Fourier series in complex form and Fourier integral

0.1 Fourier series in complex form

The Fourier series expansion of a Riemann integrablePlanetmathPlanetmath real function f on the interval[-p,p]  is

f(t)=a02+n=1(ancosnπtp+bnsinnπtp), (1)

where the coefficients are

an=1p-ppf(x)cosnπtpdt,bn=1p-ppf(x)sinnπtpdt. (2)

If one expresses the cosines and sines via Euler formulas ( with exponential functionDlmfDlmfMathworld (, the series (1) attains the form

f(t)=n=-cneinπtp. (3)

The coefficients cn could be obtained of an and bn, but they are comfortably derived directly by multiplying the equation (3) by e-imπtp and integrating it from -p to p.  One obtains

cn=12p-ppf(t)e-inπtpdt  (n=0,±1,±2,). (4)

We may say that in (3), f(t) has been dissolved to sum of harmonics (elementary waves) cneinπtp with amplitudes cn corresponding the frequencies n.

0.2 Derivation of Fourier integral

For seeing how the expansion (3) changes when  p,  we put first the expressions (4) of cn to the series (3):


By denoting  ωn:=nπp  and  Δnω:=ωn+1-ωn=πp,  the last equation takes the form


It can be shown that when  p  and thus  Δnω0,  the limiting form of this equation is

f(t)=12π-eiωt𝑑ω-f(t)e-iωt𝑑t. (5)

Here, f(t) has been represented as a Fourier integral.  It can be proved that for validity of the expansion (4) it suffices that the functionMathworldPlanetmath f is piecewise continuous on every finite interval having at most a finite amount of extremumMathworldPlanetmath points and that the integralDlmfPlanetmath



For better to compare to the Fourier series (3) and the coefficients (4), we can write (5) as

f(t)=-c(ω)eiωt𝑑ω, (6)


c(ω)=12π-f(t)e-iωt𝑑t. (7)

0.3 Fourier transform

If we denote 2πc(ω) as

F(ω)=-e-iωtf(t)𝑑t, (8)

then by (5),

f(t)=12π-eiωtF(ω)𝑑ω. (9)

F(ω) is called the Fourier transformDlmfMathworldPlanetmath of f(t).  It is an integral transformDlmfMathworld and (9) its inversePlanetmathPlanetmathPlanetmath transform.

N.B. that often one sees both the formulaMathworldPlanetmathPlanetmath (8) and the formula (9) equipped with the same constant factor 12π in front of the integral sign.


  • 1 K. Väisälä: Laplace-muunnos.  Handout Nr. 163. Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1968).
Title Fourier series in complex form and Fourier integral
Canonical name FourierSeriesInComplexFormAndFourierIntegral
Date of creation 2013-03-22 18:02:54
Last modified on 2013-03-22 18:02:54
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Derivation
Classification msc 42A38
Classification msc 42A16
Classification msc 44A55
Related topic FourierTransform
Related topic KalleVaisala
Defines Fourier integral