# Thread: Exponential Fourier Series - Trouble solving

1. ## Exponential Fourier Series - Trouble solving

Hello,

I have troubling simplifying the following problem. It is the second integral solving for c(k) of f involving [i] that is confusing.

I also was wondering if there is simpler way to solve for the exponetial fourier series for the given problem, than the way I done it. I would greatly appriciate guidance.

Thank you

2. Part of your problem is that for the $\displaystyle c_{k}$ integral, you can't use half the interval and multiply by two like you could for the $\displaystyle c_{0}$ case. You can only do that, in general, when your complete integrand is even. But the presence of that particular exponential function in the integrand precludes that. So you have to keep the full $\displaystyle [-\pi,\pi]$ interval for the limits in the $\displaystyle c_{k}$ integration, I'm afraid.

I don't know of any way to integrate $\displaystyle x^{2}e^{-ikx}$ other than by parts twice. I'm not sure I would recommend breaking the integrals up into the trig functions, because then you're going to have to do integration by parts twice on two integrals. There's no need, when working with the exponential Fourier series, to look at sin and cos individually. It's an easier integration with just the exponential in there.

So I would carry these changes through, and then we'll see what happens at the end. Sound good?

3. Originally Posted by Ackbeet
Part of your problem is that for the $\displaystyle c_{k}$ integral, you can't use half the interval and multiply by two like you could for the $\displaystyle c_{0}$ case. You can only do that, in general, when your complete integrand is even. But the presence of that particular exponential function in the integrand precludes that. So you have to keep the full $\displaystyle [-\pi,\pi]$ interval for the limits in the $\displaystyle c_{k}$ integration, I'm afraid.

I don't know of any way to integrate $\displaystyle x^{2}e^{-ikx}$ other than by parts twice. I'm not sure I would recommend breaking the integrals up into the trig functions, because then you're going to have to do integration by parts twice on two integrals. There's no need, when working with the exponential Fourier series, to look at sin and cos individually. It's an easier integration with just the exponential in there.

So I would carry these changes through, and then we'll see what happens at the end. Sound good?
I got the following expression after integrating the integral $\displaystyle [-\pi,\pi]$:

$\displaystyle c_{k} = \frac{1}{2\pi} ({-\frac{\pi^{2}}{ik}}e^{-ik\pi}}+{\frac{2\pi}{k^{2}}e^{-ik\pi}+{\frac{2}{ik^{3}}e^{-ik\pi}+\frac{\pi^{2}}{ik}}e^{ik\pi}+{\frac{2\pi}{k ^{2}}e^{ik\pi}-{\frac{2}{ik^{3}}e^{ik\pi})$

and I know that

$\displaystyle e^{ik\pi}=(-1)^{k}$

$\displaystyle \frac{1}{2\pi}(\frac{\pi^{2}}{ik}}e^{ik\pi}{-\frac{\pi^{2}}{ik}}e^{-ik\pi}})=0$

$\displaystyle \frac{1}{2\pi}({\frac{2}{ik^{3}}e^{-ik\pi}-{\frac{2}{ik^{3}}e^{ik\pi})=0$

$\displaystyle \frac{1}{2\pi}({\frac{2\pi}{k^{2}}e^{ik\pi}+{\frac {2\pi}{k^{2}}e^{-ik\pi})={\frac{1}{k^{2}}e^{ik\pi}+{\frac{1}{k^{2}} e^{-ik\pi}$

$\displaystyle c_{k}$ suppose to be $\displaystyle {\frac{{2}(-1^{k})}{k^{2}}$

How do I get:

$\displaystyle ({\frac{1}{k^{2}}e^{ik\pi}+{\frac{1}{k^{2}}e^{-ik\pi})$

to be equal $\displaystyle {\frac{{2}(-1^{k})}{k^{2}}$

I would appriciate a response. Thank you

4. Your k exponent should be outside the parentheses (otherwise it only applies to the 1). Otherwise, it looks fine.

5. Originally Posted by Ackbeet
Your k exponent should be outside the parentheses (otherwise it only applies to the 1). Otherwise, it looks fine.
Where does the 2 in $\displaystyle {\frac{{2}(-1)^{k}}{k^{2}}$ comes from. How do I get that. I just got it from the answer in the book but don't know where it comes from.

6. $\displaystyle ({\frac{1}{k^{2}}e^{ik\pi}+{\frac{1}{k^{2}}e^{-ik\pi})=\frac{1}{k^{2}}(e^{ikx}+e^{-ikx})=\frac{1}{k^{2}}(\cos(k\pi)+i\sin(k\pi)+\cos( k\pi)-i\sin(k\pi))$
$\displaystyle =\frac{2}{k^{2}}\,\cos(k\pi).$

But now $\displaystyle \cos(k\pi)=(-1)^{k}$ for all $\displaystyle k\in\mathbb{Z}.$

Does that help? It's basically all from the Euler formula, and the even-ness of cosine and the odd-ness of sine:

$\displaystyle e^{i\theta}=\cos(\theta)+i\sin(\theta).$

7. I have a question regarding solving for the exponetial Fourier series for a given function. The general formula is the following (the way I understood):

$\displaystyle \displaystyle f(x)=\displaystyle \sum_{k=-\infty}^{\infty}c_{k}e^{ik\Omega x}$

there $\displaystyle \Omega T={2\pi}$

Now, to solve for the exponetial Fourier series for the given problem above :

$\displaystyle \displaystyle f(x) = {x^{2}}$$\displaystyle ,\,\,-\pi < x \leq \pi$

Why do we solve for $\displaystyle c_{0}$ ? Its not in the general fomula.
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Where as I have a different problem where I have to solve for the exponetial Fourier series for a $\displaystyle {2\pi}$ - periodic function for the following given function:

$\displaystyle \displaystyle f(x)=e^{-x}},\,\,0 < x < {2\pi}$

In this case we just solve for $\displaystyle c_{k}$ only and not for $\displaystyle c_{0}$. Why is that?

I do appriciate for a explanation. Thank you

8. Why do we solve for $\displaystyle c_{0}$? It's not in the general formula.
I don't understand. If you have the sum

$\displaystyle \displaystyle{f(x)=\sum_{k=-\infty}^{\infty}c_{k}e^{ik\Omega x}},$

then the $\displaystyle k=0$ term has the expression $\displaystyle c_{0}$ in it. Does that qualify as "being in the general formula"?

In the problem you have there, I would think you would solve for $\displaystyle c_{0}$. Both functions you're dealing with have a nonzero average value. As I recall, the expression for the $\displaystyle c_{0}$ term looks a lot like the average value of the function over its period. *looking it up* In fact, it is the average value of the function over a period. Now if you have a function like $\displaystyle f(x)=x$ over the interval $\displaystyle (-\pi,\pi),$ and you can tell at a glance that its average value is zero, well then, you have no need to explicitly compute it. But such examples should not be considered to be the usual.

Make sense?

9. Alright, it makes sense. Thanks.

$\displaystyle \displaystyle f(x)=e^{-x}},\,\,0 < x < {2\pi}$

Do I need to compute $\displaystyle c_{0}$? If not, why?

10. Well, do you think the average value of the function over that interval is nonzero?

11. Originally Posted by Ackbeet
Well, do you think the average value of the function over that interval is nonzero?
I guess I am confused after all and don't understand.

Do you mean that:

$\displaystyle \displaystyle{f(x)=\sum_{k=-\infty}^{\infty}c_{k}e^{ik\Omega x}}\iff {f(x)=c_{0} + \sum_ {k\neq 0}c_{k}e^{ik\Omega x}}$

12. Your double implication is correct, if the understood range of $\displaystyle k$ values is the integers. All you're doing there is splitting off one term from the sum.

13. Originally Posted by Ackbeet
Now if you have a function like $\displaystyle f(x)=x$ over the interval $\displaystyle (-\pi,\pi),$ and you can tell at a glance that its average value is zero, well then, you have no need to explicitly compute it. But such examples should not be considered to be the usual.
If I compute $\displaystyle c_{0}$ for this function:

$\displaystyle f(x)=x$ over the interval $\displaystyle (-\pi,\pi)$

I would get the following:

$\displaystyle \displaystyle c_{0} =\frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) dx =\frac{2}{2\pi}\int_{0}^{\pi}x dx =\frac{\pi}{2}$ the value is $\displaystyle {\neq 0}$

14. No, the formula you used is valid for an even integrand. You have an odd integrand.

15. Originally Posted by Ackbeet
No, the formula you used is valid for an even integrand. You have an odd integrand.
Which formula do I use for an odd integrand? By the way thanks for your response.

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