Fourier series of function with arbitrary period p=2L

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Fourier series of function with arbitrary period
p2L
Instead of a period of 2?, many functions have an
arbitrary period, say a period of 2L. In order
to convert the Fourier series defined earlier for
these functions, a change of variable is
needed Replace the variable x by (?/L)x when
xL the new variable equals to ? when x -L, it
equals to - ?. Therefore, the previous formulas
can be used by simply making the change
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Even and Odd Functions
A function f(x) is even when f(x) f(-x) On the
other hand, if f(x) -f(-x), the function is an
odd function.
An even function
An odd function
f(x)
f(x)
x
x
Ex cos(x)
Ex sin(x)
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Fourier cosine and sine series
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Half Range Expansion
Expansion is useful when a function is defined
only on a given interval, say between 0 and L.
This situation is very common in real life For
example, the vibration of a guitar string occurs
only between its bridge and tension peg.
expansion