determination of Fourier coefficients

Suppose that the real function f may be presented as sum of the Fourier series:

f(x)=a02+m=0(amcosmx+bmsinmx) (1)

Therefore, f is periodic with period 2π.  For expressing the Fourier coefficients am and bm with the functionMathworldPlanetmath itself, we first multiply the series (1) by cosnx (n) and integrate from -π to π.  Supposing that we can integrate termwise, we may write

-ππf(x)cosnxdx=a02-ππcosnxdx+m=0(am-ππcosmxcosnxdx+bm-ππsinmxcosnxdx). (2)

When  n=0,  the equation (2) reads

-ππf(x)𝑑x=a022π=πa0, (3)

since in the sum of the right hand side, only the first addend is distinct from zero.

When n is a positive integer, we use the product formulas of the trigonometric identities, getting


The latter expression vanishes always, since the sine is an odd functionMathworldPlanetmath.  If  mn,  the former equals zero because the antiderivative consists of sine terms which vanish at multiples of π; only in the case  m=n  we obtain from it a non-zero result π.  Then (2) reads

-ππf(x)cosnxdx=πan (4)

to which we can include as a special case the equation (3).

By multiplying (1) by sinnx and integrating termwise, one obtains similarly

-ππf(x)sinnxdx=πbn. (5)

The equations (4) and (5) imply the formulas

an=1π-ππf(x)cosnxdx(n=0, 1, 2,)


bn=1π-ππf(x)sinnxdx(n=1, 2, 3,)

for finding the values of the Fourier coefficients of f.

Title determination of Fourier coefficients
Canonical name DeterminationOfFourierCoefficients
Date of creation 2013-03-22 18:22:47
Last modified on 2013-03-22 18:22:47
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Derivation
Classification msc 26A42
Classification msc 42A16
Synonym calculation of Fourier coefficients
Related topic UniquenessOfFourierExpansion
Related topic FourierSineAndCosineSeries
Related topic OrthogonalityOfChebyshevPolynomials