Topics for Qualifying Exam in Complex Analysis
I
Complex Plane and Elementary Function.
a)
Complex Numbers
b)
Polar Representation
c)
Stereographic Projection
d)
The Square and Square Root Functions
e)
The Exponential Function
f)
The Logarithm Function
g)
Power Functions and Phase Factors
h)
Trigonometric and Hyperbolic Functions
II
Analytic Functions
a)
Review of Basic Analysis
b)
Analytic Functions
c)
The CauchyRiemann Equations
d)
Inverse Mappings and the Jacobian
e)
Harmonic Functions
f)
Conformal Mappings
g)
Fractional Linear Transformations
III
Line Integrals and Harmonic Functions
a)
Line Integrals and Green’s Theorem
b)
Independence of Path
c)
Harmonic Conjugates
d)
The Mean Value Property
e)
The Maximum Principle
IV
Complex Integration and Analyticity
a)
Complex Line Integrals
b)
Fundamental Theorem of Calculus for Analytic Functions
c)
Cauchy’s Theorem
d)
The Cauchy Integral Formula
e)
Liouville’s Theorem
f)
Morera’s Theorem
g)
Goursat’s Theorem
h)
Complex Notation and Pompeiu’s Formula
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 Spring '11
 NA
 Exponential Function, Complex Numbers, Riemann mapping theorem

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