Create()

Usage of the procedure Create is similar to OrthogonalSeries[Create] but there are few differences.

>	s:=Create({[1,2,3], n}, SinTrigP(n,x));

>	type(s, FOURIERSERIES);

There are four types of orthogonal systems: SinTrigP, CosTrigP, CosSinTrigP and ExpTrigP2. ExpTrigP2 means that the series is defined as a sum which goes from the zero to the plus infinity (and not from the minus infinity to the plus infinity).

>	Create({[1,2,3], n}, CosTrigP(n,x));

Creating series with different period.

>	Create({[1,2,3], n}, CosTrigP(n,x), Pi);

When creating the series with both sine and cosine terms, the coefficients are specified by a list of pairs. Also it is necessary to use the option "general" to set the general coefficient of the series.

>	s:=Create({[1,[1,-1],[1/2,-1/2]], 'general'=[n,-n]}, CosSinTrigP(n,x));

Some other possibilities to specify the coefficients (see OrthogonalSeries[Create]):

>	Create({[1,[1,-1],[1/2,-1/2]]}, CosSinTrigP(n,x));

>	Create({[0=1,1=[1,-1],3=[1/2,-1/2]], 'general'=n, n=5..10}, CosSinTrigP(n,x));

>	Create({[0=1,1=[1,-1],3=[1/2,-1/2]], 'general'=n, n=5..10}, ExpTrigP2(n,x));

The data structure of the type FOURIERSERIES:

>	lprint(s);

FourierTrigSeries:-FOURIERSERIES(S)

>	print(op(1,s));

ConvertToSum()

Converting series structure to the regular sum

>	ConvertToSum(s);

>	lprint(%);

1+cos(x)-sin(x)+1/2*cos(2*x)-1/2*sin(2*x)+Sum(n*cos(n*x)-n*sin(n*x),n = 3 .. infinity)

Derivate()

Derivate the series

>	s2:=Derivate(s, 'x');

>	s3:=Create({[t], t*n}, CosTrigP(n,x));

>	Derivate(s3, 'x');

>	Derivate(s3, 't');

Evaluate(), GetPartialSum()

Evaluation

>	Evaluate(s, x=Pi/4);

>	Evaluate(s, trunc=5);

The same can be obtained by

>	GetPartialSum(s, 5);

>	Evaluate(s, x=Pi/4, trunc=5);

>	Evaluate(s, x=A, trunc=5);

ChangeBasis()

Conversion between orthogonal systems.

>

s2;

>	ChangeBasis(s2, `ExpTrigP2`(n,x));

>	ChangeBasis(%, `SinTrigP`(n,x));

Add(), ScalarMultiply()

Adding two series and multiplying the series by a scalar.

>

s;

>

s2;

>	Add(s,s2);

>	Add(s,s2, 1, 3);

>	s3:=ScalarMultiply(s2, (1-alpha));

Coefficient(), Degree(), Truncate(), Copy()

Getting the coefficients and the degree of the partial sum, copying the series.

>

s;

>	Coefficients(s,5);

>	Degree(s3);

>	Truncate(s3,7);

>	Degree(%);

Copying the series

>	s4:=Copy(s3);

SimplifyCoefficients()

Simplify the coefficients

>	s1:=Create({-1/(n-1)}, SinTrigP(n,x));

>	s2:=Create({n/(n-1)}, SinTrigP(n,x));

>	Add(s1,s2);

>	SimplifyCoefficients(%, simplify);

GetFourierSeries()

Computing the Fourier series of the given function.

>	GetFourierSeries(cos(x)^4,x=-Pi..Pi);

>	GetFourierSeries(x+signum(x),x=-Pi..Pi);

Getting the partial sum.

>	t:=Truncate(%, 4);

>	ConvertToSum(t);

Change the orthogonal system.

>	ChangeBasis(t, `ExpTrigP2`(n,x));

>	GetFourierSeries(x^2,x=-Pi..Pi, exp);

Getting the Fourier series of the odd/even periodic extension.

>	GetFourierSeries(x^2,x=0..Pi, even);

>	GetFourierSeries(x^2,x=0..Pi, even, exp);

>	GetFourierSeries(x^3,x=-Pi..Pi);

>	GetFourierSeries(x^3,x=-Pi..0, odd);

>

ExploreFourierSeriesCoefficients()

Procedure computes values of Fourier series coefficients

>	ExploreFourierSeriesCoefficients(signum(sin(2*x)), x=-Pi..Pi);

Look at coefficients in more detail

>	ExploreFourierSeriesCoefficients(signum(sin(2*x)), x=-Pi..Pi,2);

>	ExploreFourierSeriesCoefficients(signum(sin(2*x)), x=-Pi..Pi,4);

>	ExploreFourierSeriesCoefficients(signum(sin(2*x)), x=-Pi..Pi, 4, exp);

DrawPartialSum()

>	S:=GetFourierSeries(x,x=-1..1):

>	DrawPartialSum(S,3);

>	DrawPartialSum(S,3,-2..2,scaling=constrained,color=blue,thickness=3);

>

DrawPeriodicExtension()

>	DrawPeriodicExtension(x,x=-1..1,-2..2);

>	DrawPeriodicExtension(x,x=0..1,-3..3,even,thickness=3,color=blue,scaling=constrained);

>	DrawPeriodicExtension(x^2,x=-1..0,-2..2,odd,drawlimits,scaling=constrained);

>

Solving the differential equation using the Fourier series.

>	Diff(y(x),x$2)+2*y(x)=Sum(sin(n*x)/n^4,n=1..infinity);

>	S1:=Create({[0],'general'=[0,1/n^4]}, CosSinTrigP(n,x));

Create the general series

>	S2:=Create({[a0], 'general'=[aN,bN]}, CosSinTrigP(n,x));

Derivate the series two times

>	tmp1:=Derivate(Derivate(S2, 'x'), 'x');

Add the multiple of the series

>	tmp2:=Add(tmp1,S2, 1, 2);

Our equation now has the form

>

tmp2=S1;

Subtract the series

>	S:=Add(lhs(%), rhs(%), 1, -1);

>	type(S, FOURIERSERIES);

Series S should have all coefficients equal to zero. Hence a0 = 0.

Get the general coefficient.

>	Coefficients(S);

>	{%[1]=0, %[2]=0};

>	solve(%, {aN, bN});

Create the solution.

>	Y:=Create({subs(%, bN)}, SinTrigP(n,x));

Check the solution

>	Add(Derivate(Derivate(Y, 'x'), 'x'), Y, 1, 2);

>	SimplifyCoefficients(%, simplify);

>

>

>