Fourier sine and cosine series

One sees from the formulae

an =1π-ππf(x)cosnxdx,
bn =1π-ππf(x)sinnxdx

of the coefficients an and bn for the Fourier seriesMathworldPlanetmath


of the Riemann integrablePlanetmathPlanetmath real function f on the interval[-π,π],  that

  • an=2π0πf(x)cosnxdx,   bn=0n  if f is an even functionMathworldPlanetmath;

  • bn=2π0πf(x)sinnxdx,   an=0n  if f is an odd function.

Thus the Fourier series of an even function mere cosine and of an odd function mere sine .  This concerns the whole interval  [-π,π].  So e.g. one has on this interval

x 2(sinx1-sin2x2+sin3x3-+).

Remark 1.  On the half-interval[0,π]  one can in any case expand each Riemann integrable functionMathworldPlanetmath f both to a cosine series and to a sine series, irrespective of how it is defined for the negative half-interval or is it defined here at all.

Remark 2.  On an interval  [-p,p],  instead of  [-π,π],  the Fourier coefficients of the series


have the expressions

  • an=2p0pf(x)cosnπxpdx,   bn=0n  if f is an even function;

  • bn=2p0pf(x)sinnπxpdx,   an=0n  if f is an odd function.

Example.  Expand the identity function (  xx  to a Fourier cosine series on  [0,π].

This odd function may be replaced with the even function  f:x|x|.  Then we get


and, integrating by parts,


this equals to -4πn2 if n is an odd integer, but vanishes for each even n.  Thus we obtain on the interval  [0,π]  the cosine series


Chosing here  x:=0  one gets the result

π28= 1+132+152+

(cf. the entry on eta functionMathworldPlanetmath at 2).

Fourier double seriesMathworldPlanetmath.  The Fourier sine and cosine series introduced in Remark 1 on the half-interval  [0,π]  for a function of one real variable may be generalized for e.g. functions of two real variables on a rectangle  {(x,y)2  0xa, 0yb}:

f(x,y)=m=1n=1cmnsinmπxasinnπyb, (1)
f(x,y)=d004+12l=1(dl0coslπxa+d0lcoslπyb)+m=1n=1dmncosmπxacosnπyb (2)

The coefficients of the Fourier double sine series (1) are got by the double integral


where  m=1, 2, 3,  and  n=1, 2, 3,  The coefficients of the Fourier double cosine series (2) are correspondingly


where  m=0, 1, 2,  and  n=0, 1, 2,

Note.  One can use in the double series of (1) and (2) also the diagonal summing, e.g. for the double sine series as follows:


  • 1 K. Väisälä: Matematiikka IV.  Hand-out Nr. 141. Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1967).
Title Fourier sine and cosine series
Canonical name FourierSineAndCosineSeries
Date of creation 2013-03-22 15:42:20
Last modified on 2013-03-22 15:42:20
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 26
Author pahio (2872)
Entry type Topic
Classification msc 42A32
Classification msc 42A20
Classification msc 42A16
Classification msc 26A06
Related topic SubstitutionNotation
Related topic IntegralsOfEvenAndOddFunctions
Related topic CosineAtMultiplesOfStraightAngle
Related topic ExampleOfFourierSeries
Related topic DoubleSeries
Related topic UniquenessOfFourierExpansion
Related topic DeterminationOfFourierCoefficients
Related topic TwoDimensionalFourierTransforms
Defines Fourier sine series
Defines Fourier cosine series
Defines sine series
Defines cosine series
Defines half-interval
Defines Fourier double sine series
Defines Fourier double cosine series