This preview shows pages 1–2. Sign up to view the full content.
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).
Since
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 09/19/2011 for the course MATH 132 taught by Professor Grossman during the Spring '08 term at UCLA.
 Spring '08
 Grossman

Click to edit the document details