Fourier integrals
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11: 6.16 Mathematical Applications
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§6.16(i) The Gibbs Phenomenon
►Consider the Fourier series … ►Compare Figure 6.16.1. … ►It occurs with Fourier-series expansions of all piecewise continuous functions. … …12: 2.10 Sums and Sequences
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►This identity can be used to find asymptotic approximations for large when the factor changes slowly with , and is oscillatory; compare the approximation of Fourier integrals by integration by parts in §2.3(i).
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13: 1.14 Integral Transforms
14: 1.16 Distributions
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1.16.29
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1.16.38
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►The second to last equality follows from the Fourier integral formula (1.17.8).
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1.16.49
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1.16.51
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15: Bibliography P
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Fourier Series and Integral Transforms.
Cambridge University Press, Cambridge.
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16: 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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17: Errata
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Section 1.14
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There have been extensive changes in the notation used for the integral transforms defined in §1.14. These changes are applied throughout the DLMF. The following table summarizes the changes.
Transform | New | Abbreviated | Old |
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Notation | Notation | Notation | |
Fourier | |||
Fourier Cosine | |||
Fourier Sine | |||
Laplace | |||
Mellin | |||
Hilbert | |||
Stieltjes |
Previously, for the Fourier, Fourier cosine and Fourier sine transforms, either temporary local notations were used or the Fourier integrals were written out explicitly.