Fourier series

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11: 29.15 Fourier Series and Chebyshev Series

§29.15 Fourier Series and Chebyshev Series

… ►When $\nu =2n$, $m=0,1,\mathrm{\dots },n$, the Fourier series (29.6.1) terminates: … ►When $\nu =2n+1$, $m=0,1,\mathrm{\dots },n$, the Fourier series (29.6.16) terminates: … ►When $\nu =2n+1$, $m=0,1,\mathrm{\dots },n$, the Fourier series (29.6.31) terminates: … ►When $\nu =2n+1$, $m=0,1,\mathrm{\dots },n$, the Fourier series (29.6.8) terminates: …

12: 22.11 Fourier and Hyperbolic Series

§22.11 Fourier and Hyperbolic Series

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22.11.13 $${\mathrm{sn}}^2(z,k)=\frac{1}{k^2}\left(1-\frac{E}{K}\right)-\frac{2{\pi }^2}{k^2{K}^2}\sum _{n=1}^{\mathrm{\infty }}\frac{n{q}^n}{1-q^{2n}}\cos \left(2n\zeta \right).$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic 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, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\cos z$: cosine function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable

Referenced by:

§22.11

Permalink:

http://dlmf.nist.gov/22.11.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

… ►For further Fourier series see Oberhettinger (1973, pp. 23–27). …

13: 24.8 Series Expansions

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§24.8(i) Fourier Series

… ►If $n=1,2,\mathrm{\dots }$ and $0\le x\le 1$, then …

14: 23.8 Trigonometric Series and Products

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§23.8(i) Fourier Series

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15: 28.4 Fourier Series

§28.4 Fourier Series

… ►The Fourier series of the periodic Mathieu functions converge absolutely and uniformly on all compact sets in the $z$-plane. …

16: 28.34 Methods of Computation

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Solution of the systems of linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4), with the conditions (28.4.9)–(28.4.12) and (28.14.5), by boundary-value methods (§3.6) to determine the Fourier coefficients. Subsequently, the Fourier series can be summed with the aid of Clenshaw’s algorithm (§3.11(ii)). See Meixner and Schäfke (1954, §2.87). This procedure can be combined with §28.34(ii)(d).

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17: 20.2 Definitions and Periodic Properties

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§20.2(i) Fourier Series

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18: 20.14 Methods of Computation

… ►The Fourier series of §20.2(i) usually converge rapidly because of the factors $q^{{(n+\frac{1}{2})}^2}$ or $q^{n^2}$, and provide a convenient way of calculating values of ${\theta }_j\left(z\right|\tau )$. …

19: Bibliography T

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Some definite integrals and Fourier series for Jacobian elliptic functions. Z. Angew. Math. Mech. 49, pp. 95–96.

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External Links:

ISSN 0044-2267, Document, MathReview (B. C. Carlson), ZentralBlatt

Referenced by:

§22.11.

Encodings:

BibTeX

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Fourier Series. Prentice-Hall Inc., Englewood Cliffs, N.J..

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Notes:

Translated from the Russian by Richard A. Silverman. Second English edition published by Dover Publications Inc., New York, 1976.

External Links:

MathReview (R. P. Boas, Jr.), ZentralBlatt

Referenced by:

§1.8(i), §1.8(ii).

Encodings:

BibTeX

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20: 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). …