Applied Complex Analysis

Spring 2022

Time and place:

Lecture: TuTh 11:30AM-12:50PM Harriman 116

Introduction: This is an advanced mathematically rigorous course with complete proofs.  Topics covered may include:

1. The field of complex numbers  Complex conjugate, absolute value and the triangle inequality.  The distance between complex numbers.  The polar and exponential forms. Roots and powers of complex numbers.  Arguments of products and quotients.  Regions in the complex plane.

2.  Analytic Functions  Functions and Mappings. Limits Limits involving the point at infinity. Continuity and derivatives. The Cauchy-Riemann equations and differentiability of a complex function. Polar Coordinates. harmonic functions.

3. Examples of analytic functions The exponential function, logarithm, trig functions, hyperbolic trig functions

4. Integrals Contour integrals, the Cauchy-Goursat theorem and its proof, simply and multiply connected domains, Liouville's theorem and the fundamental theorem of algebra

5. Series Taylor series and Taylor's theorem, power series and its domain of convergence, Laurent series

6. Residues and poles Cauchy residue theorem, types of isolated singular points, zeros and poles

7. Additional topics as time permits

  Text Book: Complex Variables and Applications by J. W. Brown and R. V. Churchill, 9th edition  (c) 2014

Instructor: Prof. David Ebin
Math Tower 5-107
tel. 632-8283
E-mail: ebin@math.sunysb.edu
Office Hours: Tu,Th  10:00-11:30AM,  or by appointment

 Grader: Yulun Xu  E-mail: yulun.xu@stonybrook.edu

Office Hours: 

M 2:00pm-3:00pm   
Math Learning Center Hours:
  (in Math Tower S-235 or online)
       ∙ M 3:00pm-4:00pm   
       ∙ M 4:00pm-5:00pm    

Or by appointment

Homework: Homework will be assigned every week.  Doing the homework is a fundamental part of the course work.  Problems should be handed in in class on their due date.

1st assignment:  page 5, problems 5 and 6; page 8, problem 7; page 13, problems 3, 7 and 9; page 16, problems 10, 13 and 14. due   February 3.

2nd assignment: page 23, problems 6, 9 and 10; page 30, problem 7; page 34, problems 5 and 8. due February 10

3rd assignment: page 43, problem 1cd: page 54, problems 7, 9 and 11; page 61, problems 2cd, 4, 7 and 8 due February 17

4th assignment: page 70, problems 1c, 6, 7, 8ab; page 76, problems 1cd, 2b, 5, 6; page 79, problems 1,2  due February 24

5th assignment: page 84, problems 1 and 4; page 89, problems 1, 3, 5 and 6  due March 3

Midterm exam: March 10 in class

For midterm know:  Basic algebraic properties of complex numbers,  argument and modulus (absolute value), exponential form; i. e. z = reroots of complex numbers. If z not zero, then z has n nth roots.  Neighborhoods, open and closed sets, domains and regions, boundary points of a domain or region, connected sets.  Analytic functions, harmonic functions.  The real and imaginary parts of analytic functions are harmonic. Limits and continuity.  The Cauchy Riemann equations in both rectangular and polar coordinates. Proof that the real and imaginary parts of an analytic function satisfy the Cauchy-Riemann equations.  Proof and the sum difference and product of analytic functions are analytic, and the quotient is analytic at points where the denominator does not vanish. Know that   ez is many to one and log z is one to many.  Branches of log z.  The functions f(z) = zand g(z) = az and their derivatives.  sin z, cos z, sinh z and cosh z for z complex and their serivatives. Proof and sin z = 0 implies z is real.  Integrals of complex valued functions of a real variable. Contour integrals.  Bounds for moduli of contour integrals.  Proof that if f is continuous on a domain D and the integral of f on any closed contour in D is zero, then f has an antiderivative in D. Proof of the converse.  Use Green's theorem to prove the Cauchy-Goursat theorem for f(z), in the case that f(z) is analytic and f'(z) is continuous.  Compute the contour integral of 1/z around a circle C with 0 inside C and with 0 outside C. 

6th assignment: page 95, problems 7, 10; page 103, problems 6, 8; page 107, problems 3, 5, 8; page 111, problems 7, 8  due March 24

7th assignment:  page 124, problem 2; page 132, problems 9, 10, 13; page 138, problems 2, 6, 8; page 147, problem 1  due March 31

8th assignment: page 159, problems 3, 6, 7; page 170, problems 3, 4, 10; page 177, problems1, 7; page 185, problems 4, 9 due April 7

9th assignment:  page 196, problems 6, 11; page 206, problems 6, 7, 10; page 218, problems 3, 5, 6; page 224, problems 1, 3, 8, 9 due April 14

10th assignment: page 237, problems 2, 6, 7; page 242, problems 1a,c,e; page 246, problems 2; page 253, problems 1, 3, 11, 12 due April 21

11th assignment: page 264, problems 1, 2, 3, 6, 7, 9  due April 28

12th and last assignment: page 293, problem 9; page 305, problems 2, 11; page 311, problems 2, 6, 9, 11 due May 5

Final Exam:Tuesday, May17 11:15am-1:45pm

 For final know :  Everything that you had to know for the midterm.  Computation of the length of an arc.  Simply and multiply connected domains, Cauchy integral formula for functions and for derivatives, Liouville's theorem and the fundamental theorem of algebra, Maximum modulus principle and its proof, Convergence of series, formulas for power series, Taylor series and remainder, Proof that the Taylor series of an analytic functions converges, Laurant series, Computation of coefficients for Laurant series, Radius of convergence of power series,  Continuity and differentiability of power series, Proof that the uniform limit of a sequence of continuous functions is continuous,  Products of power series, Three kinds of isolated singular points of an analytic function, Residues including residues at infinity,  Evaluation of the residue at a pole of order m, Orders of zeros of analytic functions, behavior of an analytic function near an essential singular point, Improper integrals and their computations using residues,  Principle values, Argument principle and winding number, Rouche's theorem, The group of fractional linear transformations and how it acts on the extended complex plane taking lines and circles into lines and circles,  Formula for a fractional linear transformation that takes the upper half plane into the unit circle with center zero, Proof that  analytic functions are conformal transformations if their derivatives are not zero,  Harmonic functions and harmonic conjugates 

N. B. Use of calculators is not permitted in any of the examinations.

Grading Policy: The overall numerical grade will be computed by the formula   20% Homework + 30% Midterm Exam+ 50% Final Exam

Disabilities:   If you have a physical, psychological, medical, or learning disability that may impact your course work, please contact Disability Support > Services or call (631) 632-6748. They will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential. Students who require assistance during emergency evacuation are encouraged to discuss their needs with their professors and Disability Support Services. For procedures and information go to the Evacuation Guide for People with Physical Disabilities.

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