Understanding the Fourier Series Representation of Harmonic Trigonometric Functions

In this article, we will delve into the concept of Fourier series and explore how it applies to the representation of harmonic trigonometric functions, specifically ( f(t) 2 sin t ) for ( 0

What is a Fourier Series?

A Fourier series is a mathematical tool used to represent a periodic function as a sum of sine and cosine functions. It allows us to decompose a complex periodic function into its constituent frequencies, which can be particularly useful in signal processing, electrical engineering, and various scientific applications.

Fourier Series for a Harmonic Trigonometric Function

Consider the harmonic trigonometric function ( f(x) 2 sin x ) defined over the interval ( 0 [ f(x) f_0 sum_{n1}^{infty} a_n cosleft(frac{2pi n}{T} xright) sum_{n1}^{infty} b_n sinleft(frac{2pi n}{T} xright) ]

where the coefficients ( f_0 ), ( a_n ), and ( b_n ) are given by the following integrals:

[f_0 frac{1}{T} int_{0}^{T} f(x) , dx] [a_n frac{2}{T} int_{0}^{T} f(x) cosleft(frac{2pi n}{T} xright) , dx] [b_n frac{2}{T} int_{0}^{T} f(x) sinleft(frac{2pi n}{T} xright) , dx]

Calculating the Fourier Coefficients

Let's apply the above formulas to the function ( f(x) 2 sin x ) over the interval ( 0 **Calculation of ( f_0 ):** [ f_0 frac{1}{2pi} int_{0}^{2pi} 2 sin x , dx ]

The integral of ( sin x ) over a full period is zero, so:

[ f_0 0 ] **Calculation of ( a_n ):** [ a_n frac{2}{2pi} int_{0}^{2pi} 2 sin x cos(nx) , dx ]

The integral of ( sin x cos(nx) ) over a full period is zero for any integer ( n ), so:

[ a_n 0 ] **Calculation of ( b_n ):** [ b_n frac{2}{2pi} int_{0}^{2pi} 2 sin x sin(nx) , dx ]

Using the orthogonality property of sine functions, we find:

[ b_n begin{cases} frac{2}{pi} int_{0}^{2pi} sin x sin x , dx 2 quad text{if} , n 1 [4px] 0 quad text{if} , n eq 1 end{cases} ]

Final Fourier Series Representation

Substituting the calculated coefficients into the Fourier series formula, we get:

For ( b_n ): [ b_n begin{cases} 2 quad text{if} , n 1 [4px] 0 quad text{if} , n eq 1 end{cases} ] The Fourier series representation of ( f(x) ) is then: [ f(x) 2 sin x ]

Conclusion

We have demonstrated that for the function ( f(x) 2 sin x ) over the interval ( 0

Understanding Fourier series representation is crucial in many engineering and scientific applications, where periodic functions need to be analyzed or approximated. By grasping the underlying principles and calculation methods, one can effectively use Fourier series to solve complex problems.

Frequently Asked Questions (FAQs)

Why is ( f_0 ) zero for ( f(x) 2 sin x )? [ f_0 frac{1}{2pi} int_{0}^{2pi} 2 sin x , dx 0 ] Why are ( a_n ) coefficients zero? [ a_n frac{2}{2pi} int_{0}^{2pi} 2 sin x cos(nx) , dx 0 ] Why is ( b_1 2 ) and ( b_n 0 ) for ( n eq 1 )? [ b_1 frac{2}{pi} int_{0}^{2pi} sin x sin x , dx 2 ]

By answering these questions, we can further solidify our understanding of Fourier series and their applications.