Differentiating a fourier series

[Dustinsfl]

What are rules for differentiating a Fourier series?

For example, given
$$
f = \frac{4}{\pi}\sum_{n=1}^{\infty}\frac{\sin(2n-1)\theta}{2n-1} = \begin{cases} 1, & 0 < \theta < \pi\\
0, & \theta = 0, \pm\pi\\
-1, & -\pi < \theta < 0
\end{cases}
$$

Can this be differentiating term wise? If so, what conditions must be met?

 

[Dustinsfl]

In order to differentiate a Fourier sine series,

$f(\theta)$ is piecewise smooth on $[0,\pi]$$f(\theta)$ is piecewise continuous on $[0,\pi]$.$f(0) = f(\pi)$

All three are met.

$$
\frac{4}{\pi}\sum_{n = 1}^{\infty}\cos(2n-1)\theta
$$

Is this right?

 

[Sudharaka]

What are rules for differentiating a Fourier series?

For example, given
$$
f = \frac{4}{\pi}\sum_{n=1}^{\infty}\frac{\sin(2n-1)\theta}{2n-1} = \begin{cases} 1, & 0 < \theta < \pi\\
0, & \theta = 0, \pm\pi\\
-1, & -\pi < \theta < 0
\end{cases}
$$

Can this be differentiating term wise? If so, what conditions must be met?

Hi dwsmith, :)

Refer the following links to find the criteria that must be met in order for the derivative/integral of a Fourier series to be equal to the derivative/integral of the associated function.

http://www.aerostudents.com/files/partialDifferentialEquations/fourierSeries.pdf

Pauls Online Notes : Differential Equations - Convergence of Fourier Series

Kind Regards,
Sudharaka.

 

[Dustinsfl]

Hi dwsmith, :)

Refer the following links to find the criteria that must be met in order for the derivative/integral of a Fourier series to be equal to the derivative/integral of the associated function.

http://www.aerostudents.com/files/partialDifferentialEquations/fourierSeries.pdf

Pauls Online Notes : Differential Equations - Convergence of Fourier Series

Kind Regards,
Sudharaka.

I did. The derivative should be 0 but when theta is 0 the series is 4/Pi. What is wrong?

And I am lost on this part too.

Does the Fourier series in part (b) converge for any value of $\theta$ other than $\theta = \pm\frac{\pi}{2}$?
$$
f'(\theta) = \frac{4}{\pi}\sum_{n = 1}^{\infty}\cos(2n - 1)\theta = \frac{2}{\pi}\text{Re}\left(\exp\left[i(2n - 1)\theta\right]\right)
$$

 

[Sudharaka]

I did. The derivative should be 0 but when theta is 0 the series is 4/Pi. What is wrong?

And I am lost on this part too.

Does the Fourier series in part (b) converge for any value of $\theta$ other than $\theta = \pm\frac{\pi}{2}$?
$$
f'(\theta) = \frac{4}{\pi}\sum_{n = 1}^{\infty}\cos(2n - 1)\theta = \frac{2}{\pi}\text{Re}\left(\exp\left[i(2n - 1)\theta\right]\right)
$$

The function \(f\) is not continuous at \(x=0\). Therefore the derivative series converges to \(f'\) only in the intervals, \((-\pi,0)\mbox{ and }(0,\pi)\).

What do you mean by "part (b)" ?

 

[Dustinsfl]

The function \(f\) is not continuous at \(x=0\). Therefore the derivative series is only valid in the intervals, \((-\pi,0)\mbox{ and }(0,\pi)\).

What do you mean by "part (b)" ?

Part b is the derivative series.

 

[Sudharaka]

In order to differentiate a Fourier sine series,

$f(\theta)$ is piecewise smooth on $[0,\pi]$$f(\theta)$ is piecewise continuous on $[0,\pi]$.$f(0) = f(\pi)$

All three are met.

$$
\frac{4}{\pi}\sum_{n = 1}^{\infty}\cos(2n-1)\theta
$$

Is this right?

Sorry for the confusion, but I think I had overlooked something here. Note that the second condition should read, "$f(\theta)$ is continuous on $[0,\pi]$". Refer http://www.aerostudents.com/files/partialDifferentialEquations/fourierSeries.pdf. This is not satisfied by \(f\) and hence it is not differentiable term by term.

Kind Regards,
Sudharaka.

 

[Dustinsfl]

Sorry for the confusion, but I think I had overlooked something here. Note that the second condition should read, "$f(\theta)$ is continuous on $[0,\pi]$". Refer http://www.aerostudents.com/files/partialDifferentialEquations/fourierSeries.pdf. This is not satisfied by \(f\) and hence it is not differentiable term by term.

Kind Regards,
Sudharaka.

Since there is no Fourier series, it doesn't converge for theta = 0?

 

[Sudharaka]

Since there is no Fourier series, it doesn't converge for theta = 0?

The Fourier series of \(f\) is,

\[f(\theta) = \frac{4}{\pi}\sum_{n=1}^{\infty}\frac{\sin(2n-1)\theta}{2n-1} = \begin{cases} 1, & 0 < \theta < \pi\\
0, & \theta = 0, \pm\pi\\
-1, & -\pi < \theta < 0
\end{cases}\]

You cannot differentiate this series term by term to obtain the Fourier series of \(f'\) since the function \(f\) is not right/left continuous at the endpoints of \([-\pi,\pi]\). Does this clarify things for you?

 

[Dustinsfl]

The Fourier series of \(f\) is,

\[f(\theta) = \frac{4}{\pi}\sum_{n=1}^{\infty}\frac{\sin(2n-1)\theta}{2n-1} = \begin{cases} 1, & 0 < \theta < \pi\\
0, & \theta = 0, \pm\pi\\
-1, & -\pi < \theta < 0
\end{cases}\]

You cannot differentiate this series term by term to obtain the Fourier series of \(f'\) since the function \(f\) is not right/left continuous at the endpoints of \([-\pi,\pi]\). Does this clarify things for you?

No. The question ask if the derivative series converges when theta is 0. Since there is no derivative series, the series doesn't converge when theta is zero, correct?

 

[Sudharaka]

No. The question ask if the derivative series converges when theta is 0. Since there is no derivative series, the series doesn't converge when theta is zero, correct?

It is possible that in the question the term "derivative series" means the series,

\[\sum_{n=1}^{\infty}\cos(2n-1)\theta\]

This series diverges at \(\theta=0\).