I have a function $$\vec{\pi}_{B}: \Bbb R^3 \to \Bbb C^3$$ that depends parametrically on a value $B\in\Bbb R^+$. I have full knowledge of its imaginary part $$ \Im\{ \vec{\pi}_{B}\} = f(B)$$
Is there in general (or also under specific conditions) a way to use the Kramers-Kroning relation to "reconstruct" or calculate
$$\Re\{ \vec{\pi}_{B}\} \in \Bbb R^3$$
The problem emerges from some quantum mechanical considerations on the "mechanical momentum density" in a magnetic field.
My idea is to break it up componentwise somehow in something like $3\times3$ problems.
In case this is not possible, for a specific subproblem it would be already of interest to say something on the modulus $|\Re\{ \vec{\pi}_{B}\}|$ since I am specifically interested in points $\vec{r}_0$ where $$\vec{\pi}_{B=0}(\vec{r}_0)=\vec{0}$$ and on their dependency on $B$, while I know $$ \Im\{ \vec{\pi}_{B}(\vec{r}_0)\} = f(B); \forall B.$$
Most specifically the question then is if $|\Re\{ \vec{\pi}_{B}\}|$ remains $=0$ when "turning on the perturbation $B$".