1 Algebraic and Analytic Methods Topics of Discussion 1.7 Inequalities 1.9 Calculus of a Complex Variable

§1.8 Fourier Series

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Keywords:: Fourier series
Referenced by:: Erratum (V1.2.0) §1.8
Permalink:: http://dlmf.nist.gov/1.8
See also:: Annotations for Ch.1

§1.8(i) Definitions and Elementary Properties
§1.8(ii) Convergence
§1.8(iii) Integration and Differentiation
§1.8(iv) Poisson’s Summation Formula
§1.8(v) Examples

§1.8(i) Definitions and Elementary Properties

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Keywords:: Fourier series, coefficients, definition, properties
Notes:: See Protter and Morrey (1991, Chapter 10), Tolstov (1962, Chapter 1), or Titchmarsh (1962b, Chapter 13). For the Riemann–Lebesgue lemma, see Olver (1997b, p. 73).
Referenced by:: §2.10(iv), §2.4(ii), §2.5(ii), §3.11(ii), §3.11(ii), Erratum (V1.2.0) §1.8
Permalink:: http://dlmf.nist.gov/1.8.i
Modification (effective with 1.2.0):: The first sentence of this subsection was changed from “Formally,” to “Formally, if is a real- or complex-valued $2\pi$ -periodic function”.
See also:: Annotations for §1.8 and Ch.1

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

1.8.1 $f(x)=\tfrac{1}{2}a_{0}+\sum^{\infty}_{n=1}(a_{n}\cos\left(nx\right)+b_{n}\sin% \left(nx\right)),$

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Symbols:: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function and : 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

$a_{n}=\frac{1}{\pi}\int^{\pi}_{-\pi}f(x)\cos\left(nx\right)\,\mathrm{d}x,$ $n=0,1,2,\dots$ ,

$b_{n}=\frac{1}{\pi}\int^{\pi}_{-\pi}f(x)\sin\left(nx\right)\,\mathrm{d}x,$ $n=1,2,\dots$ .

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\sin\NVar{z}$ : sine function and : nonnegative integer
Referenced by:: §1.18(v), §1.8(i), §1.8(i), §18.2(viii)
Permalink:: http://dlmf.nist.gov/1.8.E2
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §1.8(i), §1.8 and Ch.1

The series (1.8.1) is called the Fourier series of , and $a_{n},b_{n}$ are the Fourier coefficients of .

If , then $b_{n}=0$ for all .

If , then $a_{n}=0$ for all .

Alternative Form

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Modification (effective with 1.2.0):: A sentence was added at the end of this paragraph.
See also:: Annotations for §1.8(i), §1.8 and Ch.1

1.8.3 $f(x)=\sum^{\infty}_{n=-\infty}c_{n}{\mathrm{e}}^{\mathrm{i}nx},$

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Symbols:: $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and : 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 $c_{n}=\frac{1}{2\pi}\int^{\pi}_{-\pi}f(x){\mathrm{e}}^{-\mathrm{i}nx}\,\mathrm% {d}x.$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and : nonnegative integer
Referenced by:: §1.17(iii), §1.18(v), §1.8(i)
Permalink:: http://dlmf.nist.gov/1.8.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

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 and $c_{0}=\frac{1}{2}a_{0}$ .

Parseval’s Formula

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Keywords:: Fourier series, Parseval’s Formula
Modification (effective with 1.2.0):: Previously this paragraph was named Bessel’s Inequality. The equation which was originally (1.8.13) was moved to (1.8.6_1).
Addition (effective with 1.2.0):: Some text was added just after (1.8.6). The equation (1.8.6_2) was added.
See also:: Annotations for §1.8(i), §1.8 and Ch.1

1.8.5 $\frac{1}{\pi}\int^{\pi}_{-\pi}{\left|f(x)\right|}^{2}\,\mathrm{d}x=\tfrac{1}{2% }{\left|a_{0}\right|}^{2}+\sum^{\infty}_{n=1}({\left|a_{n}\right|}^{2}+{\left|% b_{n}\right|}^{2}),$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : 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 $\frac{1}{2\pi}\int^{\pi}_{-\pi}{\left|f(x)\right|}^{2}\,\mathrm{d}x=\sum^{% \infty}_{n=-\infty}{\left|c_{n}\right|}^{2},$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : 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 is square-integrable on $[-\pi,\pi]$ and $a_{n},b_{n},c_{n}$ are given by (1.8.2), (1.8.4). If is also square-integrable with Fourier coefficients $a_{n}^{\prime},b_{n}^{\prime}$ or $c_{n}^{\prime}$ then

1.8.6_1 $\frac{1}{\pi}\int^{\pi}_{-\pi}f(x)\overline{g(x)}\,\mathrm{d}x=\tfrac{1}{2}a_{% 0}\overline{a_{0}^{\prime}}+\sum^{\infty}_{n=1}(a_{n}\overline{a^{\prime}_{n}}% +b_{n}\overline{b^{\prime}_{n}}),$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral and : nonnegative integer
Referenced by:: (1.8.13), (1.8.13), §1.8(i), §1.8(iv), Erratum (V1.2.0) §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 $\frac{1}{2\pi}\int^{\pi}_{-\pi}f(x)\overline{g(x)}\,\mathrm{d}x=\sum^{\infty}_% {n=-\infty}c_{n}\overline{c_{n}^{\prime}}.$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral and : nonnegative integer
Referenced by:: §1.8(i), Erratum (V1.2.0) §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

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See also:: Annotations for §1.8(i), §1.8 and Ch.1

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

1.8.7 $a_{n},b_{n},c_{n}=o\left(n^{-m}\right),$ $n\to\infty$ .

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Symbols:: $o\left(\NVar{x}\right)$ : order less than, : nonnegative integer and : 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

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Keywords:: Fourier series, uniqueness
See also:: Annotations for §1.8(i), §1.8 and Ch.1

If and are continuous, have the same period and same Fourier coefficients, then for all .

Lebesgue Constants

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Keywords:: Lebesgue constants, asymptotic behavior
See also:: Annotations for §1.8(i), §1.8 and Ch.1

1.8.8 $L_{n}=\frac{1}{\pi}\int^{\pi}_{0}\frac{\left|\sin\left(n+\frac{1}{2}\right)t% \right|}{\sin\left(\frac{1}{2}t\right)}\,\mathrm{d}t,$ $n=0,1,\dots$ .

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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 , $\int$ : integral, $\sin\NVar{z}$ : sine function, : 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 $L_{n}\sim(4/{\pi}^{2})\ln n;$

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Symbols:: $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$ : principal branch of logarithm function, : nonnegative integer and $L_{n}$ : Lebesgue constants
Permalink:: http://dlmf.nist.gov/1.8.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

see Frenzen and Wong (1986).

Riemann–Lebesgue Lemma

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Keywords:: Riemann–Lebesgue lemma
See also:: Annotations for §1.8(i), §1.8 and Ch.1

For piecewise continuous on and real $\lambda$ ,

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

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\lambda$ : real
Referenced by:: §1.8(i)
Permalink:: http://dlmf.nist.gov/1.8.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

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

§1.8(ii) Convergence

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Keywords:: Fourier series, convergence, derivatives, left-hand, right-hand
Notes:: See Tolstov (1962, p. 77).
Referenced by:: §1.17(iii)
Permalink:: http://dlmf.nist.gov/1.8.ii
See also:: Annotations for §1.8 and Ch.1

Let 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 $\tfrac{1}{2}f(x-)+\tfrac{1}{2}f(x+)$

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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 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

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Keywords:: Fourier series, differentiation, integration
Notes:: See Titchmarsh (1962b, p. 419).
Permalink:: http://dlmf.nist.gov/1.8.iii
See also:: Annotations for §1.8 and Ch.1

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

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

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\sin\NVar{z}$ : sine function and : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.8.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §1.8(iii), §1.8 and Ch.1

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 term by term converges at every point to $f^{\prime}(x)$ .

§1.8(iv) Poisson’s Summation Formula

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Keywords:: Fourier series, Poisson’s summation formula
Notes:: For Poisson’s summation formula see Rademacher (1973, pp. 71–75) and Titchmarsh (1986a, p. 61). For (1.8.16) set $f(x)={\mathrm{e}}^{-\omega x^{2}}$ in (1.8.14).
Referenced by:: §1.14(i), §20.11(i), §3.5(i), Erratum (V1.2.0) §1.8
Permalink:: http://dlmf.nist.gov/1.8.iv
Modification (effective with 1.2.0):: This subsection was previously called “Transformations”.
Rearrangement (effective with 1.2.0):: The equation which was originally (1.8.13) has been moved to (1.8.6_1). A sentence was added just after (1.8.14).
See also:: Annotations for §1.8 and Ch.1

1.8.13 Moved to (1.8.6_1).

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

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

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and : nonnegative integer
Referenced by:: §1.14(i), §1.8(iv), §1.8(iv), Erratum (V1.2.0) §1.8
Permalink:: http://dlmf.nist.gov/1.8.E14
Encodings:: TeX, pMML, png
See also:: Annotations for §1.8(iv), §1.8 and Ch.1

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

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

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral and : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.8.E15
Encodings:: TeX, pMML, png
See also:: Annotations for §1.8(iv), §1.8 and Ch.1

As a special case

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

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\Re$ : real part and : nonnegative integer
Referenced by:: §1.8(iv)
Permalink:: http://dlmf.nist.gov/1.8.E16
Encodings:: TeX, pMML, png
See also:: Annotations for §1.8(iv), §1.8 and Ch.1

§1.8(v) Examples

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Keywords:: Fourier series, compendia
Permalink:: http://dlmf.nist.gov/1.8.v
See also:: Annotations for §1.8 and Ch.1

For collections of Fourier-series expansions see Prudnikov et al. (1986a, v. 1, pp. 725–740), Gradshteyn and Ryzhik (2015, §§1.44–1.45), and Oberhettinger (1973).

1.7 Inequalities 1.9 Calculus of a Complex Variable

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