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# Compute the modulus and principal argument of given complex numbers

For each of the following complex numbers, compute the modulus and principal argument.

1. .
2. .
3. .
4. .
5. .
6. .
7. .
8. .
9. .
10. .

1. First, the modulus is given by

The principal argument is the unique such that

In this case we have , and so the principal argument satisfies

2. First, the modulus is given by

The principal argument satisfies

3. The modulus is given by

The principal argument satisfies

4. The modulus is given by

The principal argument satisfies

5. The modulus is given by

The principal argument satisfies

6. The modulus is given by

The principal argument satisfies

7. First, we write in the form ,

Therefore, the modulus is

The principal argument satisfies

8. First, we rewrite in the form ,

Therefore, the modulus is

The principal argument satisfies

9. First, we rewrite in the form ,

Therefore, the modulus is

The principal argument satisfies

10. First, we rewrite in the form ,

The modulus is

The principal argument satisfies

# Compute the absolute values of given complex numbers

For each of the following complex numbers, compute the absolute value.

1. .
2. .
3. .
4. .
5. .
6. .

1. Using the formula for the absolute value of a complex number we compute,

2. Using the formula for the absolute value of a complex number we compute,

3. Using the fact that and the formula for the absolute value of a complex number we have,

4. Using and the formula for the absolute value of a complex number we have

5. Using that we have

6. We compute,

# Compute the integral from 0 to x of f(t) for the given functions

Find a formula to compute

for all for the following function .

1. .
2. The function,

3. .
4. the maximum of 1 and .

1. We know from this exercise (Section 5.5, Exercise #13) that

2. If , then over the whole integral, and so

Then, if we have

(Since we have so this equation works. This is the form Apostol wrote these answers as in the back of the book, so I’m getting our answers to match his. I wouldn’t have written them this way otherwise.)

Finally, if we have

Since the formulas for are the same for and are the same we have

for .

3. We consider two cases. If then

If then

4. Since the maximum of 1 and is equal to 1 if and is equal to if we consider three cases (, and ).

For we have

For we have

For we have

# Find an integral formula involving the absolute value function

Prove that

for all .

Proof. We consider two cases:
Case #1: If then for all so,

The final equality follows since for .
Case #2: If . Then for all . So,

since for all

# Find the integral of the absolute value function

Show that

for all .

Proof. We consider three cases.
Case #1: If , then both sides of the equation are 0, so the equation holds.
Case #2: If , then for all so,

The last equality follows since implies .
Case #3: If , then for all so,

The final line follows since since

# Prove some continuity properties of a function

Given a function and a closed interval such that

for all .

1. Prove is continuous at every point .
2. If is integrable on prove

3. If is any point prove

1. Proof. Let be any point in . Then, for any let . Then

Thus, for any , we have whenever . Hence,

Therefore, is continuous at every point

2. Proof. Since is a constant (the value of the function evaluated at the constant ), we have

Therefore, we can compute,

The final inequality follows since, using the monotone property of the integral,

Then, since by our hypothesis on , we have

3. Proof. The proof proceeds similarly to that of part (b),

Then, since , we break the integral into two pieces (to deal with the fact that for and for ):

But now, we know implies , so we have the inequality,

# Evaluate the limit

Evaluate

Since for , and the limit is taken as approaches 0 through negative values we have,

# Evaluate the limit

Evaluate

Since for , and the limit is taken as approaches 0 through positive values, we have,

# Find the integral from -π to x of |1/2 + cos t|

For , compute the following integral:

Since we are taking the integral of an absolute value, first we want to determine on which intervals is positive and on which it is negative. We have,

Now must consider two cases. First, if , then,

In the other case, that , then,

# Compute the integral from 0 to π of |(1/2) + cos t|

Compute the following integral:

First, we note that

So, then we can compute,