Complex Analysis

1
Complex Analysis

• Exercise sheet 7
• (Avoiding mistakes in Questions 1 2)

2
Just to familiarize ourselves with the
clickerWhich English indie rock band had a
2007 single entitled Mathematics?

• Cherry Ghost
• The Killers
• Red Hot Chili Peppers
• Snow Patrol

3
Just to familiarize ourselves with the
clickerPrior to the 2008 redesign, on which
British coin can you find a German sentence that
translated to English reads I serve?

• 1p coin
• 2p coin
• 5p coin
• 10p coin
• 20p coin
• 50p coin
• 1 coin
• 2 coin
• None

4
The reverse side of the 2p coin
5
Now, lets start with Question 2

6
We start with
This is

• true.
• true, but even more is true.
• false.

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Okay, lets be more specific
This is

• true
• false

8
Remember The complex numbers are not ordered!
would mean that something like
holds which however doesnt make sense!
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By Cauchys formulae for the derivatives we have

• True
• False

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If z00 then

• True
• False
• True, but we can do more

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Or

• True
• False
• Not sure what is going on here

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To finish Using Cauchys formula for the first
derivative, we obtain
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Now, move on to
Here, we cannot use the Cauchy formulae for the
derivatives since z0 is not in the Interior of
the contour. But I think

• This thought is correct.
• Nonsense!

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That is because we can use Cauchys Theorem since
there is a domain D such that

• Correct
• Hold on a minute!

15
Yes, true, but the main point is

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

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Now, on to Question 1Gauss Fundamental Theorem
of Algebra

• Every non-constant polynomial has a zero!

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Begin of proof?

• Correct
• Mistakes in line 1 and 3
• Mistakes in line 2 and 3
• Line 3 is false

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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

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Now have a look at this calculation
where Rgtz.
Which of the three inequalities are correct?
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Which of the three inequalities are correct?

• None
• Only the first
• Only the second
• Only the third
• The first and second
• The first and third
• The second and third

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Only the first

• Second inequalityOne cannot iterate the
  inverse triangle inequality (at least not in this
  way)! See
• For the third inequality, compare

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Lets 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!)