3.2 Further exercises
Here are some further exercises to end this section.
Exercise 14
For each of the following functions , evaluate

where is any contour from
to
.
Answer
In each case, is continuous on
and has a primitive on
, so we can apply the Fundamental Theorem of Calculus to evaluate the integral using any contour
from
to
.
A primitive of
is
Hence
A primitive of
is
Hence
Let
,
. Then
and
are entire (that is,
and
are differentiable on the whole of
), and
and
are entire and hence continuous. Then, using Integration by Parts (Theorem 9), we have
Exercise 15
Evaluate the following integrals. (In each case pay special attention to the hypotheses of the theorems you use.)
,
where
is the arc of the circle
from
to
passing through 1.
, where
is as in part (a).
, where
is the unit circle
.
, where
is the circle
.
(Hint: For part (c), use the identity .)
Answer
Let
,
and
. Then
is continuous on
,
is a primitive of
on
, and
is a contour in
. Thus, by the Fundamental Theorem of Calculus,
Let
,
and
. Then
is continuous on
,
is a primitive of
on
, and
is a contour in
. Thus, by the Fundamental Theorem of Calculus,
The function
is continuous and has an entire primitive
. Thus, by the Closed Contour Theorem,
Let
,
and
. Then
is continuous on
,
is a primitive of
on
, and
is a contour in
. Thus, by the Closed Contour Theorem,
Exercise 16
Construct a grid path from to
in the domain of the function
, for each of the following cases.
,
,
Answer
The domain of is the region

The figure shows one grid path in
from 1 to 6 (there are many others).
The figure shows one grid path in
from
to
(again, there are many others).
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