Cauchy determinant
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21: 18.2 General Orthogonal Polynomials
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§18.2(ix) Moments
… ►The Hankel determinant of order is defined by and …Also define determinants by , and …The recurrence coefficients and in (18.2.11_5) can be expressed in terms of the determinants (18.2.27) and (18.2.28) by …It is to be noted that, although formally correct, the results of (18.2.30) are of little utility for numerical work, as Hankel determinants are notoriously ill-conditioned. …22: 19.17 Graphics
23: 35.6 Confluent Hypergeometric Functions of Matrix Argument
24: 19.2 Definitions
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►The integral for is well defined if , and the Cauchy principal value (§1.4(v)) of is taken if vanishes at an interior point of the integration path.
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►If , then the integral in (19.2.11) is a Cauchy principal value.
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►where the Cauchy principal value is taken if .
Formulas involving that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using .
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►The Cauchy principal value is hyperbolic:
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25: 19.6 Special Cases
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►If , then the Cauchy principal value satisfies
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►Circular and hyperbolic cases, including Cauchy principal values, are unified by using .
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►For the Cauchy principal value of when , see §19.7(iii).
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26: 1.6 Vectors and Vector-Valued Functions
27: 1.5 Calculus of Two or More Variables
28: 23.10 Addition Theorems and Other Identities
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23.10.5
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