In this lecture, we explore how complex numbers and exponentials are used in differential equations. The division of complex numbers is done by making use of the complex conjugate, and we learn how to multiply this by another number to make it real. The main focus of the lecture is the polar representation of complex numbers, which is written as r(cos(theta) + i sin(theta)). Euler's formula is introduced, where e^(i*theta) is equal to cos(theta) + i sin(theta). We then explore how this formula satisfies the exponential law and how e^(i*theta) differentiates to i*e^(i*theta).
Complex Numbers and Complex Exponentials -- Lecture 6. Learn how complex numbers and exponentials are used in ODEs.
Arthur Mattuck, 18.03 Differential Equations, Spring 2006. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed November 26, 2008). License: Creative Commons BY-NC-SA.
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