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

$f(t)={\displaystyle \frac{a_0}{2}}+{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}\left(a_n\cos {\displaystyle \frac{n\pi t}{p}}+b_n\sin {\displaystyle \frac{n\pi t}{p}}\right),$ (1)

where the coefficients are

$a_n={\displaystyle \frac{1}{p}}{\displaystyle {\int }_{-p}^p}f(x)\cos {\displaystyle \frac{n\pi t}{p}}dt,b_n={\displaystyle \frac{1}{p}}{\displaystyle {\int }_{-p}^p}f(x)\sin {\displaystyle \frac{n\pi t}{p}}dt.$ (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

$f(t)={\displaystyle \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}}{c}_n{e}^{\frac{in\pi t}{p}}.$ (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

$c_n={\displaystyle \frac{1}{2p}}{\displaystyle {\int }_{-p}^p}f(t)e^{\frac{-in\pi t}{p}}dt\mathit{\hspace{1em}\hspace{1em}}(n=0,\pm 1,\pm 2,\mathrm{\dots }).$ (4)

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

0.2 Derivation of Fourier integral

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

$$f(t)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{\frac{in\pi t}{p}}\frac{1}{2p}{\int }_{-p}^{p}f(t)e^{\frac{-in\pi t}{p}}𝑑t$$

By denoting  ${\omega }_n:=\frac{n\pi }{p}$  and  ${\mathrm{\Delta }}_n\omega :={\omega }_{n+1}-{\omega }_n=\frac{\pi }{p}$,  the last equation takes the form

$$f(t)=\frac{1}{2\pi }\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{i{\omega }_{n}t}{\mathrm{\Delta }}_n\omega {\int }_{-p}^{p}f(t)e^{-i{\omega }_{n}t}𝑑t.$$

It can be shown that when  $p\to \mathrm{\infty }$  and thus  ${\mathrm{\Delta }}_n\omega \to 0$,  the limiting form of this equation is

$f(t)={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{e}^{i\omega t}𝑑\omega {\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}f(t)e^{-i\omega t}𝑑t.$ (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

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|f(t)|𝑑t$$

converges.

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

$f(t)={\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}c(\omega )e^{i\omega t}𝑑\omega ,$ (6)

where

$c(\omega )={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}f(t)e^{-i\omega t}𝑑t.$ (7)

0.3 Fourier transform

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

$F(\omega )={\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{e}^{-i\omega t}f(t)𝑑t,$ (8)

then by (5),

$f(t)={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{e}^{i\omega t}F(\omega )𝑑\omega .$ (9)

$F(\omega )$ is called the Fourier transform of $f(t)$.  It is an integral transform and (9) 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.

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