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Fourier series in complex form and Fourier integral (Topic)

Fourier series in complex form

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

$\displaystyle f(t) = \frac{a_0}{2}+\sum_{n=1}^\infty\left(a_n\cos{\frac{n\pi t}{p}}+b_n\sin{\frac{n\pi t}{p}}\right),$ (1)
where the coefficients are
$\displaystyle a_n = \frac{1}{p}\int_{-p}^{\,p}f(x)\cos{\frac{n\pi t}{p}}\,dt, \quad b_n = \frac{1}{p}\int_{-p}^{\,p}f(x)\sin{\frac{n\pi t}{p}}\,dt.$ (2)
If one expresses the cosines and sines via Euler formulas with exponential function, the series (1) attains the form
$\displaystyle f(t) = \sum_{n=-\infty}^\infty c_ne^{\frac{in\pi t}{p}}.$ (3)
The coefficients $c_n$ could be obtained of $a_n$ and $b_n$, but they are comfortably derived directly by multiplying the equation (3) by $e^{-\frac{im\pi t}{p}}$ and integrating it from $-p$ to $p$.  One obtains
$\displaystyle c_n = \frac{1}{2p}\int_{-p}^{\,p}f(t)e^{\frac{-in\pi t}{p}}\,dt \qquad (n = 0,\,\pm1,\,\pm2,\,\ldots).$ (4)

We may say that in (3), $f(t)$ has been dissolved to sum of harmonics (elementary waves) $c_ne^{\frac{in\pi t}{p}}$ with amplitudes $c_n$ corresponding the frequencies $n$.

Derivation of Fourier integral

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

$\displaystyle f(t) = \sum_{n=-\infty}^\infty e^{\frac{in\pi t}{p}}\frac{1}{2p}\int_{-p}^{\,p}f(t)e^{\frac{-in\pi t}{p}}\,dt$
By denoting  $\omega_n := \frac{n\pi}{p}$  and  $\Delta_n\omega := \omega_{n+1}\!-\!\omega_n = \frac{\pi}{p}$,  the last equation takes the form

$\displaystyle f(t) = \frac{1}{2\pi}\sum_{n=-\infty}^\infty e^{i\omega_nt}\Delta_n\omega \int_{-p}^{\,p}f(t)e^{-i\omega_nt}\,dt.$
It can be shown that when  $p \to \infty$  and thus  $\Delta_n\omega \to 0$,  the limiting form of this equation is
$\displaystyle f(t) \,=\, \frac{1}{2\pi}\int_{-\infty}^{\,\infty} e^{i\omega t}d\omega\int_{-\infty}^{\,\infty} f(t)e^{-i\omega t}dt.$ (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 function $f$ is piecewise continuous on every finite interval having at most a finite amount of extremum points and that the integral

$\displaystyle \int_{-\infty}^{\,\infty}\vert f(t)\vert\,dt$
converges.

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

$\displaystyle f(t) \,=\, \int_{-\infty}^{\,\infty}c(\omega)e^{i\omega t}d\omega,$ (6)
where
$\displaystyle c(\omega) \,=\, \frac{1}{2\pi}\int_{-\infty}^{\,\infty}f(t)e^{-i\omega t}dt.$ (7)

Fourier transform

If we denote $2\pi c(\omega)$ as

$\displaystyle F(\omega) \,=\, \int_{-\infty}^{\,\infty} e^{-i\omega t}f(t)\,dt,$ (8)
then by (5),
$\displaystyle f(t) \,=\, \frac{1}{2\pi}\int_{-\infty}^{\,\infty}e^{i\omega t}F(\omega)\,d\omega.$ (9)
$F(\omega)$ is called the Fourier transform of $f(t)$.  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 $\displaystyle\frac{1}{\sqrt{2\pi}}$ in front of the integral sign.

Bibliography

1
K. V¨AISÄLÄ: Laplace-muunnos.  Handout Nr. 163.    Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1968).



"Fourier series in complex form and Fourier integral" is owned by pahio.

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Other names:  complex Fourier series and integral
Keywords:  Fourier series, Fourier integral

Cross-references: formula, Fourier transform, waves, function

This is version 2 of Fourier series in complex form and Fourier integral, born on 2009-04-18, modified 2009-04-18.
Object id is 650, canonical name is FourierSeriesInComplexFormAndFourierIntegral.
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Physics Classification: 02.30.-f (Function theory, analysis)
 02.30.Nw (Fourier analysis)
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