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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)=u(x_0)+iv(x_0),
(1)

then

u(x_0)=1/piPVint_(-infty)^infty(v(x)dx)/(x-x_0)
(2)
v(x_0)=-1/piPVint_(-infty)^infty(u(x)dx)/(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/piPVint_0^infty(xv(x)dx)/(x^2-x_0^2)
(4)
v(x_0)=-2/piPVint_0^infty(x_0u(x)dx)/(x^2-x_0^2).
(5)

See also

Hilbert Transform

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References

Byron, F. W. Jr. and Fuller, R. W. Mathematics of Classical and Quantum Physics. New York: Dover, p. 344, 1992.

Referenced on Wolfram|Alpha

Dispersion Relation

Cite this as:

Weisstein, Eric W. "Dispersion Relation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DispersionRelation.html

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