Finding a Fourier Series: What to Do and How?

In summary, a Fourier series is a mathematical representation of a periodic function as a sum of sine and cosine functions. It is calculated by finding the coefficients of these functions using the Fourier series formula. The purpose of finding a Fourier series is to simplify complex functions and it is commonly used in science and engineering for various applications. However, it has limitations, such as only being applicable to periodic functions with a finite number of discontinuities and potentially not accurately representing certain types of functions.
  • #1
Luongo
120
0
1. We are supposed to find a Fourier series knowing only f(t)= acos(kt)+bcos(kt)
and some values of Fourier coefficients...

please see #2 on this link http://www.math.ubc.ca/~oyilmaz/courses/m267/hmk3.pdf
2. I am using ck=1/2pi [tex]\intf(t)e-ikt[/tex]dt
3. are we supposed to convert f(t) into expnential form? i got 4 different expos thus 4 integrations, i have no idea which method i should solve it...
 
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  • #2
You only have 5 nonzero coefficients. Write out the 5 term complex series and then change it to a sine-cosine series using eiθ = cos(θ) + i sin(θ) and collecting terms.
 

FAQ: Finding a Fourier Series: What to Do and How?

What is a Fourier series?

A Fourier series is a mathematical representation of a periodic function as a sum of sine and cosine functions. It is used to decompose a complex function into simpler components, making it easier to analyze and manipulate.

How is a Fourier series calculated?

A Fourier series is calculated by finding the coefficients of the sine and cosine functions that make up the series. This is done using the Fourier series formula, which involves integrating the function over one period and solving for the coefficients.

What is the purpose of finding a Fourier series?

The purpose of finding a Fourier series is to simplify a complex periodic function into a series of simpler components. This can be used to analyze and model the behavior of the function, as well as to approximate the function with a finite number of terms.

How is a Fourier series used in science and engineering?

A Fourier series is used in many areas of science and engineering, including signal processing, image and audio compression, and solving differential equations. It is also commonly used in physics and astronomy to analyze periodic phenomena.

Are there any limitations to using a Fourier series?

While a Fourier series can be a powerful tool for analyzing periodic functions, it does have some limitations. For example, it can only be used for functions that are periodic and have a finite number of discontinuities. It also may not accurately represent functions with high-frequency components or those that are not smooth.

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