Fourier series

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21: 29.6 Fourier Series

§29.6 Fourier Series

… ►An alternative version of the Fourier series expansion (29.6.1) is given by … ►

29.6.60 $$\sum _{p=1}^{\mathrm{\infty }}{B}_{2p}{D}_{2p}=\frac{4}{\pi }{\int }_0^K{\left({\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(x,k^2)\right)}^{2}dx.$$

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Symbols:

${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $dx$: differential of $x$ , $\int$: integral, $m$: nonnegative integer, $p$: nonnegative integer, $x$: real variable, $k$: real parameter, $\nu$: real parameter, $B_{2p+1}$: coefficents and $D_{2p}$: coefficents

Permalink:

http://dlmf.nist.gov/29.6.E60

Encodings:

pMML, png, TeX

See also:

Annotations for §29.6(iv), §29.6 and Ch.29

22: 28.31 Equations of Whittaker–Hill and Ince

… ►Formal $2\pi$-periodic solutions can be constructed as Fourier series; compare §28.4: … ►

28.31.5 $$w_{o,s}(z)=\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{B}_{2\mathrm{\ell }+s}\sin (2\mathrm{\ell }+s)z,$$ $s=1,2$,

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Symbols:

$\sin z$: sine function, $z$: complex variable, $w(z)$: Mathieu’s equation solution and $B$: constant

Referenced by:

§28.31(ii)

Permalink:

http://dlmf.nist.gov/28.31.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.31(ii), §28.31 and Ch.28

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23: Bibliography V

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Fourier series representation of Ferrers function $𝖯$ .

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External Links:

FullText

Referenced by:

(14.13.1).

Encodings:

BibTeX

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24: 28.2 Definitions and Basic Properties

… ►The Fourier series of a Floquet solution ►

28.2.18 $$w(z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_{2n}{\mathrm{e}}^{\mathrm{i}(\nu +2n)z}$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: integer, $z$: complex variable, $\nu$: complex parameter, $w(z)$: Mathieu’s equation solution and $c_{2n}$: coefficients

A&S Ref:

20.3.8

Permalink:

http://dlmf.nist.gov/28.2.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(iv), §28.2 and Ch.28

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25: 22.16 Related Functions

… ►

Fourier Series

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26: 14.13 Trigonometric Expansions

… ►These Fourier series converge absolutely when . …

27: 1.17 Integral and Series Representations of the Dirac Delta

… ►

1.17.12 $$\delta \left(x-a\right)=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{\mathrm{i}(x-a)t}dt.$$

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Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $\int$: integral

Referenced by:

§1.16(viii)

Permalink:

http://dlmf.nist.gov/1.17.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §1.17(ii), §1.17 and Ch.1

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1.17.17 $$\frac{1}{2\pi }\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-\mathrm{i}ka}\left({\int }_{-\pi }^{\pi }\varphi (x){\mathrm{e}}^{\mathrm{i}kx}dx\right)=\varphi (a),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $k$: integer and $\varphi (x)$: continuous function

Permalink:

http://dlmf.nist.gov/1.17.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §1.17(iii), §1.17 and Ch.1

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28: Bibliography P

… ►

Fourier Series and Integral Transforms. Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-59209-7; 0-521-59771-4, MathReview, ZentralBlatt

Referenced by:

§1.14(vii).

Encodings:

BibTeX

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29: Bibliography O

… ►

An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

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External Links:

Document, MathReview (David L. Elliott), ZentralBlatt

Referenced by:

§3.11(ii).

Encodings:

BibTeX

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30: 21.2 Definitions

… ►This $g$-tuple Fourier series converges absolutely and uniformly on compact sets of the $𝐳$ and $𝛀$ spaces; hence $\theta \left(𝐳\right|𝛀)$ is an analytic function of (each element of) $𝐳$ and (each element of) $𝛀$. …