In the past few weeks and Thanksgiving break, I finished Unit II and moved on to Unit III. In this unit, I will primarily focus on the Fourier Series, Laplace Transformation, their connections with Differential Equations, and their applications in reality. In this blog, I will introduce the Fourier Series of periodic functions, including the trigonometry functions, the sin and cos. Continue reading
In the past two weeks, I worked on Second Order Differential Equations with constant coefficients and learned more about the Spring-Damper Model. In this blog, I will provide a brief recap of the basic knowledge, and then provide further analysis of the same model. If you are interested in my last blog, please visit here. Continue reading
In the past two weeks, I have finished Unit 1 and moved on to Unit 2. Starting here, I will be working with Second Order Differential Equations with constant coefficients. Continue reading
As mentioned in my last post, I continued to develop my algebraic skills for solving differential equations during the past two weeks. In this blog, I want to introduce Euler’s Formula, the most impressive piece of math work I have ever seen. In specific, I will dive deep into the mathematical proof of this formula and explain its broad application.
Euler’s Formula consists of a simple line:
In this equation, e represents the irrational constant 2.71828…, or more specifically , and i represents . With an irrational number and an imaginary number involved, Euler’s Formula is not easy to visualize in the regular coordinate system. In order to make this blog more comprehensive to everyone, I will offer an algebraic proof, so that we don’t have to learn the complex Cartesian and Polar coordinate system from the beginning.
To prove Euler’s Formula, we have to prove it satisfies these three fundamental properties:
- (Law of exponential)
- (Law and definition of constant e in calculus)
- Taylor’s series of the left should equal to Taylor’s series on the right as the number of terms approaches infinite.
So let’s start from the first one! Remember we are now assuming Euler’s Formula is correct and testing if it [actually] follows all of the properties.
After proving the Euler’s Formula follows exponential identity, we can move on to proving the identity of e.
The concept of constant e was first brought up by Euler himself, and is sometimes referred to as “Euler’s Number.” The core definition of e is the rate of change always equal to its self. If you want to know more about this mysterious number, please visit here. Due to its ingenious definition, e is used in multiple areas including charging/discharging a capacitor or calculating quantities related to half-life.
Now here is a simple and straightforward proof for Part II.
In the last part, we are proving the polynomials of e^iθ equals the polynomials of cos(x)+i*sin(x) from Taylor’s series. If you remember some relevant knowledge from Calculus II, this would also be straightforward.
Now, we have proved the Euler’s Formula. And I will briefly and broadly cover its applications. If you are interested in knowing more about any of these topics, please tell me and I will discuss them in the future!
Trigonometry, Fourier transformation, Taylor’s series, …. the span of the impact of Euler’s Formula goes on and on. With Euler’s Formula, we can easily model rotations in complex coordinate system or even explain the spiral movements of the starts; with Euler’s Formula, the trigonometry relationships become easier than ever; and most importantly, with Euler’s Formula, we can, for the first time in history, cross the wall between transcendental numbers and algebraic numbers.
Euler’s Formula is a piece of math theorem, but also a piece of art. Like Mona Lisa’s Smile by Leonardo da Vinci, Euler’s Formula contains much more than it seems. When rewritten as e^(iπ)+1=0, the simple line involves plus/minus, multiplication/division, exponential, trigonometry, complex number, the concept of zero, and transcendental numbers. Even within itself, there is a sea of knowledge waiting for me to explore.
To me, Euler’s Formula is just a beginning. On my journey of learning, more fascinating math topics will be closely studied and researched. What I will not forget, however, is the excitement after seeing this beautiful math work. Like a line of a poem that eulogizes our life, Euler’s Formula thoroughly depicts the beauty of math.
Fourier Transforms [Illustration]. (2010, Fall). Retrieved from https://slideplayer.com/slide/10994153/
Arthur Mattuck, Haynes Miller, Jeremy Orloff, and John Lewis. 18.03SC Differential Equations. Fall 2011. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.
Leonhard Euler [Photograph]. (n.d.). Retrieved from https://www.usna.edu/Users/math/meh/euler.html
O’Connor, J. J., & Robertson, E. F. (2001, September). The number e. Retrieved October 8, 2018, from MacTutor History of Mathematics archive website: http://www-history.mcs.st-and.ac.uk/HistTopics/e.html
* The Wikipedia example of using Fourier series to estimate periodic waves Continue reading
I’m not posting any math equations this week!
Surprised that the red block moves in sync with the green dot in a orbit?
You are using the same cup, making the same amount of coffee out of the same coffee machine. Why does your coffee cool down faster than it does in the summer? Of course you’d say that is because winter is much colder, but shouldn’t coffee cool down at some constant rate? like maybe 4°F every minute or something?