7/27/2019 Complex Analysis Section 2 Notes

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Edited: 12:46pm, August 24, 2013

JOURNAL OF INQUIRY -B ASED L EARNING IN M ATHEMATICS

A First Course in UndergraduateComplex Analysis

Richard Spindler

University of Wisconsin - Eau Claire

7/27/2019 Complex Analysis Section 2 Notes

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7/27/2019 Complex Analysis Section 2 Notes

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Problem 49. Show that z is real if and only if z = z.

Before looking at the next theorem, do the following:

Problem 50. Show that |1 + z+ z4 | 3 when | z| 1. Attempt a brief calcu-lation at showing this. Then conjecture a property that, if true, would makethis problem much easier and provide an explanation as to why it it true.

To nish the last problem, prove the following theorem by rst provingthe lemmas following the theorem.

Theorem 51 (Triangle Inequality) . z1 , z2 C , | z1 + z2 | | z1 | + | z2 | .

Lemma 52.

| z1 + z2 |2 = ( z1 + z2)( z1 + z2)= z1 z1 +( z1 z2 + z1 z2) + z2 z2

Lemma 53. z1 z2 + z1 z2 = 2 Re( z1 z2) 2| z1 || z2 |

Lemma 54. | z1 + z2 |2 (| z1 | + | z2 |)2

Now use the Triangle Inequality to prove problem (50).

You may want to wait to do the next two until you nish problem 50.

Theorem 55. For all z C , | Re( z)| | z| .

Problem 56. Show that | Re(2 + z+ z3)| 4 when | z| 1.

Richard Spindler www.jiblm.org