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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),
\end{displaymath} (1)

$\displaystyle u(x_0)$ $\textstyle =$ $\displaystyle {1\over \pi } PV \int_{-\infty}^\infty {v(x)\, dx\over x-x_0}$ (2)
$\displaystyle v(x_0)$ $\textstyle =$ $\displaystyle - {1\over \pi } PV \int_{-\infty}^\infty {u(x)\,dx\over x-x_0},$ (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
$\displaystyle u(x_0)$ $\textstyle =$ $\displaystyle {2\over \pi} PV \int_0^\infty {xv(x)\,dx\over x^2-{x_0}^2}$ (4)
$\displaystyle v(x_0)$ $\textstyle =$ $\displaystyle - {2\over \pi} PV \int_0^\infty {xu(x)\,dx\over x^2-{x_0}^2}.$ (5)

© 1996-9 Eric W. Weisstein