Kramers-Kronig relations

In mathematics and physics, the Kramers-Kronig relations describe the relation between the real and imaginary part of a certain class of complex-valued functions. The requirements for a function f(omega) to which they apply can be interpreted as that the function must represent the Fourier transform of a linear and causal physical process. If we write

Related Topics:
Mathematics - Physics - Real - Imaginary - Complex - Fourier transform - Linear - Causal

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:f(omega) = f_1(omega) + i f_2(omega),

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where f_1 and f_2 are real-valued "well-behaving" functions, then the Kramers-Kronig relations are

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:

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f_1(omega) = rac{2}{pi} int_0^{infty}

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rac{omega' f_2(omega') domega'}{omega^2 - omega'^2}

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:f_2(omega) = -rac{2 omega}{pi} int_0^{infty}

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rac{f_1(omega') domega'}{omega^2 - omega'^2}

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.

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The Kramers-Kronig relations are related to the Hilbert transform, and are most often applied on the permittivity epsilon(omega) of materials. However, it must be noticed that in this case,

Related Topics:
Hilbert transform - Permittivity

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: f(omega) = chi(omega) = epsilon(omega)/epsilon_0 - 1,

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where chi(omega) is the electric susceptibility of the material. The susceptibility can be interpreted as the Fourier transform of the time-dependent polarization in the material after an infinitely short pulsed electric field, in other words the impulse response of the polarization.

Related Topics:
Electric susceptibility - Polarization

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