# Deriving equations from fourier series representations

• seanbow
In summary, the conversation discusses the coefficients of a Fourier series representation of a function and how it may not accurately portray the actual function. The question posed is if there is a way to derive a more intuitive representation of a function given its Fourier series. The answer is that there is no known way, aside from using computer plots, and intuition is developed over time through experience.
seanbow
Say you have the coefficients $a_k$ of a Fourier series representation of some function $x(t)$. You can easily then give $x(t)$ as
$$x(t) = \sum_{k = -\infty}^{\infty} a_k e^{i k \omega_0 t}$$
But this doesn't do much good in telling you what the actual function looks like. For example, if we have
$$a_k = \frac{ \sin ( k \pi / 2)} {k \pi}$$
we can write $x(t)$ as

$x(t) = \sum_{k \neq 0} \frac{ \sin (k \pi / 2)} {k \pi} e^{i k \omega_0 t}$

but you would have a hard time telling that this is a square wave with a duty cycle of 50% unless you've previously derived the series coefficients for that exact function.

Basically, my question is: is there a way to derive a more intuitive representation of a function given its Fourier series representation?

Last edited by a moderator:
seanbow said:
Basically, my question is: is there a way to derive a more intuitive representation of a function given its Fourier series representation?
Not that I'm aware of, except by using computer power to plot the function. Additionally intuition is a rather personal property. It develops usually over time when more and more functions add to your experience.

## 1. What is the purpose of deriving equations from Fourier series representations?

The purpose of deriving equations from Fourier series representations is to express a function as a sum of sinusoidal functions with different frequencies, amplitudes, and phases. This allows us to represent complex functions in a simpler form, making it easier to analyze and solve problems in mathematics, physics, and engineering.

## 2. How do you derive an equation from a Fourier series representation?

To derive an equation from a Fourier series representation, you need to determine the coefficients of the sinusoidal functions in the series. This can be done by using the Fourier series formula, which involves integrating the function over a period and solving for the coefficients using trigonometric identities.

## 3. What is the difference between a Fourier series and a Fourier transform?

A Fourier series represents a periodic function as a sum of sinusoidal functions, while a Fourier transform represents a non-periodic function as a combination of sinusoidal functions with different frequencies. Additionally, a Fourier series has discrete coefficients, while a Fourier transform has continuous coefficients.

## 4. Can you derive equations from any type of function using Fourier series representations?

No, not all functions can be represented by Fourier series. The function must be periodic, and its period must be finite. Also, the function must satisfy certain mathematical conditions, such as being integrable and having a finite number of discontinuities within a period.

## 5. What are the applications of deriving equations from Fourier series representations?

Deriving equations from Fourier series representations has many applications in various fields such as signal processing, image and audio compression, circuit analysis, and solving differential equations. It is also used in solving partial differential equations and in the study of vibrations and waves.

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