Mathematical Tools for Physics

14—Complex Variables 446

14.7 What is a Laurent series expansion aboutz= 0to at least four terms for

sinz/z^4 ez/z^2 (1−z)

What is the residue atz= 0for each function? Assume|z|< 1. Then assume|z|> 1 and find the series.
Ans:(−e/z^3 ) +

(

(1−e)/z^2

)

+

(

(2−e)/z

)

+ (2. 5 −e) +···

14.8 By explicit integration, evaluate the integrals around the counterclockwise loops:

∫

C 1

z^2 dz

∫

C 2

z^3 dz

C 1

1 +i

1

C 2

ib a+ib

a

14.9 Evaluate the integral along the straight line fromatoa+i∞:

∫

eizdz. Takeato be real. Ans:ieia

14.10 (a) Repeat the contour integral Eq. ( 8 ), but this time push the contourdown, not up.
(b) What happens to the same integral ifais negative? And be sure to explain your answer in terms of the
contour integrals, even if you see an easier way to do it.

14.11 Carry out all the missing steps starting with Eq. ( 7 ) and leading to Eq. ( 12 ).

14.12 Sketch a graph of Eq. ( 12 ) and fork < 0 too. What is the behavior of this function in the neighborhood
ofk= 0? (Careful!)

14.13 In the integration of Eq. ( 13 ) the contourC 2 had a bump into the upper half-plane. What happens if the
bump is into the lower half-plane?

14.14 For the function in problem 7 ,ez/z^2 (1−z), do the Laurent series expansion aboutz= 0, but this time
assume|z|> 1. What is the coefficient of 1 /z now? You should have no trouble summing the series that you
get for this. Now explain why this result is as it is. Perhaps review problem 1.