Complex analysis, more than almost any other undergraduate topic in mathematics, runs the full pure/applied gamut from the most subtle, difficult, and ingenious proofs to the most direct, hands-on, engineering-based applications. This creates challenges for the instructor as much as for the very wide range of students whose various programmes require a secure grasp of complex analysis. Its techniques are indispensable to many, but skill in the use of a mathematical tool is hazardous and fallible without a sound understanding of why and when that tool is the right one to pick up. This kind of understanding develops only by combining careful exploration of ideas, analysis of proofs, and practice across a range of exercises.
