- Announcements
- Thanks for signing up!
- Introductions (4x6 cards)
- Name
- Where are you from?
- Working? If so, where, and how many hours?
- Year (Sophomore, etc.)
- Major
- Why are you taking complex analysis?
- Something special about yourself.
- The course website will be useful:
http://www.nku.edu/~longa/classes/mat430
- Syllabus (logistics, etc.)
- The plan of attack: we'll try to get through the first seven
chapters, although I may dip into eight and nine at various points....
Assignments will be monitored on the website. Keep checking on those....
- You should do the reading in advance of our discussion of
each section in class.
- For next time: sign your classroom
agreement.
- Sorry about the mix-up. Don't know how that happened. I found out about it when I was told that I was getting a desk copy of a real analysis text a week and a half ago....
- This is an old classic (I used an early version as a student).
- For your first assignment, use my photocopy of the 5th edition:
- Read Sections 1 and 2 (from handout);
- begin homework problems: pp 5--, #1, 4, 7, 9, 10, 15 (due Wednesday, 1/20)
- What do you already know of complex numbers and analysis?
- Complex numbers first appeared as solutions to polynomial
equations. It took quite awhile for negative solutions to be
accepted, so it should come as no surprise that complex
solutions took even longer!
- Diophantus thought the equation
"Absurd", as it led to the "impossible" solution
- Quadratics got the ball rolling, of course, in terms of complex numbers.
-
"absurd", of course
- Rewrite quadratics so that they're not so absurd....
-
- A little history: the "scoundrel" mathematician
Gerolamo Cardano
(aka Cardan) -- 1501-1576
- Ars_Magna --
"The Great Art"
- on algebraic equations. "Cardan was no mere plagiarist, but one who combined a measure of honest toil with his piracy." (p. 323, The History of Mathematics, Burton)
- Cardan derived the "cubic formula".
- He also considered quadratics of course, often derived as, for example, constraints: e.g., find two numbers whose sum is 10, and whose product is 40.
- In the process of using "his" formula, Cardan encountered
complex roots ("ghosts of real numbers", as Napier called
them). "Putting aside the mental tortures involved, multiply
by
....": sum 10, product 40.
- "So progresses arithmetic subtlety the end of which, as is said, is as refined as it is useless." (p. 323, The History of Mathematics, Burton)
- Rafael Bombelli (1526-1572)
- First to write the rules of negative numbers
- I think that you'll prefer the description in the first two sections of our text, which I'll hand out now....
- First to write complex numbers as sums of real and imaginary parts. "From this time on, imaginary numbers lost some of their mystical character...."
- Ars_Magna --
"The Great Art"
- Complex numbers first appeared as solutions to polynomial
equations. It took quite awhile for negative solutions to be
accepted, so it should come as no surprise that complex
solutions took even longer!
- Did you know that....
- complex analysis is essential for engineers?
- it is at the heart of applied mathematics?
- the fundamental theorem of algebra is most easily proved using complex analysis?
- complex analysis can be used to great effect in computer graphics?