# The Pierce Lab

### Introductory Methods of Applied Mathematics for the Physical Sciences

Niles A. Pierce

Olivia Pomerenk

#### Overview

Term 1 (ACM95a/100a): Complex analysis and ordinary differential equations (initial value problems). Term 2 (ACM95b/100b): Ordinary differential equations (boundary value problems) and partial differential equations.

#### Topics

##### Complex Analysis

Introduction to complex numbers, polar form, Euler's formula, complex exponential, complex logarithm, multi-valuedness, periodicity, branch points, complex exponents, trigonometric functions, inverse trigonometric functions, one-to-one mappings, Riemann surfaces, the point at infinity, stereographic projection, branch cuts, regions of the complex plane, limits and continuity, the complex derivative, the Cauchy-Riemann equations, analyticity, complex integration, parameterization, contour integrals, bounding contour integrals, 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, sequences, series, uniform convergence, Taylor series, uniqueness of analytic functions, power series, Weierstrass M-test, circle of convergence, the ratio test, continuity of power series, 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, isolated essential singularities, Picard's theorem, non-isolated essential singularities, residues, calculating residues, Cauchy's residue theorem, trigonometric integrals, improper integrals, the Cauchy principal value, the art of contour selection, Jordan's lemma, indented contours, integrals involving branch points, analytic continuation, the monodromy theorem.

##### Ordinary Differential Equations (Initial Value Problems)

Introduction to differential equations; linear first order ODEs: integrating factors, integral curves, singular points, existence and uniqueness, the view in the complex plane; homogenous second order linear initial value problems (IVPs): solution properties, the constant coefficient case, reduction of order; nonhomogeneous second order linear IVPs: variation of parameters, the delta function, Heaviside step function, Green's functions, jump conditions; Laplace transform: shifting theorems, convolution, inverting Laplace transforms with the Mellin inversion formula and the Bromwich contour; nonlinear first order ODEs: Picard's existence and uniqueness theorem; numerical methods: explicit Euler, implicit Euler and trapezoidal rule, truncation error, order of accuracy, solution error, explicit Runge-Kutta methods, generalization to first order nonlinear systems, linear stability analysis; linear equations with analytic coefficients: series solutions near ordinary points, solution behavior near singular points, regular singular points, Euler equations, solutions near regular singular points by the method of Frobenius.

#### Optional Text

J.W. Brown and R.V. Churchill. Complex Variables and Applications, 5th (\$20), 6th (\$5), 8th (\$20) or 9th (\$128) editions.

• M.J. Ablowitz and A.S. Fokas. Complex Variables: Introduction and Applications, Cambridge University Press, 1997.
• G. Birkhoff and G.-C. Rota. Ordinary Differential Equations, 3rd ed, Wiley, 1978.
• W.E. Boyce and R.C. DiPrima. Elementary Differential Equations and Boundary Value Problems, 7th ed, Wiley, 2001.
• E. Butkov, Mathematical Physics, Addison-Wesley, 1968.
• E.A. Coddington, An Introduction to Ordinary Differential Equations, Prentice-Hall, 1961.
• J.D. Lambert, Numerical Methods for Ordinary Differential Systems, Wiley, 1991.
• N. Levinson and R.M. Redheffer. Complex Variables, Holden-Day, 1970.
• E.B. Saff and A.D. Snider. Complex Variables for Mathematics, Science and Engineering, 3rd Ed., Prentice Hall, 2002.

#### Honor Code

• It is a violation of the honor code to use ACM95/100 materials from previous years.

#### Lectures

• 11:05am to 12:15pm in 119 Kerckhoff

#### Problem Sets

• Available online at 3pm
• Due at 3pm in the drop box on the north side of Steele House (east of Annenberg)
• Collaboration is encouraged but prepare your own unique solutions
• 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
• Problem Set 6 includes programming (e.g., using Matlab or Python)
• Head TA will offer a programming tutorial.

#### Exams

• The Head TA will offer a review session before each exam
• 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
• No electronic devices are permitted (e.g., no computers, calculators, tablets, phones, smartwatches, etc)

20% problem sets, 40% midterm, 40% final

#### Problem Set & Exam Schedule

Available Due
PS 1 Wednesday 1/8 Friday 1/17
PS 2 Friday 1/17 Monday 1/27
PS 3 Monday 1/27 Wednesday 2/5
PS 4 Wednesday 2/5 Friday 2/14
Midterm 7pm-10pm Tuesday 2/18
PS 5 Wednesday 2/19 Friday 2/28
PS 6 Friday 2/28 Friday 3/13
Final* Wednesday 3/18 Wednesday 3/25