5 Quadrature
    3.5.5 f ( t ) d t = h k = f ( k h ) + E h ( f ) ,
    §3.5(x) Cubature Formulas
    and the square S , given by | x | h , | y | h : …
    Table 3.5.21: Cubature formulas for disk and square.
    Diagram ( x j , y j ) w j R
    24: 14.28 Sums
    §14.28 Sums
    14.28.1 P ν ( z 1 z 2 ( z 1 2 1 ) 1 / 2 ( z 2 2 1 ) 1 / 2 cos ϕ ) = P ν ( z 1 ) P ν ( z 2 ) + 2 m = 1 ( 1 ) m Γ ( ν m + 1 ) Γ ( ν + m + 1 ) P ν m ( z 1 ) P ν m ( z 2 ) cos ( m ϕ ) ,
    where the branches of the square roots have their principal values when z 1 , z 2 ( 1 , ) and are continuous when z 1 , z 2 ( 0 , 1 ] . …
    14.28.2 n = 0 ( 2 n + 1 ) Q n ( z 1 ) P n ( z 2 ) = 1 z 1 z 2 , z 1 1 , z 2 2 ,
    §14.28(iii) Other Sums
    25: 3.11 Approximation Techniques
    §3.11(v) Least Squares Approximations
    For further information on least squares approximations, including examples, see Gautschi (1997a, Chapter 2) and Björck (1996, Chapters 1 and 2). …
    26: Errata
  • Equations (1.8.5), (1.8.6)
    1.8.5 1 π π π | f ( x ) | 2 d x = 1 2 | a 0 | 2 + n = 1 ( | a n | 2 + | b n | 2 )
    1.8.6 1 2 π π π | f ( x ) | 2 d x = n = | c n | 2

    Previously these equations were given as inequalities. For square integrable functions the inequality can be sharpened to = .

  • 27: 23.18 Modular Transformations
    where the square root has its principal value and …
    23.18.7 s ( d , c ) = r = 1 c 1 r c ( d r c d r c 1 2 ) , c > 0 .
    Here s ( d , c ) is a Dedekind sum. …
    28: 30.4 Functions of the First Kind
    30.4.4 𝖯𝗌 n m ( x , γ 2 ) = ( 1 x 2 ) 1 2 m k = 0 g k x k , 1 x 1 ,
    If f ( x ) is mean-square integrable on [ 1 , 1 ] , then formally
    30.4.9 lim N 1 1 | f ( x ) n = m N c n 𝖯𝗌 n m ( x , γ 2 ) | 2 d x = 0 .
    It is also equiconvergent with its expansion in Ferrers functions (as in (30.4.2)), that is, the difference of corresponding partial sums converges to 0 uniformly for 1 x 1 . …
    29: 19.31 Probability Distributions
    R G ( x , y , z ) and R F ( x , y , z ) occur as the expectation values, relative to a normal probability distribution in 2 or 3 , of the square root or reciprocal square root of a quadratic form. …
    19.31.1 𝐱 T 𝐀 𝐱 = r = 1 n s = 1 n a r , s x r x s ,
    30: 26.15 Permutations: Matrix Notation
    The inversion number of σ is a sum of products of pairs of entries in the matrix representation of σ :
    26.15.2 inv ( σ ) = a g h a k ,
    where the sum is over 1 g < k n and n h > 1 . … Let r j ( B ) be the number of ways of placing j nonattacking rooks on the squares of B . …
    26.15.3 R ( x , B ) = j = 0 n r j ( B ) x j .