Math 5160 Spring 2020
Tentative Schedule




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Last time around: Spring 2018


 1/22: Introduction, syllabus, characterizing $\mathbb{R}$ and $\mathbb{C}$, and

           Gamelin Sections I.1-I.4 (pages 1-18) & Fisher Sections 1.1-1.2 (pages 1-20).

	
	   
	   Note: We only briefly talked about the Riemann sphere (I.3) and glossed over Riemann surfaces (I.4).



 1/26: [VIDEOs] Complex exponentiation: Gamelin I.5 (pages 19-21) & Fisher 1.5 (pages 43-45),

           Logarithms: Gamelin I.6 (pages 21-23) & Fisher 1.5 (pages 45-48), and

           Power (multi)functions and phase factors: Gamelin I.7 (pages 24-27).

           

           Gamelin I.5 and I.6 (35 mins)

           Gamelin I.7 (30 mins)



 1/29: Trig, hyperbolic, inverse trig functions:

           Gamelin Section I-7 (pages 29-31) & Fisher Section 1.5 (pages 48-53).



 1/30: [VIDEO] Analysis background material (sequences, limits, continuity, open/closed,

          connectedness, compact, and some multivariate limits.

          

          Gamelin II.1 (90 mins)



 2/5: Jacobians, differentibility, complex differentiability, and the Cauchy-Riemann equations.

          Gamelin Sections II.2-II.3 (pages 42-46) & Fisher Section 2.1 (partial) (pages 77-81) .

       

          Note: We didn't actually finish all of II.2 and II.3.



 2/7: [VIDEO] Finishing up Gamelin II.2 and II.3 (define analytic, examples using the limit definition and C-R equations).

          Also, Gamelin II.4 (pages 51-53) on the inverse function theorem and derivatives of inverses.

          

          Gamelin II.2-3 (clean up) & II.4 (55 mins)



 2/12: Discussed some homework issues. Then Gamelin II.5 (pages 54-57) along with Fisher 2.1 (pages 81-84) & 4.1 (pages 245-247)

           covering harmonic functions and harmonic conjugates.


 
 2/14: [Video] Gamelin II.6 (pages 58-61) & II.7 (pages 63-67) and Fisher 3.4 (pages 208-218) & 3.3 (pages 196-203)

           (curves basics, angle between curves, conformal mappings, and linear fractional transformations).

           

           Gamelin II.6 (30 mins)  [You can enlarge the board view - sorry it recorded the computer screen too!]

           Gamelin II.6 (clean up) and II.7 (60 mins)



 2/19: Gamelin III.1 (pages 70-75) and III.2 part 1 (pages 76-82) and Fisher 1.6 (pages 56-73)

           covering "Calculus 3 review". We discussed parameterizing curves, line integrals, curve arithmetic, Green's theorem,

           the fundamental theorem of line integrals, path independance, and exact differential forms.



 2/21: [Video] Gamelin III.2 part 2 (pages 76-82) covering closed forms, simple connectedness, and the connection between
           exact and closed.



           Gamelin III.2 (part 2) (45 mins)



 2/23: [Video] Gamelin III.3 (pages 83-84) and Fisher 4.1 (pages 245-247) = Harmonic Conjugates

                   Gamelin III.4 (pages 85-86) and Fisher 4.1 (pages 248-249) = Mean Value Property

                   Gamelin III.5 (pages 87-88) and Fisher 3.2 (pages 191-194) plus 4.1 (pages 248-249) = Maximum Modulus Principal



           Gamelin III.3 (20 mins)

           Gamelin III.4 (20 mins)

           Gamelin III.5 (30 mins)


 
 3/12: Snow day!

           [Video] Gamelin IV.1 (pages 102-106) and Fisher 1.6 (pages 56-65) = Contour Integrals

                   Gamelin IV.2 (pages 107-109) and Fisher 2.3 (pages 106-110) [parts] = FTC for Analytic Functions

                   Gamelin IV.3 (pages 110-111) and Fisher 2.3 (pages 106-110) [parts] = Cauchy's Theorem

                   Gamelin IV.4 (pages 113-116) and Fisher 2.3 (pages 111-116) = The Cauchy Integral Formula



            Gamelin IV.1 (40 mins)

            Gamelin IV.2 (30 mins)

            Gamelin IV.3 (13 mins)

            Gamelin IV.4 (37 mins)




 3/19: Gamelin IV.5 (pages 117-119) and Fisher 2.4 (pages 130-131 plus 2.4 #18) = Cauchy Estimates & Liouville's Theorem

           Gamelin IV.6 (pages 119-122) and Fisher 2.4 (pages 129-130) = Morera's Theorem & Consequences



 3/20: [Video] Gamelin IV.7 (pages 123-124) and Fisher 2.3.1 (pages 119-123) = Goursat's Theorem

                   Gamelin IV.8 (pages 124-128) [skimmed] = Complex Notation & Synopsis of Pompeiu's Formula

                   Gamelin V.1 (pages 130-132) and Fisher 1.4 (pages 37-41) = Series Basics

                   Gamelin V.2 (pages 133-137) = Sequences and Series of Functions also Pointwise, Uniform, and Normal Convergence



           Gamelin IV.7 (22 mins)

           Gamelin IV.8 (18 mins)

           Gamelin V.1 (30 mins)

           Gamelin V.2 (55 mins)




 3/26: Gamelin V.3 (pages 138-143) and Fisher 2.2 (pages 93-100) = Power Series

           Gamelin V.4 (pages 144-147) and Fisher 2.4 (pages 123-128) = Power Series Representations of Analytic Functions



 3/29: [Video] Gamelin V.5 (pages 149-150) = Analytic at Infinity

                   Gamelin V.6 (pages 151-153) and Fisher 2.2 (pages 101-102) & 2.4 (pages 132-133) = Manipulating Power Series

                   Gamelin V.7 (pages 154-157) and Fisher 2.4 (page 128) & 3.1 (pages 171-172) = Zeros of Analytic Functions

                   Gamelin V.8 (pages 158-162) [sketch] = Analytic Continuation



           Gamelin V.5 (15 mins)

           Gamelin V.6 (22 mins)

           Gamelin V.7 (27 mins)

           Gamelin V.8 (20 mins)




 4/5: [Video] Gamelin VI.1 (pages 165-170) and Fisher 2.5 (pages 135-149) = The Laurent Decomposition

                  Gamelin VI.2 (pages 165-170) and Fisher 2.5 (pages 135-149) = Isolated Singularities



           Motivational Speech (20 mins)

           Gamelin VI.1 (45 mins)

           Gamelin VI.2 (48 mins)




 4/9: Gamelin VI.3 (page 178) and Fisher (page 152 problems 25-27) = Isolated Singularity at Infinity

          Gamelin VI.4 (pages 179-181) and Fisher 2.5 (pages 147-149) = Rational Functions & Partial Fractions



          Gamelin VI.5 (pages 182-185), VI.6 (pages 186-191) and Fisher 4.3 (pages 296-297 problems 16-23) & Chapter 5 (sort of) = Periodic Functions and Fourier Series (sketch/skipped)



          Gamelin VII.1 (pages 195-198) and Fisher 2.5 (pages 137-141), 2.6 (pages 153-154) = Residues



          Class part 1 (2 mins) [crash]
          Class part 2 (30 mins) [crash]
          Class part 3 (45 mins) [crash] to be continued...




 4/10: [Video] Gamelin VII.1 (pages 195-198) and Fisher 2.5 (pages 137-141) & 2.6 (pages 153-154) = Residues

                   Gamelin VII.2 (pages 199-202) and Fisher 2.6 (pages 154-158) Real integrals via residues part 1

                   Gamelin VII.3 (pages 203-204) and Fisher 2.6 (pages 161-162) Real integrals via residues part 2



           Gamelin VII.1 (45 mins)

           Gamelin VII.2 (20 mins)

           Gamelin VII.3 (15 mins)




 4/16: Discussing Homework #7 Related Examples



           Class part 1 (23 mins) [crash]
           Class part 2 (5 mins) [crash]
           Class part 3 (48 mins) [crash]



 4/18: [Video] Gamelin VII.4 (pages 206-207) & Fisher 2.6 (pages 162-166) = Integrals and Branch Points

                   Gamelin VII.5 (pages 209-211) = Fractional Residues

                   Gamelin VII.6 (pages 212-215) = Cauchy Principal Values

                   Gamelin VII.7 (pages 216-218) & Fisher 2.6 (pages 157-161) = Jordan's Lemma & Sine Integral

                   Gamlien VII.8 (pages 219-222) = Exterior Domains & Residues at Infinity [SKIP]



           Gamelin VII.4 (45 mins)

           Gamelin VII.5 (25 mins)

           Gamelin VII.6 (20 mins)

           Gamelin VII.7 (20 mins)




 4/23: Test #2 Clarifications

           Gamelin VIII.1 (pages 224-228) & Fisher 3.1 (pages 175-177)



           Class part 1 (32 mins) [crash]
           Class part 2 (29 mins) [crash]
           Class part 3 (15 mins) 



 4/25: [Video] Gamelin VIII.2 (pages 229-230) & Fisher 3.1 (pages 177-179) = Rouche's Theorem

                   Gamelin VIII.4 (pages 232-234) & Fisher 3.2 (pages 191-192) = Open Mapping & Inverse Function Theorems



           Gamelin VIII.2 (19 mins)

           Gamelin VIII.4 (35 mins)