Mathematical Tools for Physics

14—Complex Variables 447

14.15 In the integration of Eq. ( 13 ) the contourC 2 had a bump into the upper C?
half-plane, but the original function had no singularity at the origin, so you can start
withthiscurve and carry out the analysis. What answer do you get?

14.16 Use contour integration to evaluate Eq. ( 13 ) for the case thata < 0.
(b) Independently of this, make a change of variables in the original integral Eq. ( 13 ) in order to see if the answer
is independent ofa. In this part, consider two cases,a > 0 anda < 0.

14.17 Recalculate the residue done in Eq. ( 5 ), but economize your labor. If all that all you really want is the
coefficient of 1 /z, keep only the terms that you need in order to get it.

14.18 What is the order of all the other poles of the functioncsc^3 z, and what is the residue at each pole?

14.19 Verify the location of the roots of Eq. ( 14 ).

14.20 Verify that the Riemann surfaces work as defined for the function

√

z^2 − 1 using the alternative maps in
section14.7.

14.21 Map out the Riemann surface for

√

z(z−1)(z−2). You will need four sheets.

14.22 Map out the Riemann surface for

√

z+

√

z− 1. You will need four sheets.

14.23 Evaluate ∫

C

dz e−zz−n

whereCis a circle of radiusRabout the origin.

14.24 Evaluate ∫

C

dztanz

whereCis a circle of radiusπnabout the origin. Ans:− 4 πin