Cauchy integral formula

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11: 1.14 Integral Transforms

§1.14 Integral Transforms

… ►where the last integral denotes the Cauchy principal value (1.4.25). … ►

Parseval’s Formula

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Poisson’s Summation Formula

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Parseval-type Formulas

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12: Bibliography H

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Formulae for growth factors in expanding universes containing matter and a cosmological constant. Monthly Notices Roy. Astronom. Soc. 322 (2), pp. 419–425.

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External Links:

Document

Referenced by:

§8.24(ii).

Encodings:

BibTeX

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Asymptotic formulae in combinatory analysis. Proc. London Math. Soc. (2) 17, pp. 75–115.

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Notes:

Reprinted in Ramanujan (1962, pp. 276–309).

External Links:

Document, MathReview Entry, ZentralBlatt

Referenced by:

§27.14(iii).

Encodings:

BibTeX

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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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Notes:

Reprinted in 1993.

External Links:

ISBN 0-471-08703-3; 0-471-58986-1, MathReview (A. E. Heins), ZentralBlatt

Referenced by:

§1.14(v), Ch.3.

Encodings:

BibTeX

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An explicit formula for Bernoulli numbers. Rep. Fac. Sci. Technol. Meijo Univ. 29, pp. 1–6.

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External Links:

ZentralBlatt

Referenced by:

§24.6.

Encodings:

BibTeX

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Explicit formulas for degenerate Bernoulli numbers. Discrete Math. 162 (1-3), pp. 175–185.

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External Links:

ISSN 0012-365X, Document, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.16(i).

Encodings:

BibTeX

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13: 9.12 Scorer Functions

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§9.12(v) Connection Formulas

… ►where the last integral is a Cauchy principal value (§1.4(v)). ►

Mellin–Barnes Type Integral

… ►For other phase ranges combine these results with the connection formulas (9.12.11)–(9.12.14) and the asymptotic expansions given in §9.7. … ►

Integrals

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14: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►An inner product space $V$ is called a Hilbert space if every Cauchy sequence $\{v_n\}$ in $V$ (i. … ►The formulas in §1.18(i) are then: … ►where the integral kernel is given by … ►Now formulas (1.18.13)–(1.18.20) apply. … ►Now formulas (1.18.13)–(1.18.20) apply. …

15: 19.22 Quadratic Transformations

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§19.22(i) Complete Integrals

… ►If the last variable of $R_J$ is negative, then the Cauchy principal value is … ►

§19.22(iii) Incomplete Integrals

… ► … ►

16: 18.17 Integrals

… ►Just as the indefinite integrals (18.17.1), (18.17.3) and (18.17.4), many similar formulas can be obtained by applying (1.4.26) to the differentiation formulas (18.9.15), (18.9.16) and (18.9.19)–(18.9.28). … ►For addition formulas corresponding to (18.17.5) and (18.17.6) see (18.18.8) and (18.18.9), respectively. … ►Some of the resulting formulas are given below. … ►These integrals are Cauchy principal values (§1.4(v)). … ►Formulas (18.17.45) and (18.17.49) are integrated forms of the linearization formulas (18.18.22) and (18.18.23), respectively. …