Fourier Series – An ExampleFormulas:Let f(x) be periodic with period 2π. We want to writef(x) =A02+∞1(Akcoskx+Bksinkx).The Fourier coefficients are given by the formulasAk=1ππ-πf(x) coskx dxBk=1ππ-πf(x) sinkx dx.Moreover one has the analogue of the “Pythagorean theorem”f2=πA202+∞1(A2k+B2k).Example:Consider the functionf(x) =-1if-π < x≤01if 0< x≤πTo use the above formulas for the Fourier coefficients we split the integrals into two piecesAk=1π0-π(-1) coskx dx+π0(+1) coskx dx.andBk=1π0-π(-1) sinkx dx+π0(+1) sinkx dx.When one evaluates theAk, the two integrals cancel soAk= 0. Also,π0(+1) sinkx dx=-coskπ+ 1k=0ifkis even2kifkis oddBy a computation the first integral inBkhas the same value as this second integral. Thus,Bk=2π0ifkis even2kifkis odd=0ifkis even4kπifkis oddConsequently the desired Fourier series is
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