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# Dirichlet's theorem on both types of discontinuity

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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 Dirichlets theorem on both types of discontinuity

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#### Solution Preview

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 ...

#### Solution Summary

Dirichlet's theorem on both types of discontinuity is clearly reinforced.

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