# Fourier coefficients

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##### 2: 28.11 Expansions in Series of Mathieu Functions
28.11.3 $1=2\sum_{n=0}^{\infty}A_{0}^{2n}(q)\mathrm{ce}_{2n}\left(z,q\right),$
28.11.4 $\cos 2mz=\sum_{n=0}^{\infty}A_{2m}^{2n}(q)\mathrm{ce}_{2n}\left(z,q\right),$ $m\neq 0$,
28.11.5 $\cos(2m+1)z=\sum_{n=0}^{\infty}A_{2m+1}^{2n+1}(q)\mathrm{ce}_{2n+1}\left(z,q% \right),$
28.11.6 $\sin(2m+1)z=\sum_{n=0}^{\infty}B_{2m+1}^{2n+1}(q)\mathrm{se}_{2n+1}\left(z,q% \right),$
28.11.7 $\sin(2m+2)z=\sum_{n=0}^{\infty}B_{2m+2}^{2n+2}(q)\mathrm{se}_{2n+2}\left(z,q% \right).$
##### 3: 28.35 Tables
###### §28.35 Tables
• Ince (1932) includes eigenvalues $a_{n}$, $b_{n}$, and Fourier coefficients for $n=0$ or $1(1)6$, $q=0(1)10(2)20(4)40$; 7D. Also $\mathrm{ce}_{n}\left(x,q\right)$, $\mathrm{se}_{n}\left(x,q\right)$ for $q=0(1)10$, $x=1(1)90$, corresponding to the eigenvalues in the tables; 5D. Notation: $a_{n}=\mathit{be}_{n}-2q$, $b_{n}=\mathit{bo}_{n}-2q$.

• National Bureau of Standards (1967) includes the eigenvalues $a_{n}\left(q\right)$, $b_{n}\left(q\right)$ for $n=0(1)3$ with $q=0(.2)20(.5)37(1)100$, and $n=4(1)15$ with $q=0(2)100$; Fourier coefficients for $\mathrm{ce}_{n}\left(x,q\right)$ and $\mathrm{se}_{n}\left(x,q\right)$ for $n=0(1)15$, $n=1(1)15$, respectively, and various values of $q$ in the interval $[0,100]$; joining factors $g_{\mathit{e},n}(\sqrt{q})$, $f_{\mathit{e},n}(\sqrt{q})$ for $n=0(1)15$ with $q=0(.5\mbox{ to }10)100$ (but in a different notation). Also, eigenvalues for large values of $q$. Precision is generally 8D.

• Stratton et al. (1941) includes $b_{n}$, $b_{n}^{\prime}$, and the corresponding Fourier coefficients for $\mathrm{Se}_{n}(c,x)$ and $\mathrm{So}_{n}(c,x)$ for $n=0$ or $1(1)4$, $c=0(.1~{}\textrm{or}~{}.2)4.5$. Precision is mostly 5S. Notation: $c=2\sqrt{q}$, $b_{n}=a_{n}+2q$, $b^{\prime}_{n}=b_{n}+2q$, and for $\mathrm{Se}_{n}(c,x)$, $\mathrm{So}_{n}(c,x)$ see §28.1.

• Blanch and Clemm (1969) includes eigenvalues $a_{n}\left(q\right)$, $b_{n}\left(q\right)$ for $q=\rho e^{\mathrm{i}\phi}$, $\rho=0(.5)25$, $\phi=5^{\circ}(5^{\circ})90^{\circ}$, $n=0(1)15$; 4D. Also $a_{n}\left(q\right)$ and $b_{n}\left(q\right)$ for $q=\mathrm{i}\rho$, $\rho=0(.5)100$, $n=0(2)14$ and $n=2(2)16$, respectively; 8D. Double points for $n=0(1)15$; 8D. Graphs are included.

##### 5: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions
28.24.6 $\varepsilon_{s}\mathrm{Ie}_{2m}\left(z,h\right)=(-1)^{s}\sum_{\ell=0}^{\infty}% (-1)^{\ell}\dfrac{A_{2\ell}^{2m}(h^{2})}{A_{2s}^{2m}(h^{2})}\left(I_{\ell-s}% \left(he^{-z}\right)I_{\ell+s}\left(he^{z}\right)+I_{\ell+s}\left(he^{-z}% \right)I_{\ell-s}\left(he^{z}\right)\right),$
28.24.7 $\mathrm{Io}_{2m+2}\left(z,h\right)=(-1)^{s}\sum_{\ell=0}^{\infty}(-1)^{\ell}% \dfrac{B_{2\ell+2}^{2m+2}(h^{2})}{B_{2s+2}^{2m+2}(h^{2})}\left(I_{\ell-s}\left% (he^{-z}\right)I_{\ell+s+2}\left(he^{z}\right)-I_{\ell+s+2}\left(he^{-z}\right% )I_{\ell-s}\left(he^{z}\right)\right),$
28.24.8 $\mathrm{Ie}_{2m+1}\left(z,h\right)=(-1)^{s}\sum_{\ell=0}^{\infty}(-1)^{\ell}% \dfrac{B_{2\ell+1}^{2m+1}(h^{2})}{B_{2s+1}^{2m+1}(h^{2})}\left(I_{\ell-s}\left% (he^{-z}\right)I_{\ell+s+1}\left(he^{z}\right)+I_{\ell+s+1}\left(he^{-z}\right% )I_{\ell-s}\left(he^{z}\right)\right),$
28.24.9 $\mathrm{Io}_{2m+1}\left(z,h\right)=(-1)^{s}\sum_{\ell=0}^{\infty}(-1)^{\ell}% \frac{A_{2\ell+1}^{2m+1}(h^{2})}{A_{2s+1}^{2m+1}(h^{2})}\left(I_{\ell-s}\left(% he^{-z}\right)I_{\ell+s+1}\left(he^{z}\right)-I_{\ell+s+1}\left(he^{-z}\right)% I_{\ell-s}\left(he^{z}\right)\right),$
28.24.10 $\varepsilon_{s}\mathrm{Ke}_{2m}\left(z,h\right)=\sum_{\ell=0}^{\infty}\frac{A_% {2\ell}^{2m}(h^{2})}{A_{2s}^{2m}(h^{2})}\left(I_{\ell-s}\left(he^{-z}\right)K_% {\ell+s}\left(he^{z}\right)+I_{\ell+s}\left(he^{-z}\right)K_{\ell-s}\left(he^{% z}\right)\right),$
##### 6: 28.10 Integral Equations
With the notation of §28.4 for Fourier coefficients,
28.10.1 $\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\cos\left(2h\cos z\cos t\right)\mathrm{% ce}_{2n}\left(t,h^{2}\right)\mathrm{d}t=\frac{A_{0}^{2n}(h^{2})}{\mathrm{ce}_{% 2n}\left(\frac{1}{2}\pi,h^{2}\right)}\mathrm{ce}_{2n}\left(z,h^{2}\right),$
28.10.2 $\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\cosh\left(2h\sin z\sin t\right)\mathrm{% ce}_{2n}\left(t,h^{2}\right)\mathrm{d}t=\frac{A_{0}^{2n}(h^{2})}{\mathrm{ce}_{% 2n}\left(0,h^{2}\right)}\mathrm{ce}_{2n}\left(z,h^{2}\right),$
28.10.3 $\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\sin\left(2h\cos z\cos t\right)\mathrm{% ce}_{2n+1}\left(t,h^{2}\right)\mathrm{d}t=-\frac{hA_{1}^{2n+1}(h^{2})}{\mathrm% {ce}_{2n+1}'\left(\frac{1}{2}\pi,h^{2}\right)}\mathrm{ce}_{2n+1}\left(z,h^{2}% \right),$
28.10.4 $\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\cos z\cos t\cosh\left(2h\sin z\sin t% \right)\mathrm{ce}_{2n+1}\left(t,h^{2}\right)\mathrm{d}t=\frac{A_{1}^{2n+1}(h^% {2})}{2\mathrm{ce}_{2n+1}\left(0,h^{2}\right)}\mathrm{ce}_{2n+1}\left(z,h^{2}% \right),$
##### 8: 29.20 Methods of Computation
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. … 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. …
##### 9: 28.23 Expansions in Series of Bessel Functions
28.23.6 ${\mathrm{Mc}^{(j)}_{2m}}\left(z,h\right)=(-1)^{m}\left(\mathrm{ce}_{2m}\left(0% ,h^{2}\right)\right)^{-1}\sum_{\ell=0}^{\infty}(-1)^{\ell}A_{2\ell}^{2m}(h^{2}% ){\cal C}_{2\ell}^{(j)}(2h\cosh z),$
28.23.7 ${\mathrm{Mc}^{(j)}_{2m}}\left(z,h\right)=(-1)^{m}\left(\mathrm{ce}_{2m}\left(% \tfrac{1}{2}\pi,h^{2}\right)\right)^{-1}\sum_{\ell=0}^{\infty}A_{2\ell}^{2m}(h% ^{2}){\cal C}_{2\ell}^{(j)}(2h\sinh z),$
28.23.8 ${\mathrm{Mc}^{(j)}_{2m+1}}\left(z,h\right)=(-1)^{m}\left(\mathrm{ce}_{2m+1}% \left(0,h^{2}\right)\right)^{-1}\sum_{\ell=0}^{\infty}(-1)^{\ell}A_{2\ell+1}^{% 2m+1}(h^{2}){\cal C}_{2\ell+1}^{(j)}(2h\cosh z),$
28.23.9 ${\mathrm{Mc}^{(j)}_{2m+1}}\left(z,h\right)=(-1)^{m+1}\left(\mathrm{ce}_{2m+1}'% \left(\tfrac{1}{2}\pi,h^{2}\right)\right)^{-1}\coth z\sum_{\ell=0}^{\infty}(2% \ell+1)A_{2\ell+1}^{2m+1}(h^{2}){\cal C}_{2\ell+1}^{(j)}(2h\sinh z),$
28.23.10 ${\mathrm{Ms}^{(j)}_{2m+1}}\left(z,h\right)=(-1)^{m}\left(\mathrm{se}_{2m+1}'% \left(0,h^{2}\right)\right)^{-1}\tanh z\sum_{\ell=0}^{\infty}(-1)^{\ell}(2\ell% +1)B_{2\ell+1}^{2m+1}(h^{2}){\cal C}_{2\ell+1}^{(j)}(2h\cosh z),$
##### 10: 27.17 Other Applications
Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series. …