COMPLEX ANALYSIS

Prerequisites & course information

Requisites: MATH 321 or 521 or graduate/professional standing or member of the Pre-Masters Mathematics (Visiting International) Program Course Designation: Core GenEd - Natural Science & Wellness Breadth - Natural Science Level - Advanced L&S Credit - L&S Liberal Arts and Science Grad 50% - Counts toward 50% graduate coursework requirement Repeatable for Credit: No Last Taught: Fall 2025 Learning Outcomes: 1. Recall and state the formal definitions of the mathematical objects and their properties used in complex analysis (e.g., principle branches, conjugates, holomorphic and analytic functions, etc.). Audience: Both Grad & Undergrad 2. Use such definitions to argue that a mathematical object does or does not have the condition of being a particular type or having a particular property (e.g., that a function is complex differentiable, that an infinite product converges, classification of singularities, etc.). Audience: Both Grad & Undergrad 3. Recall and state the standard theorems of complex analysis (e.g., Cauchy's Integral formula, the maximum modulus principle, the Riemann mapping Theorem), and recall the arguments for these theorems and the underlying logic of their proofs. Audience: Both Grad & Undergrad 4. Use such theorems in the context of longer arguments by examining their premises (e.g., computation of real integrals using path integration techniques, computation of residues of meromorphic functions, that a given function has an analytic continuation, etc.). Audience: Both Grad & Undergrad 5. Prove or disprove statements related to the above definitions, properties, and theorems using techniques of mathematical argument (direct methods, indirect methods, constructing examples and counterexamples, induction, etc.). Audience: Both Grad & Undergrad 6. Convey arguments in oral and written forms using English and appropriate mathematical terminology, notation and grammar. Audience: Both Grad & Undergrad 7. Identify applications of course content in current areas of research. Audience: Graduate Learning Outcomes: 1. Recall and state the formal definitions of the mathematical objects and their properties used in complex analysis (e.g., principle branches, conjugates, holomorphic and analytic functions, etc.). Audience: Both Grad & Undergrad 2. Use such definitions to argue that a mathematical object does or does not have the condition of being a particular type or having a particular property (e.g., that a function is complex differentiable, that an infinite product converges, classification of singularities, etc.). Audience: Both Grad & Undergrad 3. Recall and state the standard theorems of complex analysis (e.g., Cauchy's Integral formula, the maximum modulus principle, the Riemann mapping Theorem), and recall the arguments for these theorems and the underlying logic of their proofs. Audience: Both Grad & Undergrad 4. Use such theorems in the context of longer arguments by examining their premises (e.g., computation of real integrals using path integration techniques, computation of residues of meromorphic functions, that a given function has an analytic continuation, etc.). Audience: Both Grad & Undergrad 5. Prove or disprove statements related to the above definitions, properties, and theorems using techniques of mathematical argument (direct methods, indirect methods, constructing examples and counterexamples, induction, etc.). Audience: Both Grad & Undergrad 6. Convey arguments in oral and written forms using English and appropriate mathematical terminology, notation and grammar. Audience: Both Grad & Undergrad 7. Identify applications of course content in current areas of research. Audience: Graduate