Dispersion Relation

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

$$f(x_0) \equiv u(x_0)+iv(x_0),$$ (1)

then

$$u(x_0) = {1\over \pi } PV \int_{-\infty}^\infty {v(x)\, dx\over x-x_0}$$ (2)

$$v(x_0) = - {1\over \pi } PV \int_{-\infty}^\infty {u(x)\,dx\over x-x_0},$$ (3)

where $PV$ denotes the Cauchy Principal Value and $u(x_0)$ and $v(x_0)$ are Hilbert Transforms of each other. If the Complex function is symmetric such that $f(-x) = f^*(x)$, then

$$u(x_0) = {2\over \pi} PV \int_0^\infty {xv(x)\,dx\over x^2-{x_0}^2}$$ (4)

$$v(x_0) = - {2\over \pi} PV \int_0^\infty {xu(x)\,dx\over x^2-{x_0}^2}.$$ (5)