# Fourier coefficients

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##### 11: 1.8 Fourier Series
The series (1.8.1) is called the Fourier series of $f(x)$, and $a_{n},b_{n}$ are the Fourier coefficients of $f(x)$. … If $f(x)$ and $g(x)$ are continuous, have the same period and same Fourier coefficients, then $f(x)=g(x)$ for all $x$. … If $a_{n}$ and $b_{n}$ are the Fourier coefficients of a piecewise continuous function $f(x)$ on $[0,2\pi]$, then … when $f(x)$ and $g(x)$ are square-integrable and $a_{n},b_{n}$ and $a^{\prime}_{n},b^{\prime}_{n}$ are their respective Fourier coefficients. …
##### 12: 28.34 Methods of Computation
• (d)

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).

• ##### 13: 28.22 Connection Formulas
28.22.5 $g_{\mathit{e},2m}(h)=(-1)^{m}\sqrt{\dfrac{2}{\pi}}\dfrac{\mathrm{ce}_{2m}\left% (\frac{1}{2}\pi,h^{2}\right)}{A_{0}^{2m}(h^{2})},$
28.22.6 $g_{\mathit{e},2m+1}(h)=(-1)^{m+1}\sqrt{\frac{2}{\pi}}\dfrac{\mathrm{ce}_{2m+1}% '\left(\frac{1}{2}\pi,h^{2}\right)}{hA_{1}^{2m+1}(h^{2})},$
28.22.7 $g_{\mathit{o},2m+1}(h)=(-1)^{m}\sqrt{\dfrac{2}{\pi}}\dfrac{\mathrm{se}_{2m+1}% \left(\frac{1}{2}\pi,h^{2}\right)}{hB_{1}^{2m+1}(h^{2})},$
28.22.8 $g_{\mathit{o},2m+2}(h)=(-1)^{m+1}\sqrt{\dfrac{2}{\pi}}\dfrac{\mathrm{se}_{2m+2% }'\left(\frac{1}{2}\pi,h^{2}\right)}{h^{2}B_{2}^{2m+2}(h^{2})},$
##### 14: 28.28 Integrals, Integral Representations, and Integral Equations
28.28.16 $\int_{0}^{\infty}\sin\left(2h\cos y\cosh t\right)\mathrm{Ce}_{2n}\left(t,h^{2}% \right)\mathrm{d}t=-\dfrac{\pi A_{0}^{2n}(h^{2})}{2\mathrm{ce}_{2n}\left(\frac% {1}{2}\pi,h^{2}\right)}\*\left(\mathrm{ce}_{2n}\left(y,h^{2}\right)\mp\dfrac{2% }{\pi C_{2n}(h^{2})}\mathrm{fe}_{2n}\left(y,h^{2}\right)\right),$
28.28.20 $\dfrac{2}{\pi}\int_{0}^{\pi}\mathcal{C}^{(j)}_{2\ell}(2hR)\cos\left(2\ell\phi% \right)\mathrm{ce}_{2m}\left(t,h^{2}\right)\mathrm{d}t=\varepsilon_{\ell}(-1)^% {\ell+m}A^{2m}_{2\ell}(h^{2}){\mathrm{Mc}^{(j)}_{2m}}\left(z,h\right),$
28.28.21 $\dfrac{4}{\pi}\int_{0}^{\pi/2}\mathcal{C}^{(j)}_{2\ell+1}(2hR)\cos\left((2\ell% +1)\phi\right)\mathrm{ce}_{2m+1}\left(t,h^{2}\right)\mathrm{d}t=(-1)^{\ell+m}A% ^{2m+1}_{2\ell+1}(h^{2}){\mathrm{Mc}^{(j)}_{2m+1}}\left(z,h\right),$
28.28.22 $\dfrac{4}{\pi}\int_{0}^{\pi/2}\mathcal{C}^{(j)}_{2\ell+1}(2hR)\sin\left((2\ell% +1)\phi\right)\mathrm{se}_{2m+1}\left(t,h^{2}\right)\mathrm{d}t=(-1)^{\ell+m}B% ^{2m+1}_{2\ell+1}(h^{2}){\mathrm{Ms}^{(j)}_{2m+1}}\left(z,h\right),$
28.28.23 $\dfrac{2}{\pi}\int_{0}^{\pi}\mathcal{C}^{(j)}_{2\ell+2}(2hR)\sin\left((2\ell+2% )\phi\right)\mathrm{se}_{2m+2}\left(t,h^{2}\right)\mathrm{d}t=(-1)^{\ell+m}B^{% 2m+2}_{2\ell+2}(h^{2}){\mathrm{Ms}^{(j)}_{2m+2}}\left(z,h\right).$
##### 15: Bibliography W
• G. Wolf (2008) On the asymptotic behavior of the Fourier coefficients of Mathieu functions. J. Res. Nat. Inst. Standards Tech. 113 (1), pp. 11–15.
• ##### 16: 28.29 Definitions and Basic Properties
For this purpose the discriminant can be expressed as an infinite determinant involving the Fourier coefficients of $Q(x)$; see Magnus and Winkler (1966, §2.3, pp. 28–36). …
##### 18: Bibliography L
• J. N. Lyness (1971) Adjusted forms of the Fourier coefficient asymptotic expansion and applications in numerical quadrature. Math. Comp. 25 (113), pp. 87–104.
• ##### 19: 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 $w_{\mathit{o},s}(z)=\sum_{\ell=0}^{\infty}B_{2\ell+s}\sin(2\ell+s)z,$ $s=1,2$,
where the coefficients satisfy …
##### 20: 28.2 Definitions and Basic Properties
28.2.18 $w(z)=\sum_{n=-\infty}^{\infty}c_{2n}e^{\mathrm{i}(\nu+2n)z}$