Problem 1(a) (5 points): Let
gamma_1
be the curve
gamma_1(theta) = exp(i theta), 0 <= theta <= 3
pi
.
Compute the integral of
z
dz
along
gamma_1
, where
z
is the conjugate of
z
.
Answer: Making the change of variables
z = exp(i theta)
dz = i exp(i theta) d theta
and observing that
z
= exp( i theta)
the integral becomes that of
exp(i theta) i exp(i theta) d theta
from 0 to 3pi.
The exponentials cancel, so you are just integrating
i dtheta
from 0 to 3 pi, which is 3 pi i.
Some common errors:
Ignoring the conjugation, and writing
z
as
exp(i theta)
instead of
exp( i theta)
.
(Or ignoring
the z
term altogether).
Using the fundamental theorem of calculus, and trying to press
z^2/2
or
z
^2/2
into service as an
antiderivative for z
.
(Actually z
does not have an antiderivative, because it is not analytic, and only
analytic functions have antiderivatives).
Other comments:
It is possible to "guess" the answer as 3 pi i, based on reasoning by analogy with similar examples.
However, this type of reasoning can be dangerous.
For instance, suppose we are integrating z
on the
curve
z = 2 exp(i theta), 0 <= theta <= 2 pi.
Can you guess the answer by inspection?
Make a guess, and then check your answer.
Problem 1(b) (10 points):
Let gamma_2 be the line segment from 1 to 1i.
Compute the integral of dz/z on
gamma_2.
Answer: We shall apply the fundamental theorem of calculus.
We want an antiderivative of dz/z which is
analytic on gamma_2.
The principal branch
Log(z)
won't work, because it's not analytic on the negative real
line, and gamma_2 just touches this line.
However, a branch such as
Log_(0,2 pi] (z)
is analytic
on gamma_2 (it is only nonanalytic at 0 and the positive real axis, which is not a problem).
So by the
Fundamental theorem of calculus the answer is
Log_(0,2 pi](1i)  Log_(0,2 pi](1).
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 Spring '08
 Grossman
 Holomorphic function, Logarithm

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