10. Appendix

620 Appendix B
two-dimensional and a possible set of basis function has the form: x 2 y 2 and
z 2 [x 2 y 2 ]/2. In this case y 2 and z 2 will not change sign under C2(x) rotation.
To achieve the result with the Si wave functions we find that [Y Y] and
[ZZ] will not change sign under the C2(x) rotation. Thus the normalized linear
combination for the E irreducible representations are: {[XX][YY]}/2
and {[2Z 2Z] [X X] [Y Y]}/ √ 12.
Solution to Problem 4.5
Assume that the imaginary part of an analytic function F(E) is given by:
Im[F(E)] [(E E ′ ) 2 ° 2 ] 1 . (4.5a)
Applying the Kramers-Kronig relation (4.56), we find that the real part of
F(E) is given by:
Re[f (E)] 1
apple P
1
apple P
∞
∞ ∞
∞
Im[F(z)]
z E dz
(1)
(z E)[(z E ′ ) 2 ° 2 ] dz
(4.5b)
One way to calculate this definite integral is to rewrite the integrand as a sum:
1/(z E)[(z E ′ ) 2 ° 2 ] [A/(z E)] [B/(z E ′ i°)]
[C/(z E ′ i°)]
(4.5c)
where i2 (1), A 1/[(E E ′ )2 ° 2 ], B (i/2°)[1/(E E ′ i°)]
and C (i/2°)[1/(E E ′ i°)]. For each term in (4.5c) the corresponding
integration in (4.5b) can be performed with the help of a contour integral. As
an example, let us consider:
∞
dz
P
∞ z (E ′ (4.5d)
i°)
First, we will simplify the integral with a change in variable: x z E ′ so that
(4.5d) becomes:
∞
dx
P
(4.5e)
∞ x i°
The integrand in (4.5e) has a pole at x i°. To obtain its principal value we
consider an integral over a closed contour C in the complex z ′ -plane where
z ′ x iy: