# Fourier-series solutions

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##### 2: 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 $w_{\mathit{o},s}(z)=\sum_{\ell=0}^{\infty}B_{2\ell+s}\sin(2\ell+s)z,$ $s=1,2$,
##### 3: 28.2 Definitions and Basic Properties
The Fourier series of a Floquet solution
28.2.18 $w(z)=\sum_{n=-\infty}^{\infty}c_{2n}e^{\mathrm{i}(\nu+2n)z}$
##### 4: 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).

• ##### 5: 29.6 Fourier Series
###### §29.6 FourierSeries
When $\nu\neq 2n$, where $n$ is a nonnegative integer, it follows from §2.9(i) that for any value of $H$ the system (29.6.4)–(29.6.6) has a unique recessive solution $A_{0},A_{2},A_{4},\dots$; furthermore … In the special case $\nu=2n$, $m=0,1,\dots,n$, there is a unique nontrivial solution with the property $A_{2p}=0$, $p=n+1,n+2,\dots$. This solution can be constructed from (29.6.4) by backward recursion, starting with $A_{2n+2}=0$ and an arbitrary nonzero value of $A_{2n}$, followed by normalization via (29.6.5) and (29.6.6). … An alternative version of the Fourier series expansion (29.6.1) is given by …
##### 6: Bibliography V
• H. Van de Vel (1969) On the series expansion method for computing incomplete elliptic integrals of the first and second kinds. Math. Comp. 23 (105), pp. 61–69.
• C. Van Loan (1992) Computational Frameworks for the Fast Fourier Transform. Frontiers in Applied Mathematics, Vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
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• H. Volkmer (2021) Fourier series representation of Ferrers function ${\sf P}$ .
• ##### 7: Bibliography O
• F. Oberhettinger (1990) Tables of Fourier Transforms and Fourier Transforms of Distributions. Springer-Verlag, Berlin.
• F. Oberhettinger (1973) Fourier Expansions. A Collection of Formulas. Academic Press, New York-London.
• A. B. Olde Daalhuis (2004a) Inverse factorial-series solutions of difference equations. Proc. Edinb. Math. Soc. (2) 47 (2), pp. 421–448.
• J. Oliver (1977) 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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• ##### 8: Bibliography T
• I. C. Tang (1969) Some definite integrals and Fourier series for Jacobian elliptic functions. Z. Angew. Math. Mech. 49, pp. 95–96.
• A. Terras (1999) Fourier Analysis on Finite Groups and Applications. London Mathematical Society Student Texts, Vol. 43, Cambridge University Press, Cambridge.
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• ##### 9: Bibliography S
• O. A. Sharafeddin, H. F. Bowen, D. J. Kouri, and D. K. Hoffman (1992) Numerical evaluation of spherical Bessel transforms via fast Fourier transforms. J. Comput. Phys. 100 (2), pp. 294–296.
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• ##### 10: Bibliography C
• S. M. Candel (1981) An algorithm for the Fourier-Bessel transform. Comput. Phys. Comm. 23 (4), pp. 343–353.
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• C. W. Clenshaw (1957) The numerical solution of linear differential equations in Chebyshev series. Proc. Cambridge Philos. Soc. 53 (1), pp. 134–149.
• H. S. Cohl (2013a) Fourier, Gegenbauer and Jacobi expansions for a power-law fundamental solution of the polyharmonic equation and polyspherical addition theorems. SIGMA Symmetry Integrability Geom. Methods Appl. 9, pp. Paper 042, 26.
• J. W. Cooley and J. W. Tukey (1965) An algorithm for the machine calculation of complex Fourier series. Math. Comp. 19 (90), pp. 297–301.