The Pierce Lab

ACM 95a/ACM 100a

Introductory Methods of Applied Mathematics

Instructor

Niles A. Pierce
165 Broad
Open office hours

Overview

The first term introduces the methods of complex analysis.

Topics

Part 1

Introduction to complex numbers, polar form, Euler's formula, the complex exponential, trigonometric functions, deMoivre's formula, integer powers and roots, the complex logarithm, multiple-valuedness, periodicity, complex exponents, inverse trigonometric functions, one-to-one mappings, Riemann surfaces, the point at infinity, the stereographic projection, branch points and branch cuts, branch point at infinity, regions of the complex plane, limits and continuity, the complex derivative, analyticity, the Cauchy-Riemann equations, complex integration, parameterization, contours, fundamental theorem for contour integration, equivalence between existence of antiderivatives, vanishing of closed contour integrals and independence of path, the Cauchy-Goursat theorem, extensions to self-intersecting contours and multiply-connected domains, deformation of contours, Cauchy integral formula, derivatives of analytic functions, generalized Cauchy integral formula, Morera's theorem, Cauchy's inequality, Liouville's theorem, harmonic functions, harmonic conjugates, potential flow applications.

Part 2

Uniform convergence, Taylor series, uniqueness of analytic functions, power series, Weierstrass M-test, circle of convergence, the ratio test, integration of power series, analyticity of power series, differentiation of power series, uniqueness of power series, arithmetic operations on power series, Laurent series, zeros of analytic functions, isolated singularities, removable singularities, poles, essential singularities, Picard's theorem, non-isolated essential singularities, residues, calculating residues, Cauchy's residue theorem, real trigonometric integrals, improper integrals, the Cauchy principal value, Jordan's lemma, indented contours, integrals involving branch points, winding number, meromorphic functions, the argument principle, analytic continuation, the monodromy theorem, conformal mapping, angle preservation, local scaling, critical points, open mapping property, inverse mappings, solving the Laplace equation by conformal mapping of harmonic functions.

Lecture Handouts

Regions and limits
Cauchy-Goursat
Midterm material
Taylor series
Singularities
Conformal mapping
Final material

Primary Text

E.B. Saff and A.D. Snider. Complex Variables for Mathematics, Science and Engineering, 3rd Ed., Prentice Hall, 2002.

Reserve Texts

  • M.J. Ablowitz and A.S. Fokas. Complex Variables: Introduction and Applications, Cambridge University Press, 1997.
  • J.W. Brown and R.V. Churchill. Complex Variables and Applications, 6th Ed., McGraw Hill, 1996.
  • N. Levinson and R.M. Redheffer. Complex Variables, Holden-Day, 1970.

Problem Sets

  • Available online at 3pm
  • Due at 3pm in the slot of Firestone 303
  • Collaboration is encouraged but prepare your own unique solutions
  • It is a violation of the honor code to use ACM95/100 materials from previous years except for the abridged version of Sean Mach's text book available from the ACM95/100 Underground web site
  • If you need help, ask a TA...
  • If you still need help, ask the instructor
  • Extensions only in exceptional circumstances: see the Head TA
  • Accepted without extension for 50% credit up to one week late
  • Solutions available online on the evening of the due date
  • Please report suspected errors in problems or solutions to the Head TA

Exams

  • Calculators are not permitted
  • Closed-book: with the exception of official lecture handouts and problem set questions, only material written in your own hand or typed by your own hand may be used during exams
  • Extensions only with permission of the Dean
  • The Head TA will offer a review session before each exam

Grading

25% problem sets, 37.5% midterm, 37.5% final

Problem Set & Exam Schedule

Available Due
PS 1 Friday 10/3 Friday 10/10
PS 2 Friday 10/10 Friday 10/17
PS 3 Friday 10/17 Friday 10/24
PS 4 Friday 10/24 Friday 10/31
Midterm Friday 10/31 Tuesday 11/4
PS 5 Tuesday 11/4 Friday 11/14
PS 6 Friday 11/14 Tuesday 11/25
PS 7 Tuesday 11/25 Friday 12/5
Final Tuesday 12/9 Friday 12/12

Teaching Assistants

Head TA: and the ACM95/100 Underground

TA Office hour Location Section Location
9pm ThursAvery Lower Fishbowl2pm Thurs119 Downs
2pm ThursRed Door2pm Tues119 Downs
2pm MonRed Door4pm Thurs119 Downs
5pm TuesRed Door2pm Wed119 Downs
9pm ThursAvery Lower Fishbowl11am Tues11 Downs
9am Fri212 Firestone9am Thurs11 Downs
10am Wed304 Firestone11am Thurs11 Downs
8pm MonRuddock Lounge3pm Mon11 Downs
4pm Mon13 Steele basement1pm Tues11 Downs
6pm Thurs220 SFL4pm Tues119 Downs

Attend any sections of your choosing. Please leave your official section unchanged.