Fourier cosine series

A Fourier cosine series has the formIt represents the even function obtained by reflecting its data across the origin.

For $L>0$ and $0\le x\le L$, set ${\mu }_n=(2n+1)\pi /(2L)$. The cosine modes satisfy a homogeneous Neumann boundary condition at zero and a homogeneous Dirichlet boundary condition at $L$. Their squared norm is $L/2$, and integration by parts givesThus their Fourier coefficients are $2/(L{\mu }_n^2)$, proving the displayed formula by Sturm-Liouville theory. One can also obtain completeness by extending evenly across zero and oddly across $L$: the resulting continuous $4L$-periodic extension has only odd-index cosine modes. The $n^{-2}$ coefficient bound proves uniform convergence, including both endpoints. This expansion supplies the initial transient for a heat equation with mixed endpoint conditions.

The even extension of a function $f:[0,L]\to \mathbb{R}$ is $F_e(x)=f(|x|)$ for $-L\le x\le L$. It is an even function. Its $2L$-periodic extension is the function expanded by the half-range Fourier cosine series.

For $b>0$ and $-b\le y\le b$, the even quadratic polynomial $b^2-y^2$ has the displayed expansion in even Dirichlet eigenfunctions. Put $k_n=(2n+1)\pi /(2b)$. Their squared norm on $[-b,b]$ is $b$. Two integrations by parts, using $\cos (k_{n}b)=0$, giveDivision by $b$ gives the displayed Fourier coefficients. The cubic decay gives uniform convergence. The half-integer frequencies arise from requiring both evenness and zero values at the endpoints, rather than from a periodic boundary condition. This expansion is useful in rectangular duct flow.

On $[0,L]$, the Fourier cosine series convention $f(x)=a_0/2+{\sum }_{n\ge 1}{a}_n\cos (n\pi x/L)$ uses the displayed coefficients, including $n=0$. Orthogonality of the cosine modes gives these formulas for square-integrable functions; the expansion is interpreted in the L2 norm until stronger convergence is established.

For a function prescribed on $0<x<L$, its half-range Fourier cosine series is the Fourier series of the even extension, made $2L$-periodic. Its coefficients are $a_n=(2/L){\int }_0^{L}f(x)\cos (n\pi x/L)dx$, including $n=0$. At a jump the usual piecewise-smooth convergence theorem gives the average of the two one-sided limits of the periodic extension.

For a positive integer $m$, take $f(x)=\sin (mx)$ on $0<x<\pi$ and form its half-range Fourier cosine series. The product-to-sum identity givesThus $a_n=4m/[\pi (m^2-n^2)]$ for opposite parity and zero for the same parity. The exceptional integral $a_m=(2/\pi ){\int }_0^{\pi }\sin (mx)\cos (mx)dx$ is zero. The constant term is $a_0/2$, not $a_0$. These coefficients represent the even periodic extension, not the original sine wave on negative arguments. Their quadratic decay gives absolute convergence.