Cauchy
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11: 6.2 Definitions and Interrelations
12: 2.10 Sums and Sequences
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βΊFor an extension to integrals with Cauchy principal values see Elliott (1998).
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βΊand Cauchy’s theorem, we have
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βΊThese problems can be brought within the scope of §2.4 by means of Cauchy’s integral formula
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βΊBy allowing the contour in Cauchy’s formula to expand, we find that
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13: 9.10 Integrals
14: Bibliography H
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Applied and Computational Complex Analysis. Vol. 3: Discrete Fourier Analysis—Cauchy Integrals—Construction of Conformal Maps—Univalent Functions.
Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons Inc.], New York.
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15: 18.40 Methods of Computation
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18.40.6
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16: 1.14 Integral Transforms
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1.14.3
βΊwhere the last integral denotes the Cauchy principal value (1.4.25).
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1.14.41
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1.14.44
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17: 9.12 Scorer Functions
18: 19.20 Special Cases
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βΊCases encountered in dynamical problems are usually circular; hyperbolic cases include Cauchy principal values.
If are permuted so that , then the Cauchy principal value of is given by
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19: 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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20: 4.2 Definitions
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