Fourier series without integration

In summary, to find the Fourier series for $h(\theta)$ without doing any integration, you would need to know the Fourier series for the functions $f(\theta) = \theta$ and $g(\theta)$ and use the fact that $h(\theta) = \frac12f(\theta) + \frac\pi2g(\theta)$. This would result in the series $\sum\limits_{n = 1}^{\infty}(-1)^{n + 1}\frac{\sin n\theta}{n} + \frac{1}{2}\sum\limits_{n = 1}^{\infty}\frac{\sin(2n - 1)\theta}{2n
  • #1
Dustinsfl
2,281
5
Let
$$
h(\theta) = \begin{cases}
\frac{1}{2}(\theta + \pi), & 0 < \theta < \pi\\
0, & \theta = 0, \pm\pi\\
\frac{1}{2}(\theta - \pi), & -\pi < \theta < 0
\end{cases}
$$
How can I find the Fourier series without doing any integration?
 
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  • #2
dwsmith said:
Let
$$
h(\theta) = \begin{cases}
\frac{1}{2}(\theta + \pi), & 0 < \theta < \pi\\
0, & \theta = 0, \pm\pi\\
\frac{1}{2}(\theta - \pi), & -\pi < \theta < 0
\end{cases}
$$
How can I find the Fourier series without doing any integration?
I think the only way to do this would be if you already happen to know the Fourier series for the functions $$ f(\theta) = \theta, \qquad g(\theta) = \begin{cases} 1 & (0 < \theta < \pi) \\ 0 & (\theta = 0, \pm\pi) \\ -1 & (-\pi < \theta < 0) \end{cases}.$$ Then you can use the fact that $h(\theta) = \frac12f(\theta) + \frac\pi2g(\theta)$ to write down the answer.
 
  • #3
Opalg said:
I think the only way to do this would be if you already happen to know the Fourier series for the functions $$ f(\theta) = \theta, \qquad g(\theta) = \begin{cases} 1 & (0 < \theta < \pi) \\ 0 & (\theta = 0, \pm\pi) \\ -1 & (-\pi < \theta < 0) \end{cases}.$$ Then you can use the fact that $h(\theta) = \frac12f(\theta) + \frac\pi2g(\theta)$ to write down the answer.

Then
$$
h(\theta) = \sum\limits_{n = 1}^{\infty}(-1)^{n + 1}\frac{\sin n\theta}{n} + \frac{1}{2}\sum\limits_{n = 1}^{\infty}\frac{\sin(2n - 1)\theta}{2n - 1}.
$$
 
Last edited:

FAQ: Fourier series without integration

What is a Fourier series without integration?

A Fourier series without integration is a mathematical representation of a periodic function using a combination of sine and cosine functions. It is used to approximate a function by decomposing it into simpler trigonometric functions.

What is the formula for a Fourier series without integration?

The formula for a Fourier series without integration is given by:
f(x) = a0 + ∑n=1 (ancos(nx) + bnsin(nx)),
where a0, an, and bn are the coefficients of the series.

What is the purpose of using a Fourier series without integration?

The purpose of using a Fourier series without integration is to represent a function in terms of simpler trigonometric functions, making it easier to analyze and manipulate mathematically. It can also be used for data compression and signal processing.

What are the limitations of a Fourier series without integration?

One limitation of a Fourier series without integration is that it can only be used for representing periodic functions. It also may not accurately represent functions with sharp corners or discontinuities. Additionally, as the number of terms in the series increases, the computation becomes more complex and may require a large number of terms to accurately approximate the function.

What are some applications of a Fourier series without integration?

A Fourier series without integration has various applications in mathematics, physics, engineering, and other fields. It is commonly used in signal processing, data compression, and solving partial differential equations. It is also used in sound and image analysis, circuit design, and in studying the behavior of physical systems.

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