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Covariant Green's function for wave equation

  1. Aug 25, 2009 #1
    This comes from Jackson's Classical Electrodynamics 3rd edition, page 613. He finds the Green's function for the covariant form of the wave equation as:

    [tex]D(z) = -1/(2\pi)^{4}\int d^{4}k\: \frac{e^{-ik\cdot z}}{k\cdot k}[/tex]

    Where z = x - x' the 4 vector difference, [tex] k\cdot z = k_0z_0 - \mathbf{k \cdot z}[/tex]

    He then performs the integral over k0 first by considering it as a complex variable to give:

    [tex]D(z) = -1/(2\pi)^{4}\int d^{3}k e^{i\mathbf{k\cdot z}}\int_{-\infty }^{\infty } dk_0\: \frac{e^{-ik_0z_0}}{k^2_0 - \kappa^2}[/tex]

    where [tex] \kappa = |\mathbf{k}|[/tex]

    But then he says for [tex]z_0>0, e^{-ik_0z_0}[/tex] increases without limit in the upper half plane.

    Is this correct? There's a minus in the exponential so i would have thought it's the exact opposite.

    He considers a contour r in the upper half of the k0 plane from +oo to -oo, closed by a semicircle also in the upper half of k0 and says that for z0 < 0, the resulting integral vanishes, whereas for z0 > 0, the integral over k0 is:

    [tex]\oint_{r} dk_0\: \frac{e^{-ik_0z_0}}{k^2_0 - \kappa^2} = -2\pi i\: Res \left (\frac{e^{-ik_0z_0}}{k^2_0 - \kappa^2} \right) = \frac{-2\pi}{\kappa} sin(\kappa z_0)[/tex]

    The Green function is then:

    [tex]D_r(z) = \frac{\theta(z_0)}{(2\pi)^3} \int d^3k \: e^{i\mathbf{k \cdot z}}\ \frac{sin(\kappa z_0)}{\kappa}[/tex]

    Where does the [tex]\theta(z_0)[/tex] come from?

    Thanks in advance for your interest.
  2. jcsd
  3. Aug 25, 2009 #2
    Thats the Heaviside step function, which is zero for [itex]z_0 <0[/itex] and unity for [itex]z_0 > 0[/itex]. The integral is zero for [itex]z_0 <0[/itex] as you pointed out yourself, so this is just a convenient way of writing it.
  4. Aug 25, 2009 #3

    George Jones

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    Take [itex]z_0 = a + ib[/itex] with [itex]a[/itex] and [itex]b[/itex] real, and [itex]b > 0[/itex] (upper half-plane). Then

    [tex]e^{-i k_0 z_0} =e^{-i a k_0} e^{b k_0},[/tex]

    which blows up as [itex]b[/itex] becomes large.
  5. Aug 25, 2009 #4

    George Jones

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    Oops, I wrote this without reading. Often, [itex]z[/itex] denotes a complex intregration variable, and I made the mistake of taking this to be so here. What I wrote above should, however, make clear what happens for complex [itex]k[/itex] and [itex]z_0[/itex] a positive real.
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