Introduction to the basic tools of the theory of complex variables, such as complex differentiation and the Cauchy-Rieman equations, analytic functions, complex series, complex contour integration, residues, poles and conformational mapping. Prerequisite: Grade of C or higher in MATH 222.
Prerequisite(s) / Corequisite(s):
Grade of C or higher in MATH 222.
Course Rotation for Day Program:
Offered even Spring.
Most current editions of the following:
Complex Variables and Applications
By Brown, James and Churchill, Ruel (McGraw-Hill Publishing) Recommended
Fundamentals of Complex Analysis
By Saff, E.B., Snider, A.D. (Prentice Hall) Recommended
To establish connections between real and complex variable analysis.
To descrbe and explain procedural and conceptual aspects of basic complex analysis notions such as analytical functions, complex differentiation and integration, complex series, residues and poles.
To master formal proofs of fundamental complex analysis results.
To realize the significance of abstract complex analysis notions through applications to real world problems.
Find the modulus and the principle argument of a complex number.
Analyze mapping properties of power, exponential and logarithmic functions.
Compute limits and derivatives of functions of a complex variable.
Demonstrate knowledge of the Cauchy-Riemann equations and their relations to differentiability of functions of a complex variable.
Verify that a give function is analytic or harmonic.
Define power, exponential, logarithmic and trigonometric functions of a complex variable.
Define and evaluate contour integrals.
Formulate and prove the Cauchy-Goursat theorem.
Use the Cauchy integral formula to evaluate integrals along simple closed contours.
State the maximum modulus principle.
Obtain the Taylor and Maclaurin series representations of analytic functions.
Find the Laurent series for functions of a complex variable.
Define the residue and use it to evaluate integrals along simple closed contours.
Use residues to compute improper integrals.
Master the notions of linear and linear fractional transformations, and mappings of the upper half plane onto the open disk.
Define the concept of a conformal mapping and relate it to harmonic functions.
Residues and poles
Applications of residues
Mapping by elementary functions
Recommended maximum class size for this course: 20
NOTE: The intention of this master course syllabus is to provide an outline of the contents of this course, as specified by
the faculty of Columbia College, regardless of who teaches the course, when it is taught, or where it is taught. Faculty members teaching this
course for Columbia College are expected to facilitate learning pursuant to the course objectives and cover the subjects listed in the topical
outline. However, instructors are also encouraged to cover additional topics of interest so long as those topics are relevant to the course's
subject. The master syllabus is, therefore, prescriptive in nature but also allows for a diversity of individual approaches to course material.