Complex Numbers and Complex Exponentials
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.
More info at: http://ocw.mit.edu/terms
- Input-Response Models - Input-Response Models
- Notes - Notes and exercises written by Prof. Mattuck
- Lecture Notes - Lecture Notes
- Recitation Problems - Recitation Problems and classwork exercises
- Recitation Solutions - Recitation problem solutions
- Complex Exponential Java Applet - An interactive Java Applet to discover complex exponentials.
- Complex Number Definitions - Complex number definitions and functions.
- Supplementary Notes - Chapter 4 Supplementary Notes written by Prof. Miller
- What are complex numbers?
- What are complex exponentials?
- What is a complex conjugate?
- What is Euler's Formula?
- What is the exponential law?
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Staff Review
In this video, complex numbers and complex exponentials are discussed and used in relation to differential equations. Complex exponential equations are discussed in depth, so a decent background in complex / imaginary numbers will be very beneficial when watching this lesson.
