#103 In each of Problem, write the Fourier integral representation

In each of Problem, write the Fourier integral representation - Math

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Free Chegg Question

In each of Problem, write the Fourier integral representation (14.1) of the function and determine what this integral converges to.

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Free Chegg Answer

Let 
The graph of is

Certainly  is piecewise smooth



Therefore

The Fourier coefficients of f are

Since the graph is symmetric with respect to the origin, so it is an odd function.
Therefore is an odd function
Hence

Therefore

Now

Since is odd function and is odd function, so is even function.
Therefore


The Fourier integral of  is

is discontinuous at the points so, at that points, we take an average of left and right-hand limits i.e., and 

Then