Solving the Fourier cosine series

In summary: You could certainly do that, but it would be confusing to anyone reading your work. Also, it doesn't matter if you use f(x) or x^2 in the end result, as long as it's clear what you're referring to.
  • #1
chwala
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Homework Statement
Kindly see attached (just need clarification on highlighted part).
Relevant Equations
Fourier cosine series
1672350334369.png


My question is; is showing the highlighted step necessary? given the fact that ##\sin (nπ)=0##? My question is in general i.e when solving such questions do i have to bother with showing the highlighted part...

secondly,

1672350483726.png


Can i have ##f(x)## in place of ##x^2##? Generally, on problems to do with Fourier series- what is usually indicated is ##f(x)##... or it does not matter? yes, i know that ##f(x)=x^2##. My question is in reference to the general widely used notation on such problems.

Cheers.
 
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  • #2
chwala said:
My question is; is showing the highlighted step necessary? given the fact that sin⁡(nπ)=0? My question is in general i.e when solving such questions do i have to bother with showing the highlighted part...
Yes, it's necessary, IMO. Otherwise, the transition from the line above the highlighted line to the one below it would be harder to follow. Several people have made comments in some of your threads that it was difficult to follow your work because of omitted steps.
chwala said:
Can i have f(x) in place of ##x^2##?
Why would you want to? Since you're finding the Fourier series of ##x^2##, why hide this fact by calling it f(x)?
 
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FAQ: Solving the Fourier cosine series

What is the Fourier cosine series?

The Fourier cosine series is a mathematical technique used to represent a function as a sum of cosine functions. It is a special case of the Fourier series, which is used to represent a periodic function as a sum of sine and cosine functions.

Why is the Fourier cosine series important?

The Fourier cosine series is important because it allows us to approximate any function as a sum of simpler cosine functions. This can be useful in many applications, such as signal processing, image compression, and solving differential equations.

How do you solve the Fourier cosine series?

To solve the Fourier cosine series, we use a formula called the Fourier cosine coefficients formula. This formula involves integrating the function we want to represent with respect to cosine functions over a specific interval. The resulting coefficients can then be used to construct the series.

What is the difference between the Fourier cosine series and the Fourier series?

The main difference between the Fourier cosine series and the Fourier series is that the former only uses cosine functions, while the latter uses both sine and cosine functions. This means that the Fourier cosine series can only represent even functions, while the Fourier series can represent both even and odd functions.

Can the Fourier cosine series represent any function?

No, the Fourier cosine series can only represent even functions. If a function is not even, we would need to use the Fourier series, which includes both sine and cosine functions, to represent it accurately.

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