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Up: PHY531-PHY532 Classical Electrodynamics

PHY532 Problem Set 4. Due April 19, 2001

1.
Jackson 7.24. You should assume that $\epsilon$(0)$\ge$1 and that $\epsilon$($\omega$) goes to 1 at high frequencies. One way to attack parts a and b is to remember that the difference in the number of zeroes and poles of a function f (z) analytic except for simple poles inside a contour is given by the integral around the contour of

$\displaystyle{\textstyle\frac{1}{2\pi i}}$$\displaystyle\oint$dz$\displaystyle{\frac{d \ln f(z)}{dz}}$ . (1)

The derivative of the logarithm is analytic except at the zeroes and poles of f (z) where it has simple poles
$\displaystyle{\frac{d}{dz}}$ln[a(z - z0)] = $\displaystyle{\textstyle\frac{1}{z-z_0}}$   
$\displaystyle{\frac{d}{dz}}$ln[a(z - z0)- 1] = - $\displaystyle{\textstyle\frac{1}{z-z_0}}$ . (2)
Notice that the residue is + 1 for zeroes and - 1 for poles, so the sum of the residues is the difference in the number of zeroes and poles. You can use this to determine the number of times f (z) takes on any particular value inside a contour.

2.
Jackson 8.2

3.
Jackson 8.4

4.
Jackson 8.6

5.
Jackson 8.19


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