### Become an OU student

Introduction to complex analysis

Start this free course now. Just create an account and sign in. Enrol and complete the course for a free statement of participation or digital badge if available.

# 2.4 Further exercises

Here are some further exercises to end this section.

## Exercise 8

Evaluate the following integrals (using the standard parametrisation of the path in each case).

• a.

• i.,

• ii.,

• iii.,

where is the line segment from 1 to .

• b.

• i.,

• ii.,

where is the unit circle .

• c.

• i.,

• ii.,

where is the upper half of the circle with centre 0 and radius 2 traversed from 2 to .

• a.The standard parametrisation of , the line segment from 1 to , is

hence

• i.Here , and

• ii.Here , and

(Note that this integral is different from , which from part (a)(i) is 0.)

• iii.Here , and

(Again, note that this is different from .)

• b.We set out this solution in a similar style to Example 4.

The standard parametrisation of , the unit circle , is

hence

• i.Here , and

• ii.Here , and

• c.The standard parametrisation of , the upper half of the circle with centre 0 and radius 2, traversed from 2 to , is

hence

• i.Here , and

• ii.Here , and

## Exercise 9

Evaluate

for each of the following contours  from 0 to .

• a., where is the line segment from 0 to and is the line segment from to .

We choose to use the standard parametrisations

Then , . Hence

• b., where is the line segment from 0 to 1 and is the line segment from 1 to .

We choose to use the standard parametrisations

Then , . Hence

(Note that the integrals in parts (a) and (b) have different values.)