3.6 Obtain Fourier Series approximation

3.6.1 Example 1
3.6.2 Example 2
3.6.3 Example 3

3.6.1 Example 1

3.6.1.1 Maple
3.6.1.2 Mathematica

Obtain Fourier Series approximation of $$f(x)=e^{-|x|}$$ for $$-1<x<1$$

3.6.1.1 Maple
restart;
f:=x->exp(-abs(x));
f_approx:=OrthogonalExpansions:-FourierSeries(f(x),x=-1..1,infinity ):
f_approx:=subs(i=n,f_approx);



$1-{\mathrm e}^{-1}+\sum _{n=1}^{\infty } \left (-\frac {2 \left (\left (-1\right )^{n} {\mathrm e}^{-1}-1\right ) \cos \! \left (\pi n x \right )}{\pi ^{2} n^{2}+1}\right )$

3.6.1.2 Mathematica

Mathematica does not have a buildin function to give general series expression as the above with Maple. There is a user written package and answer here https://mathematica.stackexchange.com/questions/149468/a-more-convenient-fourier-series which provides this.

In Mathematica it is possible to obtain the terms using the command FourierSeries . For example the terms $$n=0,n=-1,n=1$$ can be obtained using

expr = Exp[-Abs[x]];
FourierSeries[expr, x, 1, FourierParameters -> {1, Pi}]



$\frac {(1+e) e^{-i \pi x}}{e+e \pi ^2}+\frac {(1+e) e^{i \pi x}}{e+e \pi ^2}+\frac {e-1}{e}$

see https://reference.wolfram.com/language/ref/FourierSeries.html for deļ¬nitions of FourierParameters used above.

3.6.2 Example 2

3.6.2.1 Maple

Obtain Fourier Series approximation of $f\left ( x\right ) =\left \{ \begin {array} [c]{ccc}\frac {2xh}{L} & & 0\leq x\leq \frac {L}{2}\\ \frac {2h\left ( L-x\right ) }{L} & & \frac {L}{2}\leq x\leq L \end {array} \right .$

For $$0<x<L$$

3.6.2.1 Maple
restart;
f:=x->piecewise(0<x and x<L/2,2*x*h/L, L/2<x and x<L, 2*h*(L-x)/L);
f_approx:=OrthogonalExpansions:-FourierSeries(f(x),x=0..L,infinity ):
f_approx:=subs(i=n,f_approx):
simplify(%) assuming L>0



$2 \left (\sum _{n=1}^{\infty } \frac {h \left (\left (-1\right )^{n}-1\right ) \cos \left (\frac {2 \pi n x}{L}\right )}{\pi ^{2} n^{2}}\right )+\frac {h}{2}$

3.6.3 Example 3

3.6.3.1 Maple

Obtain Fourier Series approximation of $$\cosh x$$ for $$-1<x<1$$.

3.6.3.1 Maple
restart;
f:=x->cosh(x);
f_approx:=OrthogonalExpansions:-FourierSeries(f(x),x=-1..1,infinity ):
f_approx:=subs(i=n,f_approx);
convert(%,trig);



$\sinh \left (1\right )+\left (\sum _{n=1}^{\infty } \frac {2 \sinh \left (1\right ) \left (-1\right )^{n} \cos \left (\pi n x \right )}{\pi ^{2} n^{2}+1}\right )$