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.
To understand the Fourier Series of periodic functions, we first need to clarify the definition of the word “period.” Conventionally, a function is periodic when it follows a certain pattern, and here is the mathematical definition:
- A function f(t) is periodic when f(t)=f(t+P), where P represents the period bigger than 0 and t is the independent variable.
Furthermore, when f(t) is a periodic function with a period P, then it also has the periods of 2P, 3P, …, nP, where n is an integer. In this system, we have the base angular frequency .
Then, we have the core statement of the Fourier Series for today: any periodic function with can be written as linear combinations of sin(m*x) and cos(n*x). Personally, this concept is very similar to Taylor Series or Maclaurin Series, which states that any functions can be written as a power function. If you are interested in this or want to review it in detail, please visit here.
Now, here are the formulas for the Fourier Series. This how we transform the periodic function into combinations of sin and cos.
In this blog, I am not proving these formulas as they are unnecessarily complicated for an introduction; instead, I will discuss an example to demonstrate why I believe the Fourier Series is similar to Taylor Series.
Imagine a function defined by
We can easily graph it as
Now if we simply plug in the formulas of the Fourier Series, we will find
In this example, I am using x and k instead of t and n just to distinguish the notations.
If we start to test our new function from k=1, we will have
which looks like:
But if now we consider more terms and have k=99, we will have the new f(x) looks like:
As a result, we know that the approximation of the Fourier Series becomes closer and closer to real values as we involve more terms. This is exactly the same as the Taylor Series!
Here is a moving gif that illustrates this process with more details. Note that the function f(x) is different from the example above, so this is only a demonstration.
In this blog, we went through the core concepts and definitions of the Fourier Series and looked at an example. The Fourier Series is, in fact, closely related to many theorems in Physics and Mathematics. In the following weeks, I will continue my study on the Fourier Series and its implications. In the longer future, I will keep on building my knowledge on the Fourier Series as thoroughly comprehending this concept is crucial to my future study on math.
Dawkins, P. (Ed.). (2018, May 31). Taylor Series. Retrieved November 27, 2018, from Paul’s Online Notes website: http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx
Mattuck, A., Miller, H., Orloff, J., & Lewis, J. (2011, Fall). 18.03SC Differential Equations. Retrieved November 5, 2018, from MIT Open Course Ware website: https://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/index.htm
18.03SCF11 text: Examples [Illustration]. (2011, Fall). Retrieved from https://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-iii-fourier-series-and-laplace-transform/fourier-series-basics/MIT18_03SCF11_s21_5text.pdf
Maple worksheets on Fourier series [Illustration]. (n.d.). Retrieved from http://www.peterstone.name/Maplepgs/fourier.html