# If we make use of eulers formula we can write the

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Unformatted text preview: 10. 76 3.3.4. Find the Fourier sine series of the following function deﬁned on the interval [0, 1]: f (t) = 5t(1 − t). 3.3.5.(For students with access to Mathematica) Find numerical approximations to the ﬁrst ten coeﬃcients of the Fourier sine series for the function f (t) = t + t2 − 2t3 , deﬁned for t in the interval [0, 1], by running the following Mathematica program f[n ] := 2 NIntegrate[(t + t∧2 - 2 t∧3) Sin[n Pi t], {t,0,1}]; b = Table[f[n], {n,1,10}] 3.4 Complex version of Fourier series* We have seen that if f : R → R is a well-behaved function which is periodic of period 2π , f can be expanded in a Fourier series f (t) = a0 + a1 cos t + a2 cos(2t) + . . . 2 +b1 sin t + b2 sin(2t) + . . . . We say that this is the real form of the Fourier series. It is often convenient to recast this Fourier series in complex form by means of the Euler formula, which states that eiθ = cos θ + i sin θ. It follows from this formula that eiθ + e−iθ = 2 cos θ, e−iθ + e−iθ = 2i sin θ, or eiθ + e−iθ eiθ + e−iθ , sin...
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## This document was uploaded on 01/12/2014.

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