425 Complex Variables, Spring 1997

Instructor; Alan Howard

The course followed the textbook:
S.D. Fisher, Complex Variables, 2nd edition.

We essentially covered the first 2 chapters plus sections 3.1 and 3.2.
(See table of contents below.)

I originally planned to cover all of the first 3 chapters plus, but I
ran out of time. I think, in retrospect, that I could have done that by
organizing better, going a little faster, and being less repetitive.
But I tried to get the students to participate by calling on them to
solve problems, review previous material, etc., and that took much
time.

There were two tests (one take-home and one in class) during the
semester and a two-part final (first part, take home; second part, in
class).

Chapter 1. 	The Complex Plane	1

	1.1.	The Complex Numbers and the Complex Plane	1
	1.1.1* A Formal View of the Complex Numbers	10
	1.2.	Sone Geometry	12	
	1.3.	Subsets of the Plane	22
	1.4.	Functions and Limits	30
	1.5	The Exponential, Logarithm, and	
		Trigonometric Functions	43
	1.6.	Line Integrals and Green's Theorem	56

Chapter 2.	Basic Properties of Analytic Functions	77

	2.1.	Analytic and Harmonic Functions;	
		The CauchyÐRiemann Equations	77
	2.1.1* Flows, Fields, and Analytic Functions	86
	2.2.	Power Series	93
	2.3.	Cauchy's Theorem and Cauchy's Formula	106
	2.3.1* The CauchyÐGoursat Theorem	119
	2.4.	Consequences of Cauchy's Formula	123
	2.5.	Isolated Singularities	135
	2.6.	The Residue Theorem and Its Application to the
		Evaluation of Definite Integrals	153

Chapter 3.	Analytic Functions as Mappings	171

	3.1.	The Zeros of an Analytic Function	171
	3.1.1* The Stability of Solutions of a System of Linear
		Differential Equations	183
	3.2.	Maximum Modulus and Mean Value	191
	3.3.	Linear Fractional Transformations	196
	3.4.	Conformal Mapping	208
	3.4.1* Conformal Mapping and Flows	219
	3.5.	The Riemann Mapping Theorem and 
		SchwarzÐChristoffel Transformations	224
	

Chapter 4.	Analytic and Harmonic Functions in Applications	245
	
	4.1.	Harmonic Functions	245
	4.2.	Harmonic Functions as Solutions 
		to Physical Problems	254
	4.3.	Integral Representations of Harmonic Functions	284
	4.4.	Boundary-Value Problems	298
	4.5.	Impulse Functions and the
		Green's Function of a Domain	309

Chapter 5.	Transform Methods	318

	5.1	The Fourier Transform: Basisc Properties	318
	5.2	Formulas Relating u  and uÊ^ 	335
	5.3.	The Laplace Transform	346
	5.4.	Applications of the Laplace Transform 
		to Differential Equations	356
	5.5.	The Z-Transform 	365
	5.5.1* The Stabiity of a Discrete Linear System	374