Mathematical Physics Vol 1

6.2 The fundamental convergence theorem for Fourier series

315

6.2.2 Expansion of functions into Fourier series on the interval ( − π , π ) In tis special case ℓ = π , so we obtain that

∞ ∑ k = 1

a 0 2

f ( x )=

( a k cos kx + b k sin kx ) ,

+

where the coefficients are determined by the following expressions

π Z

π Z

π Z

1 π

1 π

1 π

a 0 =

f ( x ) d x , a k =

f ( x ) cos kx d x , b k =

f ( x ) sin kx d x .

− π

− π

− π

6.2.3 Expansion of functions into Fourier series on the interval ( 0 ,ℓ ) . Extension of the half-interval

In various problems of physics and technology, there is a need to expand a function into a Fourier series in a finite interval, ( 0 ,ℓ ) . This can be achieved by translating the coordinate system (i.e. introducing a substitution) for ℓ/ 2, and then extending, in the general case, a continuous function f ( x ) , as previously explained (choosing ℓ as the period). Another, more practical way, to do this is the following. Assume that the period is 2 ℓ and extend the initial function to the interval ( − ℓ, 0 ) . Since the function is given only on the interval ( 0 ,ℓ ) , we can further extend it to the interval ( − ℓ, 0 ) , so that its extension is either an even or an odd function. This procedure is demonstrated on Fig. 6.3.

Figure 6.3: Extension of non-periodic function.

This extension is convenient, because in these cases the expansion of a function into a Fourier series is reduced to a Fourier sine or cosine series (due to the parity or oddness of the extended function), and we thus stay in the interval in which the initial function is defined.

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