Subject: Study list for Exam Beta
Hi All
I will ask you to prove Cauchy-Goursat, or some part thereof, but
only when the loop is a rectangle.
The exam will mostly ask you for proofs of theorems that we have
done recently. Note: Some little-steps in a proof I have, below,
dignified with the name of Lemma.
Please know how to prove these:
Triangle Inequality thm.
Cauchy-Goursat Thm.
Shrinking-Circle Thm (see below).
CIF Thm (for a general K).
Cauchy's Inequality Thm (This uses M_R(w)).
Liouville's thm.
FTA (Fund. thm of Algebra).
Polynomial Blow-up Lemma (see below).
Modulus-constancy Lemma.
MMP thm (Maximum modulus Principle).
================================================================
SHRINKING-CIRCLE THM: Suppose f is continuous (no analyticity
required) in a neighborhood of a point w. Let I_r denote the contour
integral, around the circle C_r(w), of f(z)/[z-w].
Then the following limit exists and equality holds.
Limit_{r -> 0} I_r
equals
[2*Pi*i] times f(w).
----------------
POLYNOMIAL BLOW-UP LEMMA: Suppose p() is a non-constant polynomial.
Then p(z) goes to infinity as z goes to infinity.
----------------
MODULUS-CONSTANCY LEMMA: Suppose f analytic on a ball, and |f| is
constant on the ball. Then f itself is constant on the ball.
----------------
Also know how to compute specific contour integrals using CIF,
justifying the application of the theorem.
Bring colored pencils for drawing good pictures. You may want to
bring graph paper, and/or blank paper, for pictures.
Know how to do the problems on P.136. Also, know some EXAMPLES of
interesting analytic functions.
Come prepared to STATE and SOLVE an interesting problem (Complex
Analysis) that you have created or read in a book.
The homework problems on our webpage, and the E problems, are also
good references.
Best Wishes,
Prof. Jonathan
http://www.math.ufl.edu/~squash/course.complexanalysis.html
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