Any pair of equations giving the real part of a function as an integral of its imaginary part and the imaginary part as an integral of its real part. Dispersion relationships imply causality in physics. Let
(1)
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then
(2)
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(3)
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where denotes the Cauchy principal value and and are Hilbert transforms of each other. If the complex function is symmetric such that , then
(4)
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(5)
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