This course introduces students to the fundamental concepts and methods of applied mathematics. The course consists of two parts: complex analysis and ordinary differential equations (initial value problems). The list of topics include:
• Complex Analysis: complex numbers, complex plane, Euler’s formula, regions in the complex plane, functions of complex variable, limits and continuity, stereographic projection, complex derivative, the Cauchy-Riemann equations, analytic functions, complex exponential, trigonometric functions, logarithmic function, branches, branch cuts, brunch points, power function, roots of unity, contour integrals, equivalence theorem, the Cauchy-Goursat theorem, deformation of paths, Cauchy’s integral formula, derivatives of analytic functions, Morera’s theorem, sequences and series, Taylor series, power series, circle of convergence, ratio test, Cauchy-Hadamard formula, continuity, integration, and analyticity of power series, Laurent series, zeros and singularities of analytic functions, residues, Cauchy’s residue theorem, improper integrals, Jordan’s lemma, analytic continuation.
• Ordinary Differential Equations (initial value problems): differential equations, general terminology, 1st order liners ODEs, 2nd order linear ODEs, existence and uniqueness, superposition principle, Wronskian, fundamental set of solutions, Abel’s theorem, reduction of order, variation of parameters, Green’s functions, the Laplace transform, applications to initial value problems, shifting theorems, initial value problems with discontinuous and impulsive forcing, convolution integral, the Mellin inversion formula, linear vs nonlinear ODEs, numerical methods for 1st order nonlinear IVPs, Euler’s method, backward Euler’s method, approximation errors, Heun’s method, Runge-Kutta methods, Adams methods, nst order nonlinear IVPs, power series solutions, Airy’s equations, ordinary and singular points, Fuchs’ theorem, Euler’s equations, the method of Frobenius.

• Ma 1 abc, Ma 2 or equivalents.
• Some familiarity with MATLAB, e.g. ACM 11, is desired.

Each lecture will be accompanied by two practice problems: a somewhat easier, more practical Problem A, and a more difficult, more conceptual Problem B. The main goal of the practice problems is threefold: to help you better understand the material covered in the corresponding lecture, to help you prepare to solve problems in problem sets and exams, and to accommodate the diversity of students’ math backgrounds by providing both easier and more challenging problems. These problems are for self-practice: they will not be graded, and the solutions (posted on Piazza) also illustrate the expected level of rigor for problem sets and exams.

Your final grade will be based on your total score. Your total score is a weighted average of Problem Sets (60%), First-Half Exam (10%), Second-Half Exam (10%), and the Final exam (20%). You can increase your total score by up to 5% if you participate actively in Piazza discussions in the Q&A section. Every answer submitted before TAs or instructor answer, which is later endorsed as a “good answer” by TAs or instructor, gets 1% of the total score. Also, if you are interested in being a TA next year, try to be active on Piazza and help other students by answering their questions. There are no fixed thresholds for grades, but if your total score is 90% (80%, 70%, 60%), you are guaranteed at least “A” (“B”, “C”, “D”).

There will be seven Problem Sets. Problems (and solutions) will be posted on Piazza. For assignment and due dates see “Important Dates” below. Late submissions will not be accepted for any reason, but the Problem Set with the lowest score will be dropped and not counted toward your total score. Submitting wrong files or files in a wrong format is considered as a late submission. Extensions may be granted for academic, personal, or medical reasons. For extensions, please email the Head TA.