# §1.8 Fourier Series

## §1.8(i) Definitions and Elementary Properties

Formally, if $f(x)$ is a real- or complex-valued $2\pi$-periodic function,

 1.8.1 $f(x)=\tfrac{1}{2}a_{0}+\sum^{\infty}_{n=1}(a_{n}\cos\left(nx\right)+b_{n}\sin% \left(nx\right)),$ ⓘ Symbols: $\cos\NVar{z}$: cosine function, $\sin\NVar{z}$: sine function and $n$: nonnegative integer Referenced by: §1.18(v), §1.8(i), §1.8(i), §1.8(ii), §18.2(viii) Permalink: http://dlmf.nist.gov/1.8.E1 Encodings: TeX, pMML, png See also: Annotations for §1.8(i), §1.8 and Ch.1
 1.8.2 $\displaystyle a_{n}$ $\displaystyle=\frac{1}{\pi}\int^{\pi}_{-\pi}f(x)\cos\left(nx\right)\,\mathrm{d% }x,$ $n=0,1,2,\dots$, $\displaystyle b_{n}$ $\displaystyle=\frac{1}{\pi}\int^{\pi}_{-\pi}f(x)\sin\left(nx\right)\,\mathrm{d% }x,$ $n=1,2,\dots$.

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 $f(-x)=f(x)$, then $b_{n}=0$ for all $n$.

If $f(-x)=-f(x)$, then $a_{n}=0$ for all $n$.

### Alternative Form

 1.8.3 $f(x)=\sum^{\infty}_{n=-\infty}c_{n}{\mathrm{e}}^{\mathrm{i}nx},$ ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $n$: nonnegative integer Referenced by: §1.17(iii), §1.18(v) Permalink: http://dlmf.nist.gov/1.8.E3 Encodings: TeX, pMML, png See also: Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1
 1.8.4 $c_{n}=\frac{1}{2\pi}\int^{\pi}_{-\pi}f(x){\mathrm{e}}^{-\mathrm{i}nx}\,\mathrm% {d}x.$

Here $c_{n}$ is related to $a_{n}$ and $b_{n}$ in (1.8.1), (1.8.2) by $c_{n}=\frac{1}{2}(a_{n}-\mathrm{i}b_{n})$, $c_{-n}=\frac{1}{2}(a_{n}+\mathrm{i}b_{n})$ for $n>0$ and $c_{0}=\frac{1}{2}a_{0}$.

### Parseval’s Formula

 1.8.5 $\frac{1}{\pi}\int^{\pi}_{-\pi}{\left|f(x)\right|}^{2}\,\mathrm{d}x=\tfrac{1}{2% }{\left|a_{0}\right|}^{2}+\sum^{\infty}_{n=1}({\left|a_{n}\right|}^{2}+{\left|% b_{n}\right|}^{2}),$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $n$: nonnegative integer and $\left|\NVar{x}\right|$: absolute value of $\NVar{x}$ Referenced by: Erratum (V1.2.0) for Equations (1.8.5), (1.8.6) Permalink: http://dlmf.nist.gov/1.8.E5 Encodings: TeX, pMML, png Modification (effective with 1.2.0): Previously this equation was given as an inequality. For square integrable functions this inequality $\geq$ can be sharpened to $=$. See also: Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1
 1.8.6 $\frac{1}{2\pi}\int^{\pi}_{-\pi}{\left|f(x)\right|}^{2}\,\mathrm{d}x=\sum^{% \infty}_{n=-\infty}{\left|c_{n}\right|}^{2},$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $n$: nonnegative integer and $\left|\NVar{x}\right|$: absolute value of $\NVar{x}$ Referenced by: §1.8(i), Erratum (V1.2.0) for Equations (1.8.5), (1.8.6) Permalink: http://dlmf.nist.gov/1.8.E6 Encodings: TeX, pMML, png Modification (effective with 1.2.0): Previously this equation was given as an inequality. For square integrable functions this inequality $\geq$ can be sharpened to $=$. See also: Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

where $f(x)$ is square-integrable on $[-\pi,\pi]$ and $a_{n},b_{n},c_{n}$ are given by (1.8.2), (1.8.4). If $g(x)$ is also square-integrable with Fourier coefficients $a_{n}^{\prime},b_{n}^{\prime}$ or $c_{n}^{\prime}$ then

 1.8.6_1 $\frac{1}{\pi}\int^{\pi}_{-\pi}f(x)\overline{g(x)}\,\mathrm{d}x=\tfrac{1}{2}a_{% 0}\overline{a_{0}^{\prime}}+\sum^{\infty}_{n=1}(a_{n}\overline{a^{\prime}_{n}}% +b_{n}\overline{b^{\prime}_{n}}),$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$: complex conjugate, $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $n$: nonnegative integer Referenced by: (1.8.13), (1.8.13), §1.8(i), §1.8(iv), Erratum (V1.2.0) Section 1.8 Permalink: http://dlmf.nist.gov/1.8.E6_1 Encodings: TeX, pMML, png Rearrangement (effective with 1.2.0): This equation, which was originally (1.8.13), was moved here. See also: Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1
 1.8.6_2 $\frac{1}{2\pi}\int^{\pi}_{-\pi}f(x)\overline{g(x)}\,\mathrm{d}x=\sum^{\infty}_% {n=-\infty}c_{n}\overline{c_{n}^{\prime}}.$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$: complex conjugate, $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $n$: nonnegative integer Referenced by: §1.8(i), Erratum (V1.2.0) Section 1.8 Permalink: http://dlmf.nist.gov/1.8.E6_2 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

### Asymptotic Estimates of Coefficients

If $f(x)$ is of period $2\pi$, and $f^{(m)}(x)$ is piecewise continuous, then

 1.8.7 $a_{n},b_{n},c_{n}=o\left(n^{-m}\right),$ $n\to\infty$. ⓘ Symbols: $o\left(\NVar{x}\right)$: order less than, $m$: nonnegative integer and $n$: nonnegative integer Referenced by: §3.11(ii) Permalink: http://dlmf.nist.gov/1.8.E7 Encodings: TeX, pMML, png See also: Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

### Uniqueness of Fourier Series

If $f(x)$ and $g(x)$ are continuous, have the same period and same Fourier coefficients, then $f(x)=g(x)$ for all $x$.

### Lebesgue Constants

 1.8.8 $L_{n}=\frac{1}{\pi}\int^{\pi}_{0}\frac{\left|\sin\left(n+\frac{1}{2}\right)t% \right|}{\sin\left(\frac{1}{2}t\right)}\,\mathrm{d}t,$ $n=0,1,\dots$. ⓘ Defines: $L_{n}$: Lebesgue constants (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\sin\NVar{z}$: sine function, $n$: nonnegative integer and $\left|\NVar{x}\right|$: absolute value of $\NVar{x}$ Permalink: http://dlmf.nist.gov/1.8.E8 Encodings: TeX, pMML, png See also: Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

As $n\to\infty$

 1.8.9 $L_{n}\sim(4/{\pi}^{2})\ln n;$

see Frenzen and Wong (1986).

### Riemann–Lebesgue Lemma

For $f(x)$ piecewise continuous on $[a,b]$ and real $\lambda$,

 1.8.10 $\int^{b}_{a}f(x){\mathrm{e}}^{\mathrm{i}\lambda x}\,\mathrm{d}x\to 0,$ as $\lambda\to\infty$. ⓘ

(1.8.10) continues to apply if either $a$ or $b$ or both are infinite and/or $f(x)$ has finitely many singularities in $(a,b)$, provided that the integral converges uniformly (§1.5(iv)) at $a,b$, and the singularities for all sufficiently large $\lambda$.

## §1.8(ii) Convergence

Let $f(x)$ be an absolutely integrable function of period $2\pi$, and continuous except at a finite number of points in any bounded interval. Then the series (1.8.1) converges to the sum

 1.8.11 $\tfrac{1}{2}f(x-)+\tfrac{1}{2}f(x+)$ ⓘ Permalink: http://dlmf.nist.gov/1.8.E11 Encodings: TeX, pMML, png See also: Annotations for §1.8(ii), §1.8 and Ch.1

at every point at which $f(x)$ has both a left-hand derivative (that is, (1.4.4) applies when $h\to 0-$) and a right-hand derivative (that is, (1.4.4) applies when $h\to 0+$). The convergence is non-uniform, however, at points where $f(x-)\neq f(x+)$; see §6.16(i).

For other tests for convergence see Titchmarsh (1962b, pp. 405–410).

## §1.8(iii) Integration and Differentiation

If $a_{n}$ and $b_{n}$ are the Fourier coefficients of a piecewise continuous function $f(x)$ on $[0,2\pi]$, then

 1.8.12 $\int^{x}_{0}(f(t)-\tfrac{1}{2}a_{0})\,\mathrm{d}t=\sum^{\infty}_{n=1}\frac{a_{% n}\sin\left(nx\right)+b_{n}(1-\cos\left(nx\right))}{n},$ $0\leq x\leq 2\pi$.

If a function $f(x)\in C^{2}[0,2\pi]$ is periodic, with period $2\pi$, then the series obtained by differentiating the Fourier series for $f(x)$ term by term converges at every point to $f^{\prime}(x)$.

## §1.8(iv) Poisson’s Summation Formula

 1.8.13 Moved to (1.8.6_1). ⓘ Referenced by: (1.8.6_1), §1.8(i), §1.8(iv) Permalink: http://dlmf.nist.gov/1.8.E13 Rearrangement (effective with 1.2.0): This equation has been moved to (1.8.6_1). See also: Annotations for §1.8(iv), §1.8 and Ch.1

Suppose that $f(x)$ is twice continuously differentiable and $f(x)$ and $\left|f^{\prime\prime}(x)\right|$ are integrable over $(-\infty,\infty)$. Then

 1.8.14 $\sum^{\infty}_{n=-\infty}f(x+n)=\sum^{\infty}_{n=-\infty}{\mathrm{e}}^{2\pi% \mathrm{i}nx}\int^{\infty}_{-\infty}f(t){\mathrm{e}}^{-2\pi\mathrm{i}nt}\,% \mathrm{d}t.$

It follows from definition (1.14.1) that the integral in (1.8.14) is equal to $\sqrt{2\pi}\mathscr{F}\left(f\right)\left(-2\pi n\right)$.

An alternative formulation is as follows. Suppose that $f(x)$ is continuous and of bounded variation on $[0,\infty)$. Suppose also that $f(x)$ is integrable on $[0,\infty)$ and $f(x)\to 0$ as $x\to\infty$. Then

 1.8.15 $\tfrac{1}{2}f(0)+\sum^{\infty}_{n=1}f(n)=\int^{\infty}_{0}f(x)\,\mathrm{d}x+2% \sum^{\infty}_{n=1}\int^{\infty}_{0}f(x)\cos\left(2\pi nx\right)\,\mathrm{d}x.$

As a special case

 1.8.16 $\sum_{n=-\infty}^{\infty}{\mathrm{e}}^{-(n+x)^{2}\omega}={\sqrt{\frac{\pi}{% \omega}}\*\left(1+2\sum_{n=1}^{\infty}{\mathrm{e}}^{-n^{2}{\pi}^{2}/\omega}% \cos\left(2n\pi x\right)\right)},$ $\Re\omega>0$.

## §1.8(v) Examples

For collections of Fourier-series expansions see Prudnikov et al. (1986a, v. 1, pp. 725–740), Gradshteyn and Ryzhik (2000, pp. 45–49), and Oberhettinger (1973).