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

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

The derivative of the logarithm is analytic except at the zeroes and poles of f (z) where it has simple poles

$${\frac{d}{dz}} {\textstyle\frac{1}{z-z_0}}$$ =

$${\frac{d}{dz}} {\textstyle\frac{1}{z-z_0}}$$ = (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