Mathematical Tools for Physics

14—Complex Variables 440

going around the second branch point will introduce a second identical factor. As(−1)^2 = +1, then when you
you return toz 0 the function is back at

√

3 , you have returned to the base point and this whole loop is closed.
If this were the sum of two square roots instead of their product, this wouldn’t work. You’ll need four sheets to
map that surface. See problem 22.

These cuts are rather awkward, and now that I know the general shape of the surface it’s possible to arrange
the maps into a more orderly atlas. Here are two better ways to draw the maps. They’re much easier to work
with.

0

1

a

b

b

a

c

d

d

c or

0

1

e

f

f

e

I used the dash-dot line to indicate the cuts. In the right pair, the base point is on the right-hand solid line
of sheet #0. In the left pair, the base point is on thecpart of sheet #0. See problem 20.

14.8 Other Integrals
There are many more integrals that you can do using the residue theorem, and some of these involve branch
points. In some cases, the integrand you’re trying to integrate has a branch point already built into it. In other
cases you can pull some tricks and artificially introduce a branch point to facilitate the integration. That doesn’t
sound likely, but it can happen.

Example 8
The integral

∫∞

0 dxx/(a+x)

(^3). You can do this by elementary methods (very easily in fact), but I’ll use it to
demonstrate a contour method. This integral is from zero to infinity and it isn’t even, so the previous tricks don’t
seem to apply. Instead, consider the integral (a > 0 )
∫∞
0
dxlnx
x
(a+x)^3
and you see that right away, I’m creating a branch point where there wasn’t one before.