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FourierSeries package Karel Srot, xsrot@math.muni.cz, www.math.muni.cz/~xsrot/fourierseries Faculty of Science, Masaryk University, Czech Republic This package concept differs from other Fourier series packages. It provides new data structure for the representation of trigonometric series and few procedures to manipulate trigonometric series and to compute Fourier series. Some procedures and data structure are (for better understanding) similar (but not exactly the same) to OrthogonalSeries package so in many cases you can use the OrthogonalSeries package help files. restart: with(FourierSeries); NzFJJEFkZEc2IkksQ2hhbmdlQmFzaXNHRiRJLUNvZWZmaWNpZW50c0dGJEktQ29udmVydFRvU3VtR0YkSSVDb3B5R0YkSSdDcmVhdGVHRiRJJ0RlZ3JlZUdGJEkpRGVyaXZhdGVHRiRJKUV2YWx1YXRlR0YkSS5GT1VSSUVSU0VSSUVTR0YkSTFHZXRGb3VyaWVyU2VyaWVzR0YkSTdTRVJJRVNPUlRIT0dPTkFMU1lTVEVNR0YkSS9TY2FsYXJNdWx0aXBseUdGJEk1U2ltcGxpZnlDb2VmZmljaWVudHNHRiRJKVRydW5jYXRlR0Yk
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Create()</Font></Text-field> Usage of the procedure Create is similar to OrthogonalSeries[Create] but there are few differences. s:=Create({[1,2,3], n}, SinTrigP(n,x)); LUkuRk9VUklFUlNFUklFU0dJJTotMzRHUVhDOlxQcm9ncmFtfkZpbGVzXE1hcGxlfjExXHRvb2xib3hcRm91cmllclxsaWJcbWFwbGUubGliNiI2I0kiU0dGJg== type(s, FOURIERSERIES); SSV0cnVlRyUqcHJvdGVjdGVkRw== 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)); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== Creating series with different period. Create({[1,2,3], n}, CosTrigP(n,x), Pi); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== 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)); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== Some other possibilities to specify the coefficients (see OrthogonalSeries[Create]): Create({[1,[1,-1],[1/2,-1/2]]}, CosSinTrigP(n,x)); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== Create({[0=1,1=[1,-1],3=[1/2,-1/2]], 'general'=n, n=5..10}, CosSinTrigP(n,x)); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== Create({[0=1,1=[1,-1],3=[1/2,-1/2]], 'general'=n, n=5..10}, ExpTrigP2(n,x)); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== The data structure of the type FOURIERSERIES: lprint(s); FourierSeries:-FOURIERSERIES(S) print(op(1,s)); SSJTRzYi
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">ConvertToSum()</Font></Text-field> Converting series structure to the regular sum ConvertToSum(s); LC4iIiJGIy1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSJ4R0YpRiMtSSRzaW5HRiZGKiEiIi1GJTYjLCRGKyIiIyNGI0YyLUYtRjAjRi5GMi1JJFN1bUdGJjYkLCYqJkkibkdGKUYjLUYlNiMqJkY7RiNGK0YjRiNGIyomRjtGIy1GLUY9RiNGLi9GOzsiIiRJKWluZmluaXR5R0YnRiM= 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)
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Derivate()</Font></Text-field> Derivate the series s2:=Derivate(s, 'x'); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== s3:=Create({[t], t*n}, CosTrigP(n,x)); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== Derivate(s3, 'x'); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== Derivate(s3, 't'); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA==
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Evaluate()</Font></Text-field> Evaluation Evaluate(s, x=Pi/4); LCYjIiIiIiIjRiQtSSRzdW1HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JCwmKiZJIm5HRitGJC1JJGNvc0dGKDYjLCQqJkYvRiRJI1BpR0YpRiQjRiQiIiVGJEYkKiZGL0YkLUkkc2luR0YoRjJGJCEiIi9GLzsiIiRJKWluZmluaXR5R0YpRiQ= Evaluate(s, x=Pi/4, trunc=5); LCYjISIoIiIjIiIiKiRGJSNGJkYlISIk Evaluate(s, x=A, trunc=5); LDgiIiJGIy1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSJBR0YpRiMtSSRzaW5HRiZGKiEiIi1GJTYjLCRGKyIiIyNGI0YyLUYtRjAjRi5GMi1GJTYjLCRGKyIiJEY5LUYtRjchIiQtRiU2IywkRisiIiVGPy1GLUY9ISIlLUYlNiMsJEYrIiImRkUtRi1GQyEiJg==
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">ChangeBasis()</Font></Text-field> Conversion between orthogonal systems. s2; LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== ChangeBasis(s2, `ExpTrigP2`(n,x)); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== ChangeBasis(%, `SinTrigP`(n,x)); SVZDYW5ub3R+Y29udmVydH50b35TaW5UcmlnUCx+Y29udmVydGluZ350b35Db3NTaW5UcmlnUEc2Ig== LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA==
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Add(), ScalarMultiply()</Font></Text-field> Adding two series and multiplying the series by a scalar. s; LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== s2; LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== Add(s,s2); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== Add(s,s2, 1, 3); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== s3:=ScalarMultiply(s2, (1-alpha)); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kjUzJHRig=
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Coefficient(), Degree(), Truncate(), Copy()</Font></Text-field> Getting the coefficients and the degree of the partial sum, copying the series. s; LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== Coefficients(s,5); NyQiIiYhIiY= Degree(s3); SSlpbmZpbml0eUclKnByb3RlY3RlZEc= Truncate(s3,7); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kjUzJHRig= Degree(%); IiIo Copying the series s4:=Copy(s3); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kjUzJHRig=
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">SimplifyCoefficients()</Font></Text-field> Simplify the coefficients s1:=Create({-1/(n-1)}, SinTrigP(n,x)); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== s2:=Create({n/(n-1)}, SinTrigP(n,x)); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== Add(s1,s2); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== SimplifyCoefficients(%, simplify); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kjUzJHRig=
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">GetFourierSeries()</Font></Text-field> Computing the Fourier series of the given function. GetFourierSeries(cos(x)^4,x=-Pi..Pi); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== GetFourierSeries(x+signum(x),x=-Pi..Pi); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== Getting the partial sum. t:=Truncate(%, 4); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kjUzJHRig= ConvertToSum(t); LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQsJCoqSSNQaUdGJSEiIiwoKUYsSSJuR0YnIiIiKiZGLkYwRitGMEYwRixGMEYwRi9GLC1JJHNpbkdGJDYjKiZGL0YwSSJ4R0YnRjBGMCEiIy9GLztGMCIiJQ== Change the orthogonal system. ChangeBasis(t, `ExpTrigP2`(n,x)); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== GetFourierSeries(x^2,x=-Pi..Pi, exp); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== Getting the Fourier series of the odd/even periodic extension. GetFourierSeries(x^2,x=0..Pi, even); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== GetFourierSeries(x^2,x=0..Pi, even, exp); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== GetFourierSeries(x^3,x=-Pi..Pi); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== GetFourierSeries(x^3,x=-Pi..0, odd); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA==
<Text-field style="Text" family="Arial" bold="true" size="18" layout="Heading 1"><Font family="Arial" bold="true" size="18">Solving the differential equation using the Fourier series.</Font></Text-field> Diff(y(x),x$2)+2*y(x)=Sum(sin(n*x)/n^4,n=1..infinity); LywmLUklRGlmZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlLUkieUdGKTYjSSJ4R0YpRi5GLiIiIkYrIiIjLUkkU3VtR0YmNiQqJi1JJHNpbkdGJjYjKiZJIm5HRilGL0YuRi9GL0Y5ISIlL0Y5O0YvSSlpbmZpbml0eUdGJw== S1:=Create({[0],'general'=[0,1/n^4]}, CosSinTrigP(n,x)); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== Create the general series S2:=Create({[a0], 'general'=[aN,bN]}, CosSinTrigP(n,x)); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== Derivate the series two times tmp1:=Derivate(Derivate(S2, 'x'), 'x'); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== Add the multiple of the series tmp2:=Add(tmp1,S2, 1, 2); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== Our equation now has the form tmp2=S1; Ly1JLkZPVVJJRVJTRVJJRVNHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lRzYiSS5Gb3VyaWVyU2VyaWVzR0YpNiNJIlNHRilGIw== Subtract the series S:=Add(lhs(%), rhs(%), 1, -1); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== type(S, FOURIERSERIES); SSV0cnVlRyUqcHJvdGVjdGVkRw== Series S should have all coefficients equal to zero. Hence a0 = 0. Get the general coefficient. Coefficients(S); NyQsJkkjYU5HNiIiIiMqJkkibkdGJUYmRiQiIiIhIiIsKCokRighIiVGKkkjYk5HRiVGJiomRihGJkYuRilGKg== {%[1]=0, %[2]=0}; PCQvLCZJI2FORzYiIiIjKiZJIm5HRiZGJ0YlIiIiISIiIiIhLywoKiRGKSEiJUYrSSNiTkdGJkYnKiZGKUYnRjFGKkYrRiw= solve(%, {aN, bN}); PCQvSSNhTkc2IiIiIS9JI2JOR0YlLCQqJkkibkdGJSEiJSwmISIjIiIiKiRGKyIiI0YvISIiRjI= Create the solution. Y:=Create({subs(%, bN)}, SinTrigP(n,x)); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== Check the solution Add(Derivate(Derivate(Y, 'x'), 'x'), Y, 1, 2); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kiU0dGKA== SimplifyCoefficients(%, simplify); LUkuRk9VUklFUlNFUklFU0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLkZvdXJpZXJTZXJpZXNHRig2I0kjUzJHRig=
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