Cauchy sum
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11—20 of 20 matching pages
11: 19.21 Connection Formulas
12: 19.22 Quadratic Transformations
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19.22.9
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19.22.10
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19.22.12
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►If the last variable of is negative, then the Cauchy principal value is
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19.22.14
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13: 1.3 Determinants, Linear Operators, and Spectral Expansions
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1.3.4
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1.3.9
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►for every distinct pair of , or when one of the factors vanishes.
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Cauchy Determinant
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1.3.19
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14: 3.5 Quadrature
15: 1.2 Elementary Algebra
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►which for is the Cauchy-Schwartz inequality
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►If then, depending on , there is either no solution or there are infinitely many solutions, being the sum of a particular solution of (1.2.61) and any solution of .
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►Nonzero vectors are linearly independent if implies that all coefficients are zero.
…The sum of all multiplicities is .
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►Thus is the sum of the (counted according to their multiplicities) eigenvalues of .
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16: 9.12 Scorer Functions
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9.12.15
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9.12.16
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9.12.23
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►where the last integral is a Cauchy principal value (§1.4(v)).
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9.12.29
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17: Bibliography G
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The solution of Cauchy’s problem for two totally hyperbolic linear differential equations by means of Riesz integrals.
Ann. of Math. (2) 48 (4), pp. 785–826.
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Positive sums of the classical orthogonal polynomials.
SIAM J. Math. Anal. 8 (3), pp. 423–447.
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Representations of Integers as Sums of Squares.
Springer-Verlag, New York.
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18: 3.4 Differentiation
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3.4.1
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3.4.7
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3.4.11
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3.4.15
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►If can be extended analytically into the complex plane, then from Cauchy’s integral formula (§1.9(iii))
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19: 19.29 Reduction of General Elliptic Integrals
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►The Cauchy principal value is taken when or is real and negative.
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19.29.12
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►The only cases that are integrals of the third kind are those in which at least one with is a negative integer and those in which and is a positive integer.
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19.29.18
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20: 19.36 Methods of Computation
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►When the differences are moderately small, the iteration is stopped, the elementary symmetric functions of certain differences are calculated, and a polynomial consisting of a fixed number of terms of the sum in (19.19.7) is evaluated.
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►All cases of , , , and are computed by essentially the same procedure (after transforming Cauchy principal values by means of (19.20.14) and (19.2.20)).
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19.36.13
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