conjugate Poisson integral

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1: 1.15 Summability Methods

… ►is the conjugate Poisson integral of $f(x)$. … …

2: 1.14 Integral Transforms

§1.14 Integral Transforms

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

Poisson’s Summation Formula

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§1.14(viii) Compendia

►For more extensive tables of the integral transforms of this section and tables of other integral transforms, see Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2000), Marichev (1983), Oberhettinger (1972, 1974, 1990), Oberhettinger and Badii (1973), Oberhettinger and Higgins (1961), Prudnikov et al. (1986a, b, 1990, 1992a, 1992b).

3: 1.8 Fourier Series

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1.8.6_1 $$\frac{1}{\pi }{\int }_{-\pi }^{\pi }f(x)\overline{g(x)}dx=\frac{1}{2}{a}_0\overline{a_0^{\prime }}+\sum _{n=1}^{\mathrm{\infty }}(a_n\overline{a_n^{\prime }}+b_n\overline{b_n^{\prime }}),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $dx$: differential of $x$ , $\int$: integral and $n$: nonnegative integer

Referenced by:

(1.8.13), (1.8.13), §1.8(i), §1.8(iv), Erratum (V1.2.0) Section 1.8

Permalink:

http://dlmf.nist.gov/1.8.E6_1

Encodings:

pMML, png, TeX

Rearrangement (effective with 1.2.0):

This equation, which was originally (1.8.13), was moved here.

See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

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1.8.6_2 $$\frac{1}{2\pi }{\int }_{-\pi }^{\pi }f(x)\overline{g(x)}dx=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_n\overline{c_n^{\prime }}.$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $dx$: differential of $x$ , $\int$: integral and $n$: nonnegative integer

Referenced by:

§1.8(i), Erratum (V1.2.0) Section 1.8

Permalink:

http://dlmf.nist.gov/1.8.E6_2

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

… ►(1.8.10) continues to apply if either $a$ or $b$ or both are infinite and/or $f(x)$ has finitely many singularities in $(a,b)$, provided that the integral converges uniformly (§1.5(iv)) at $a,b$, and the singularities for all sufficiently large $\lambda$. … ►

§1.8(iv) Poisson’s Summation Formula

… ►It follows from definition (1.14.1) that the integral in (1.8.14) is equal to $\sqrt{2\pi }ℱ\left(f\right)\left(-2\pi n\right)$. …

4: Errata

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18.28.8 $$\frac{1}{2\pi }{\int }_0^{\pi }{Q}_n(\cos \theta ;a,b|q)Q_m(\cos \theta ;a,b|q){\left|\frac{{({\mathrm{e}}^{2i\theta };q)}_{\mathrm{\infty }}}{{(a{\mathrm{e}}^{\mathrm{i}\theta },b{\mathrm{e}}^{\mathrm{i}\theta };q)}_{\mathrm{\infty }}}\right|}^{2}d\theta =\frac{{\delta }_{n,m}}{{(q^{n+1},ab{q}^n;q)}_{\mathrm{\infty }}},$$ $a,b\in ℝ$ or $a=\overline{b}$; $ab\ne 1$; $|a|,|b|\le 1$

The constraint which originally stated that “” has been updated to be “$ab\ne 1$”.

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The following additions were made in Chapter 1:

• New subsections, 1.2(v) Matrices, Vectors, Scalar Products, and Norms and 1.2(vi) Square Matrices, with Equations (1.2.27)–(1.2.77).

• The title of this section was changed from “Determinants” to “Determinants, Linear Operators, and Spectral Expansions”. An extra paragraph just below (1.3.7). New subsection, 1.3(iv) Matrices as Linear Operators, with Equations (1.3.20), (1.3.21).

• In 1.4(v), three new paragraphs on Stieltjes integrals/measure, with Equations (1.4.23_1)–(1.4.23_3).

• In Subsection 1.8(i), the title of the paragraph “Bessel’s Inequality” was changed to “Parseval’s Formula”. We give the relation between the real and the complex coefficients, and include more general versions of Parseval’s Formula, Equations (1.8.6_1), (1.8.6_2). The title of Subsection 1.8(iv) was changed from “Transformations” to “Poisson’s Summation Formula”, and we added an extra remark just below (1.8.14).

• New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).

• New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms, with Equations (1.13.26)–(1.13.31).

• Another form of Parseval’s formula, (1.14.7_5).

• We include several extra remarks and Equations (1.16.3_5), (1.16.9_5). New subsection, 1.16(ix) References for Section 1.16.

• Two extra paragraphs in Subsection 1.17(ii) Integral Representations, with Equations (1.17.12_1), (1.17.12_2); Subsection 1.17(iv) Mathematical Definitions is almost completely rewritten.

• An entire new section, 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions, including new subsections, 1.18(i)–1.18(x), and several equations, (1.18.1)–(1.18.71).

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The entry for $\ast$ to represent complex conjugation was removed (see Version 1.0.19).

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The notation and markup for complex conjugation has been made more consistent in §§1.17(iii), 9.9(i), 10.11, 10.34, 10.63(ii), 12.11(ii), 13.7(ii), 14.30(ii), 23.5(iv), 28.12(ii), 31.15(iii), 34.3(vii), 36.2(iii), 36.2(iv), 36.8, 36.11.

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18.33.3 $${\varphi }_n^{\ast }(z)=z^n\overline{{\varphi }_n({\overline{z}}^{-1})}={\kappa }_n+\sum _{\mathrm{\ell }=1}^n{\overline{\kappa }}_{n,n-\mathrm{\ell }}{z}^{\mathrm{\ell }}$$

Originally this equation was written incorrectly as ${\varphi }_n^{\ast }(z)={\kappa }_n{z}^n+{\sum }_{\mathrm{\ell }=1}^n{\overline{\kappa }}_{n,n-\mathrm{\ell }}{z}^{n-\mathrm{\ell }}$. Also, the equality ${\varphi }_n^{\ast }(z)=z^n\overline{{\varphi }_n({\overline{z}}^{-1})}$ has been added.

Reported 2014-10-03 by Roderick Wong.

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