Fourier Integrals

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Fourier Integrals

• For non-periodic applications (or a specialized
  Fourier series when the period of the function is
  infinite L??)

L
-L
L??
-L?-?
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Fourier Cosine Sine Integrals
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Example
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f10 integrate from 0 to 10 f100 integrate from 0
to 100 g(x) the real function
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Similar to Fourier series approximation, the
Fourier integral approximation improves as the
integration limit increases. It is expected that
the integral will converges to the real function
when the integration limit is increased to
infinity. Physical interpretation The higher
the integration limit means more higher frequency
sinusoidal components have been included in the
approximation. (similar effect has been observed
when larger n is used in Fourier series
approximation) This suggests that w can be
interpreted as the frequency of each of the
sinusoidal wave used to approximate the real
function. Suggestion A(w) can be interpreted as
the amplitude function of the specific sinusoidal
wave. (similar to the Fourier coefficient in
Fourier series expansion)
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Fourier Cosine Transform
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Fourier Sine Transform