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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 wellbehaved 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.
 Winter '14
 Equations

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