Syllabus for the Midterm Exam
for Prof. W. Kahan's Math. 185 section #2
289 Cory Hall, 1:15 pm. - 2, 30 Oct. 2006
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The exam's subject matter includes everything in
chapters I to VII incl. of the course text
"Notes on Complex Function Theory"
by Donald Sarason
(1994, Henry Helson, Berkeley)
plus the following notes posted on the class web
page :
30Aug06S.pdf Cnfrml.pdf Conics.pdf DSpr2p43.pdf
Derivative.pdf Errata.pdf Ex19.pdf Mobius.pdf
except for text printed in small (10-pt.) type.
Here is a list of topics covered in the course so far:
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Cauchy-Riemann equations
Derivatives as limits or confluent divided differences
Inverse functions and implicit differentiation
Conformal maps, Critical points
"Conformal" <=> "Holomorphic" => "Harmonic"
Harmonic functions and their conjugates
Recovery of harmonic conjugates by integration of derivatives
Recovery of harmonic conjugates without integration
Maximum of harmonic functions occurs on a boundary
(Ill-)determination of a harmonic function by its values
on an uncountable set of points.
Branch-points and slitted domains of Principal Values
Elementary transcendental functions and their inverses
Conformal maps of circles and lines to conic sections
Bilinear-Rational / Linear Fractional / Mšbius Functions
Clircles to clircles
Isomorphic to Quotient Group of 2-by-2 matrices ...
Characterization by its fixed-points
Preservation of Cross Ratios
Stereographic Projection and the Riemann Sphere
Chordal distance
Inversion/Reflection in a circle or line
Series
Absolute Convergence, Uniform Convergence
Power series, radius of convergence
Algebra (sums, products, quotients, ...) of series
Term-by-term derivatives and integrals
Integrals of complex functions
Length of a curve
Integrals of Complex Analytic Functions
Independence of path of integration
Cauchy integrals are holomorphic
Cauchy's Integral Formula for an Analytic Function's value
inside a closed curve in terms of its values on the curve
Liouville's Theorem about bounded entire functions
Gauss's Theorem that polynomials have zeros
Analytic functions' zeros are isolated
Analytic functions are (ill-)determined by their values
on an uncountable point-set.
Uniform convergence of analytic functions to another
Maximum modulus principle, D'Alembert's principle
Schwarz's Lemma bounds the magnitude of an analytic function
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The midterm exam will be a closed-book exam to
which you must bring no notes, texts, papers,
calculating nor communicating instruments like
cell-phones. Blank paper will be supplied for
both scratch use and for submission of material
to be graded, so bring no paper of your own.
The washrooms on Cory Hall's second floor are
closed for renovation, so do whatever need be
done to preclude visits to a washroom or water
fountain during the exam's 45 min.
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