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Cauchy sum

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1: 17.5 ϕ 0 0 , ϕ 0 1 , ϕ 1 1 Functions
Cauchy’s Sum
2: 1.7 Inequalities
Cauchy–Schwarz Inequality
Cauchy–Schwarz Inequality
3: 2.10 Sums and Sequences
§2.10 Sums and Sequences
For an extension to integrals with Cauchy principal values see Elliott (1998). … and Cauchy’s theorem, we have … These problems can be brought within the scope of §2.4 by means of Cauchy’s integral formula … By allowing the contour in Cauchy’s formula to expand, we find that …
4: 4.10 Integrals
4.10.7 0 x d t ln t = li ( x ) , x > 1 .
The left-hand side of (4.10.7) is a Cauchy principal value (§1.4(v)). …
5: 19.8 Quadratic Transformations
19.8.6 E ( k ) = π 2 M ( 1 , k ) ( a 0 2 - n = 0 2 n - 1 c n 2 ) = K ( k ) ( a 1 2 - n = 2 2 n - 1 c n 2 ) , - < k 2 < 1 , a 0 = 1 , g 0 = k ,
If α 2 > 1 , then the Cauchy principal value is
6: 2.3 Integrals of a Real Variable
is finite and bounded for n = 0 , 1 , 2 , , then the n th error term (that is, the difference between the integral and n th partial sum in (2.3.2)) is bounded in absolute value by | q ( n ) ( 0 ) / ( x n ( x - σ n ) ) | when x exceeds both 0 and σ n . …
2.3.4 a b e i x t q ( t ) d t e i a x s = 0 q ( s ) ( a ) ( i x ) s + 1 - e i b x s = 0 q ( s ) ( b ) ( i x ) s + 1 , x + .
  • (b)

    As t a +

    2.3.14
    p ( t ) p ( a ) + s = 0 p s ( t - a ) s + μ ,
    q ( t ) s = 0 q s ( t - a ) s + λ - 1 ,

    and the expansion for p ( t ) is differentiable. Again λ and μ are positive constants. Also p 0 > 0 (consistent with (a)).

  • But if (d) applies, then the second sum is absent. …
    2.3.31 f ( α , w ) = s = 0 ϕ s ( α ) ( w - a ) s ,
    7: 19.21 Connection Formulas
    19.21.11 6 R G ( x , y , z ) = 3 ( x + y + z ) R F ( x , y , z ) - x 2 R D ( y , z , x ) = x ( y + z ) R D ( y , z , x ) ,
    The latter case allows evaluation of Cauchy principal values (see (19.20.14)). …
    8: 19.22 Quadratic Transformations
    19.22.9 4 π R G ( 0 , a 0 2 , g 0 2 ) = 1 M ( a 0 , g 0 ) ( a 0 2 - n = 0 2 n - 1 c n 2 ) = 1 M ( a 0 , g 0 ) ( a 1 2 - n = 2 2 n - 1 c n 2 ) ,
    19.22.10 R D ( 0 , g 0 2 , a 0 2 ) = 3 π 4 M ( a 0 , g 0 ) a 0 2 n = 0 Q n ,
    19.22.12 R J ( 0 , g 0 2 , a 0 2 , p 0 2 ) = 3 π 4 M ( a 0 , g 0 ) p 0 2 n = 0 Q n ,
    If the last variable of R J is negative, then the Cauchy principal value is
    19.22.14 R J ( 0 , g 0 2 , a 0 2 , - q 0 2 ) = - 3 π 4 M ( a 0 , g 0 ) ( q 0 2 + a 0 2 ) ( 2 + a 0 2 - g 0 2 q 0 2 + g 0 2 n = 0 Q n ) ,
    9: 1.3 Determinants
    1.3.4 det [ a j k ] = = 1 n a j A j .
    1.3.9 det [ a j k ] 2 ( k = 1 n a 1 k 2 ) ( k = 1 n a 2 k 2 ) ( k = 1 n a n k 2 ) .
    for every distinct pair of j , k , or when one of the factors k = 1 n a j k 2 vanishes. …
    Cauchy Determinant
    1.3.19 j , k = - | a j , k - δ j , k |
    10: 3.5 Quadrature
    3.5.5 - f ( t ) d t = h k = - f ( k h ) + E h ( f ) ,
    3.5.12 G 0 ( 1 2 h ) = 1 2 G 0 ( h ) + 1 2 h k = 0 n - 1 f ( x 0 + ( k + 1 2 ) h ) ,
    3.5.15 a b f ( x ) w ( x ) d x = k = 1 n w k f ( x k ) + E n ( f ) ,
    3.5.46 f ( x ) = 1 π - f ( t ) t - x d t , x ,
    where the integral is the Cauchy principal value. …