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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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f 10 integrate from 0 to 10 f 100 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

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