determination of Fourier coefficients

Suppose that the real function $ f$ may be presented as sum of the Fourier series:

$\displaystyle f(x) \;=\; \frac{a_0}{2}+\sum_{m=0}^\infty(a_m\cos{mx}+b_m\sin{mx})$ (1)

Therefore, $ f$ is periodic with period $ 2\pi$ . For expressing the Fourier coefficients $ a_m$ and $ b_m$ with the function itself, we first multiply the series (1) by $ \cos{nx}$ ( $ n \in \mathbb{Z}$ ) and integrate from $ -\pi$ to $ \pi$ . Supposing that we can integrate termwise, we may write

$\displaystyle \int_{-\pi}^\pi\!f(x)\cos{nx}\,dx \,=\, \frac{a_0}{2}\!\int_{-\pi...
...^\pi\!\cos{mx}\cos{nx}\,dx+b_m\!\int_{-\pi}^\pi\!\sin{mx}\cos{nx}\,dx\right)\!.$ (2)

When $ n = 0$ , the equation (2) reads

$\displaystyle \int_{-\pi}^\pi f(x)\,dx = \frac{a_0}{2}\cdot2\pi = \pi a_0,$ (3)

since in the sum of the right hand side, only the first addend is distinct from zero.

When $ n$ is a positive integer, we use the product formulas of the trigonometric identities, getting

$\displaystyle \int_{-\pi}^\pi\cos{mx}\cos{nx}\,dx
= \frac{1}{2}\int_{-\pi}^\pi[\cos(m-n)x+\cos(m+n)x]\,dx,$

$\displaystyle \int_{-\pi}^\pi\sin{mx}\cos{nx}\,dx
= \frac{1}{2}\int_{-\pi}^\pi[\sin(m-n)x+\sin(m+n)x]\,dx.$

The latter expression vanishes always, since the sine is an odd function. If $ m \neq n$ , the former equals zero because the antiderivative consists of sine terms which vanish at multiples of $ \pi$ ; only in the case $ m = n$ we obtain from it a non-zero result $ \pi$ . Then (2) reads

$\displaystyle \int_{-\pi}^\pi f(x)\cos{nx}\,dx = \pi a_n$ (4)

to which we can include as a special case the equation (3).

By multiplying (1) by $ \sin{nx}$ and integrating termwise, one obtains similarly

$\displaystyle \int_{-\pi}^\pi f(x)\sin{nx}\,dx = \pi b_n.$ (5)

The equations (4) and (5) imply the formulas

$\displaystyle a_n \;=\; \frac{1}{\pi}\int_{-\pi}^\pi f(x)\cos{nx}\,dx \quad (n = 0,\,1,\,2,\,\ldots)$

and

$\displaystyle b_n \;=\; \frac{1}{\pi}\int_{-\pi}^\pi f(x)\sin{nx}\,dx \quad (n = 1,\,2,\,3,\,\ldots)$

for finding the values of the Fourier coefficients of $ f$ .



Contributors to this entry (in most recent order):

As of this snapshot date, this entry was owned by pahio.