# 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

 $\displaystyle\int_{-\pi}^{\pi}\!f(x)\cos{nx}\,dx\,=\,\frac{a_{0}}{2}\!\int_{-% \pi}^{\pi}\!\cos{nx}\,dx+\!\sum_{m=0}^{\infty}\!\left(a_{m}\!\int_{-\pi}^{\pi}% \!\cos{mx}\cos{nx}\,dx+b_{m}\!\int_{-\pi}^{\pi}\!\sin{mx}\cos{nx}\,dx\right)\!.$ (2)

When  $n=0$,  the equation (2) reads

 $\displaystyle\int_{-\pi}^{\pi}f(x)\,dx=\frac{a_{0}}{2}\cdot 2\pi=\pi a_{0},$ (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

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

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

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

and

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

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

Title determination of Fourier coefficients DeterminationOfFourierCoefficients 2013-03-22 18:22:47 2013-03-22 18:22:47 pahio (2872) pahio (2872) 7 pahio (2872) Derivation msc 26A42 msc 42A16 calculation of Fourier coefficients UniquenessOfFourierExpansion FourierSineAndCosineSeries OrthogonalityOfChebyshevPolynomials