Complex variables throughout the Curriculum

We offer many specific detailed examples, several of which are new, that instructors can use (in lecture or as student projects) to revitalize the role of complex variables throughout the curriculum. We conclude with three primary recommendations: revise the syllabus of Calculus II to allow early introductions of complex numbers and linear algebra, include complex variables and some infinite-dimensional linear maps in linear algebra courses, and spice up complex variable courses by better connecting them with the mathematics used in engineering in physics.