Kramers-Kronig: Solving for Dielectric Permeability E'(w)

[Spy2008]

In formula Kramers-Kronig 82,9 Landau - Lifgarbages page 390 volume 8 for dielectric permeability have added 2 composed 4*pisigma/W (omega) which is responsible for a pole in zero in a conductor. A question: the First part of imaginary dielectric permeability E’’(w)=-1/pi*∱E’(w’)/(w’-w) dw’ where E-dielectric permeability should remain constant, but in it was gone-1 why, and hardly above in the formula 82,7 for dielectric was not gone-1?
The question of the teacher in that that it was not necessary but if who can prove that it influences something or something depends on it will be plus:). Using any materials. Here were, what variants:
" On the mathematician the conclusion assumes to neglect integral on infinite to a floor of a circle for what from function subtract its value on infinity, and it just (see at the same LL) 1. " On other that due to-1 integral to converge better and it is not necessary to look what frequency, here, if it is possible to become more in detail at which frequencies noticeably participation-1
If it is possible write the variants as can influence-1 physically, mathematical, for example if all таки with-1 to converge better that it is possible to paint if not difficultly or to give the reference to the literature where is painted, simply the teacher should to something be shown, that-1 is though any sense, instead of in words. In advance ALL THANKS.

 

[ZapperZ]

Just so you know, your post here is very difficult to understand. I'm guessing that you used a translator of some kind, which means that you may use the same one reading the responses on here. That will make it very tough to communicate effectively on both sides.

Zz.

 

[charng]

The Kramers-Kronig relations are a powerful tool in understanding the relationship between the real and imaginary parts of a complex function. In the case of dielectric permeability, the Kramers-Kronig relations allow us to relate the real part of the dielectric permeability (E') to its imaginary part (E'') through integrals.

In the formula given, the term 4*pisigma/W (omega) is added to account for the presence of a pole at zero frequency in conductors. This is important because it allows us to accurately describe the behavior of materials at low frequencies, where the dielectric permeability is typically complex.

Now, let's address the question about the first part of the imaginary dielectric permeability, E''(w). In the formula, we see that it is given by the integral of E'(w')/(w'-w) dw' over all frequencies. The teacher's question is why this integral evaluates to -1/pi when it should remain constant.

The answer lies in the nature of the integral. When we integrate over all frequencies, we are essentially summing up the contributions of all frequencies to the imaginary part of the dielectric permeability. In this case, we are subtracting the value of E'(w') at infinity, which is assumed to be zero. This is because in the limit of high frequencies, the dielectric permeability tends to zero.

So, why is -1/pi the value that this integral evaluates to? This is due to the mathematical properties of the function being integrated. As the teacher suggested, the integral converges better when we subtract its value at infinity. This means that the value of the integral is more accurate and less dependent on the specific frequency range being considered. In this case, the value of -1/pi is a result of the mathematical properties of the function being integrated.

As for the physical implications of this result, it is difficult to say without more context. However, the fact that the integral evaluates to a constant value regardless of frequency suggests that the behavior of the material at low frequencies is not significantly affected by the presence of the pole at zero frequency. This could be further explored through experimentation or by consulting relevant literature.

In summary, the value of -1/pi in the Kramers-Kronig relations for dielectric permeability is a result of the mathematical properties of the function being integrated. It is important in accurately describing the behavior of materials at low frequencies and