Fourier series in complex form and Fourier integral
0.1 Fourier series in complex form
The Fourier series expansion of a Riemann integrable real function f on the interval [-p,p] is
f(t)=a02+∞∑n=1(ancosnπtp+bnsinnπtp), | (1) |
where the coefficients are
an=1p∫p-pf(x)cosnπtpdt,bn=1p∫p-pf(x)sinnπtpdt. | (2) |
If one expresses the cosines and sines via Euler formulas (http://planetmath.org/ComplexSineAndCosine) with exponential function (http://planetmath.org/ComplexExponentialFunction), 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∫p-pf(t)e-inπtpdt | (4) |
We may say that in (3), has been dissolved to sum of harmonics (elementary waves) with amplitudes corresponding the frequencies .
0.2 Derivation of Fourier integral
For seeing how the expansion (3) changes when , we put first the expressions (4) of to the series (3):
By denoting and , the last equation takes the form
It can be shown that when and thus , the limiting form of this equation is
(5) |
Here, has been represented as a Fourier integral. It can be proved that for validity of the expansion (4) it suffices that the function is piecewise continuous on every finite interval having at most a finite amount of extremum
points and that the integral
For better to compare to the Fourier series (3) and the coefficients (4), we can write (5) as
(6) |
where
(7) |
0.3 Fourier transform
If we denote as
(8) |
then by (5),
(9) |
is called the Fourier transform of . It is an integral transform
and (9) its inverse
transform.
N.B. that often one sees both the formula (8) and the formula (9) equipped with the same constant factor in front of the integral sign.
References
- 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 |