Fourier-series expansions

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1: 6.16 Mathematical Applications

… ►Compare Figure 6.16.1. … ►It occurs with Fourier-series expansions of all piecewise continuous functions. … …

2: 27.10 Periodic Number-Theoretic Functions

… ►is a periodic function of $n(modk)$ and has the finite Fourier-series expansion …

3: 1.8 Fourier Series

… ►For collections of Fourier-series expansions see Prudnikov et al. (1986a, v. 1, pp. 725–740), Gradshteyn and Ryzhik (2000, pp. 45–49), and Oberhettinger (1973). …

4: 28.2 Definitions and Basic Properties

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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:

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See also:

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

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5: 3.11 Approximation Techniques

… ►In fact, (3.11.11) is the Fourier-series expansion of $f(\cos \theta )$; compare (3.11.6) and §1.8(i). …

6: 29.6 Fourier Series

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

7: 22.11 Fourier and Hyperbolic Series

§22.11 Fourier and Hyperbolic Series

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8: 28.5 Second Solutions ${\mathrm{fe}}_n$, ${\mathrm{ge}}_n$

… ►For further information on $C_n(q)$, $S_n(q)$, and expansions of $f_n(z,q)$, $g_n(z,q)$ in Fourier series or in series of ${\mathrm{ce}}_n$, ${\mathrm{se}}_n$ functions, see McLachlan (1947, Chapter VII) or Meixner and Schäfke (1954, §2.72). …

9: 24.8 Series Expansions

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§24.8(i) Fourier Series

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10: 29.20 Methods of Computation

… ►Initial approximations to the eigenvalues can be found, for example, from the asymptotic expansions supplied in §29.7(i). Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6. …The Fourier series may be summed using Clenshaw’s algorithm; see §3.11(ii). … ►A third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv). … ►The corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. …