Fourier exercise 5

: Math 2730 Sequences and Series: Fourier exercise 5
   By Owen Douglas Bowkett on Tuesday, April 15, 2008 - 08:11 pm: Edit Post

I am at a loss how to solve this question. I want to do a transformation on the two parts of f(x) so that they are symmetric about the Y-axis, and then I can use the usual formula to solve. Say I make g(x) = {abs value}(x/2). Both parts of f(x) have been rotated clockwise by the same angle, are symmetric about the Y-axis and so I can use the formula to solve g(x).

Can we do this? Is there an easier solution?


   By Sasho on Tuesday, April 15, 2008 - 09:47 pm: Edit Post

Your approach is unclear to me. The simplest would be to proceed straightforward and simply evaluate the integrals given in the formulas for the coefficients [that is, 1/pi integral (from -pi to pi) of the f(x) cosnx , n=0,1,2,... and the 1/pi integral (from -pi to pi) of the f(x) sinnx, n=1, 2,...].

   By Owen Douglas Bowkett on Wednesday, April 16, 2008 - 12:46 pm: Edit Post

Thanks Professor Kalajdzievski. I solved the exercise. I had inadvertently assumed that, since problems 1-3 involved symmetric functions which made finding their solutions much simpler, question 5 must also involve symmetry and, therefore, the function must be transformed in some manner prior to tackling the integrals. Simply following the definition for the Fourier series coefficients, of course, produced the answer.

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