Dirichlet's theorem on both types of discontinuity

Please see the attachment.

a) Sketch the periodic function y=ex, -2< x <2 and y(x)=y(x+4) for values of x from -6 to 6. State the period and whether the function is odd even or neither.
b) Give the Fourier series for the odd periodic extension of: y=ex, 0< x <2
c) Confirm Dirichlet's theorem on both types of discontinuity

Solution:

a) Since it is given that , the period is T = 4. We will plot the graph of and this graph will be repeated on (-6, -2) and (2, 6), according to the figure below:

Relevant intersections:
Since the graph shows no symmetry about Oy axis or the origin, the function such defined is neither odd, nor even.

b) The odd periodic extension of will have the graph like in the next figure:

Analytically, the odd periodic function such defined will have the expression:
( 1)

The function was built by adding a symmetric branch about the origin on
(-2, 0) and then, by repeating periodically the graph.

The expansion in Fourier series will contain only "sine" terms:
( 2)
where
( 3)
and
( 4)
Replacing with and T = 4, we will get:
, ( 5)
We have now to compute the coefficients bn, using a repeated integration by parts:







The Fourier series of the odd function defined by (1) will be:
( 6)
c) The Dirichlet's theorem says that, if f is piecewise monotonic with a finite number of discontinuities and x0 is a point of discontinuity, then the Fourier series in x0 converges to .
We will check that for xo = 0 and xo = 2 (or xo = -2), by substituting in (6):
(i)
But and so that

and the Dirichlet's theorem is verified.
(ii)
But and so that

and the Dirichlet's theorem is verified again.

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