Sunday, 17 April 2016

Finding Fourier series y=|1-x|



I've been struggling with finding Fourier series of the given function for a while now
enter image description here



I've calculated my coefficients using formulas :










Though my results does approximate function sufficiently enough (see the picture), I'm pretty sure that I did something wrong, as online calculators give me different values for coefficients, and the sum of some series(like (-1)^n/n^2) does not match expected values(integrals are calculated correctly). Is there something wrong with how I approach solving those integrals for coefficients or maybe there are some other pitfalls that I could have encountered? I would really appreciate it if you could help me out at least with calculating . Thank you in advance.



enter image description here



Resulting Furier series




enter image description here


Answer



Here are my calculations. I haven't thoroughly checked them, but hopefully they will help.



an=12[12(1x)cosnπ2x21(1x)cosnπ2x]12[2(x1)nπsinnπ2x+4n2π2cosnπ2x|21+2(x1)nπsinnπ2x+4n2π2cosnπ2x|21]



Evaluated at x=2, and x=2,



sinnπ2x=0cosnπ2x=(1)n



at x=1,(x1)=0



cosnπ2x=0 if n is odd, 1 if n2(mod4), 1 if n4(mod4)



an=4n2π2,8n2π2,4n2π2,0 when n1,2,3,4 respectively.



bn=12[12(1x)sinnπ2x21(1x)sinnπ2x]12[4n2π2sinnπ2x2(x1)nπcosnπ2x]




Evaluated at:



x=22nπcosnπ(1)n2nπ



x=26nπcosnπ(1)n6nπ



x=14n2π2sinnπ24n2π2,0,4n2π2,0



bn=4nπ4n2π,4nπ,4nπ+4n2π,4nπ


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