## COMPLEX CALCULUS

 Course Name COMPLEX CALCULUS Course ID MCS21QCM Department Mathematics Subject Calculus Can you take this course more than once? No Periods per Day 1.0 Special Permission No Eligibility One of the following is true:All of the following are true:Student has passed AP CALCULUS AB 1 OF 2 (MCS21X)One of the following is true:Student has passed AP CALCULUS BC 2 OF 2 (MCS44X)Student is taking AP CALCULUS AB 2 OF 2 (MCS22X)All of the following are true:Student has passed AP CALCULUS BC 1 OF 2 (MCS43X)One of the following is true:Student has passed AP CALCULUS BC 2 OF 2 (MCS44X)Student is taking AP CALCULUS BC 2 OF 2 (MCS44X)All of the following are true:Student is taking DIFFERENTIAL EQUATIONS (MCS66C) Fulfills the following graduation requirements Math Also in the following groups Syllabus No Syllabus Found

#### Description

Complex Calculus is a year-long elective course for advanced mathematics students, taught at the college level.

The aim of complex calculus is to investigate the ways in which ordinary calculus changes (or remains essentially the same) when we replace real-valued functions of a real variable with complex-valued functions of a complex variable. It turns out that there are many remarkable and unexpected differences. For example, in real calculus, a function can be once differentiable while failing to be twice differentiable. In complex calculus, a once differentiable function is automatically infinitely many times differentiable.

As part of the study of the complex plane, a good deal of topology will be covered; this is the study of continuous deformations of shape. Some highlights will include the Cauchy Integral Formula, the winding number and its applications to topology (such as the famous Brouwer Fixed Point Theorem), a rigorous proof of the Fundamental Theorem of Algebra, the calculus of residues, the Casorati-Weierstrass Theorem, and Rouché's Theorem (also known as the ""Dog-Walking Theorem""). ",Full year course