Applying Kramers-Kronig relation to a simple damped oscillator

Question

I just discovered the Kramers-Kronig relation and am trying to apply it to a simple damped oscillator of the form subjected to an impulse at $t=0$, which is a causal system:

$$m\ddot x + c\dot x + k x = \delta(t).$$

In the time-domain, the response can be decomposed in odd/even parts as $x(t) = \operatorname{sign}(t)h_0(t) + h_0(t)$. The Kramers-Kronig relation implies that because the signal is causal, the real part of the Fourier transform of a solution $x(t)$ is equal to the Hilbert transform of the imaginary part of the Fourier transform of $x(t)$. That is what I am trying to illustrate here (and I am afraid that it would lead to a constraint between $m$, $c$ and $k$, which would probably suggest a misunderstanding on my side...).

The Fourier transform of the ODE gives $$ (-m \omega^2 +k + ic\omega )\hat x = 1$$

so

$$\hat x = \dfrac{1}{-m \omega^2 +k + ic\omega } = \dfrac{k-m\omega^2}{(k-m\omega^2)^2 + (c\omega )^2} + i \dfrac{-c\omega}{(k-m\omega^2)^2 + (c\omega )^2} $$

If I am not mistaken, the KK relation would imply that the Hilbert transform of $\dfrac{-c\omega}{(k-m\omega^2)^2 + (c\omega )^2} $ is $\dfrac{k-m\omega^2}{(k-m\omega^2)^2 + (c\omega )^2}$, ie

$$\dfrac{1}{\pi}\int_{-\infty}^\infty -\dfrac{1}{u-\omega}\dfrac{c\omega}{(k-m\omega^2)^2 + (c\omega )^2} d\omega \stackrel{?}{=} \dfrac{k-m u^2}{(k-m u^2)^2 + (cu )^2}$$ which does not hold (take for instance $u=m=c=k=1$, the integral does not converge). Edit Actually, it does hold for $m=c=k=1$, in the sense of the principal value (see hint and answer) ! I was not able to verify for arbitrary $m,c,k$.

Mathematica code for whoever is interested:

m = k = c = 1;
LHS  = Assuming[Element[u, Reals], 1/Pi*Integrate[-1/(u - w)*c*
      w/((k - m*w^2)^2 + (c^2*w^2)), {w, -Infinity, Infinity}, 
     PrincipalValue -> True]];
RHS = (k - m*u^2)/((k - m*u^2)^2 + c^2*u^2);
LHS == RHS // Simplify

Answer 1

This work!

$$x=\frac{1}{-m{\omega}^{2}+k+ic\omega}=\Re{(x(\omega))}+i\,\Im{(x(\omega))}$$ with: $$\Re{(x(\omega))}={\frac {-m{\omega}^{2}+k}{ \left( -m{\omega}^{2}+k \right) ^{2}+{c}^{2 }{\omega}^{2}}}\tag 1$$ and $$\Im{(x(\omega))}=-{\frac {c\omega}{ \left( -m{\omega}^{2}+k \right) ^{2}+{c}^{2}{\omega }^{2}}} \tag 2$$

Kramers-Kronig relation :

$$I_R=-\frac{2}{\pi}\int_0^\inf \frac{\omega\,\Im{(x(\omega))}}{\omega^2-s^2}\,d\omega$$

for $m=1\,,k=1\,,c=1$

we get:

$$I_R=\frac{1-s^2}{s^4-s^2+1}$$

this is the value of equation (1) for $\Re{(x(s))|_{m=1\,,k=1\,,c=1}}$

$$I_I=\frac{2}{\pi}\int_0^\inf \frac{s\,\Re{(x(\omega))}}{\omega^2-s^2}\,d\omega$$

we get:

$$I_I=-\frac{s}{s^4-s^2+1}$$

this is the value of equation (2) for $\Im{(x(s))|_{m=1\,,k=1\,,c=1}}$

to get the right answer you must take the Cauchy-Principal -Value of the integral result :

$$I= \text{Cauchy-Principal -Value}\,\left[\int_0^\inf (...)\,d\omega\right]$$

I did the calculation with Maple symbolic program