sine transform
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1: 1.14 Integral Transforms
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§1.14(ii) Fourier Cosine and Sine Transforms
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1.14.10
►In this subsection we let , , , and .
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Inversion
… ► …2: 15.17 Mathematical Applications
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►Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform.
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3: 6.14 Integrals
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§6.14(i) Laplace Transforms
…4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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Example 2: Sine and Cosine Transforms,
►The Fourier cosine and sine transform pairs (1.14.9) & (1.14.11) and (1.14.10) & (1.14.12) can be easily obtained from (1.18.57) as for the Bessel functions reduce to the trigonometric functions, see (10.16.1). … ►For even in this yields the Fourier cosine transform pair (1.14.9) & (1.14.11), and for odd the Fourier sine transform pair (1.14.10) & (1.14.12). … …5: 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.
6: Bibliography B
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Discrete Cosine and Sine Transforms. General Properties, Fast Algorithms and Integer Approximations.
Elsevier/Academic Press, Amsterdam.
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7: 19.7 Connection Formulas
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Reciprocal-Modulus Transformation
… ►Imaginary-Modulus Transformation
… ►Imaginary-Argument Transformation
►With , … ►For two further transformations of this type see Erdélyi et al. (1953b, p. 316). …8: 7.14 Integrals
9: Guide to Searching the DLMF
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Table 1: Query Examples
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Query | Matching records contain |
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"Fourier transform" and series |
both the phrase “Fourier transform” and the word “series”. |
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Fourier (transform or series) |
at least one of “Fourier transform” or “Fourier series”. |
1/(2pi) and "Fourier transform" |
both and the phrase “Fourier transform”. |
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