Fourier series in complex form and Fourier integral
The Fourier series expansion of a Riemann integrable real function
on the interval
is
 |
(1) |
where the coefficients are
 |
(2) |
If one expresses the cosines and sines via Euler formulas with exponential function, the series (1) attains the form
 |
(3) |
The coefficients
could be obtained of
and
, but they are comfortably derived directly by multiplying the equation (3) by
and integrating it from
to
. One obtains
 |
(4) |
We may say that in (3),
has been dissolved to sum of harmonics (elementary waves)
with amplitudes
corresponding the frequencies
.
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
converges.
For better to compare to the Fourier series (3) and the coefficients (4), we can write (5) as
 |
(6) |
where
 |
(7) |
If we denote
as
 |
(8) |
then by (5),
 |
(9) |
is called the Fourier transform of
. It is an integral transform and (9) represents 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.
- 1
- K. V¨AISÄLÄ: Laplace-muunnos. Handout Nr. 163. Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1968).
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