Complex Analysis

Exercise sheet 7

(Avoiding mistakes in Questions 1 & 2)

Just to familiarize ourselves with the clicker:

Which English indie rock band had a 2007 single entitled “Mathematics”?

1 2 3 4

45%41%

9%5%

1. Cherry Ghost

2. The Killers

3. Red Hot Chili Peppers

4. Snow Patrol

Just to familiarize ourselves with the clicker:

Prior to the 2008 redesign, on which British coin can you find a German sentence that

translated to English reads “I serve”?

1 2 3 4 5 6 7 8 9

11%

7%

0%

4%

29%

11%

14%

7%

18%

1. 1p coin

2. 2p coin

3. 5p coin

4. 10p coin

5. 20p coin

6. 50p coin

7. £1 coin

8. £2 coin

9. None

The reverse side of the 2p coin:

Now, let’s start with Question 2:

We start with

This is

1 2 3

25%

33%

42%

1. true.

2. true, but even more is true.

3. false.

Okay, let’s be more specific:

This is

1 2

65%

35%

1. true

2. false

Remember: The complex numbers are not ordered!

would mean that something like

holds which however doesn’t make sense!

By Cauchy’s formulae for the derivatives we have:

1 2

52%

48%

1. True

2. False

If z0=0 then

1 2 3

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57%

17%

1. True

2. False

3. True, but we can do more

Or:

1 2 3

19%

46%

35%1. True

2. False

3. Not sure what is going on here

To finish: Using Cauchy’s formula for the first derivative, we obtain

Now, move on toHere, we cannot use the Cauchy formulae for the derivatives since z0 is not in the Interior of the contour. But I think

1 2

46%

54%

1. This thought is correct.

2. Nonsense!

That is because we can use Cauchy’s Theorem since there is a

domain D such that

1 2

75%

25%

1. Correct

2. Hold on a minute!

Yes, true, but the main point is:

and since z0 is outside the contour we can apply Cauchy’s Theorem to get 0 for the path integral in question.

Now, on to Question 1:Gauss’ Fundamental Theorem of

Algebra

Every non-constant polynomial has a zero!

Begin of proof?

1 2 3 4

40%

28%24%

8%

1. Correct

2. Mistakes in line 1 and 3

3. Mistakes in line 2 and 3

4. Line 3 is false

There are two problems:

• Again, the complex numbers are not ordered, thus do not write(what does infinity mean here anyway?)

• You cannot bound it that way, see the following real example:

Now have a look at this calculation:

where R>|z|.

Which of the three inequalities are correct?

Which of the three inequalities are correct?

1 2 3 4 5 6 7

12%

15%

4%

19%

8%

35%

8%

1. None

2. Only the first

3. Only the second

4. Only the third

5. The first and second

6. The first and third

7. The second and third

Only the first…

• Second inequality:One cannot “iterate” the inverse triangle inequality (at least not in this way)! See

• For the third inequality, compare

Let’s thank all the people that contributed their errors in this exercise sheet so that nobody

has to do them in the final exam again!

(Please, at least, come up with new ones!)