finding fourier cosine coefficients

Find the Fourier cosine coefficients of

e^x

e^x = \frac{1}{2}a_0 + \displaystyle\sum_{n=1}^\infty a_n cos \frac{n \pi x}{L}

Differentiating yields:

e^x = - \displaystyle\sum_{n=1}^\infty \frac{n \pi}{L}a_n sin \frac{n \pi x}{L}

,

the Fourier sine series of e^x. Differentiating again yields

e^x = - \displaystyle\sum_{n=1}^\infty (\frac{n \pi}{L})^2 a_n cos \frac{n \pi x}{L}

,

Since equations 1 and 3 both give Fourier cosine series of e^x, they must be identical. Thus,

a_o = 0

and

a_n = 0.

Can anyone please explain step by step what is wrong with this? I'm supposed to correct the mistakes and then find

a_n

without using the typical technique but I'm so confused!

Any mistakes at all you can see, please point them out!

Thanks in advance :smile:

Reply 1

DFranklin

hohoho11

Find the Fourier cosine coefficients of

e^x

e^x = \frac{1}{2}a_0 + \displaystyle\sum_{n=1}^\infty a_n cos \frac{n \pi x}{L}

Differentiating yields:

e^x = - \displaystyle\sum_{n=1}^\infty \frac{n \pi}{L}a_n sin \frac{n \pi x}{L}

,

the Fourier sine series of e^x. Differentiating again yields

e^x = - \displaystyle\sum_{n=1}^\infty (\frac{n \pi}{L})^2 a_n cos \frac{n \pi x}{L}

,

Since equations 1 and 3 both give Fourier cosine series of e^x, they must be identical. Thus,

a_o = 0

and

a_n = 0.

Can anyone please explain step by step what is wrong with this?You have no reason to think equation 3 converges, and if it doesn't then it doesn't equal e^x.

Reply 2

hohoho11

Sorry DFranklin I don't understand how you know, or why equation 3 does not converge...could you possibly elaborate a bit more? Also could you please explain how I can go about correcting this to obtain the correct value for

a_n

?

Thanks once again.

Reply 3

insparato

What i think he's saying is you should test whether equation 3 converges/diverges.

Reply 4

DFranklin

insparato

What i think he's saying is you should test whether equation 3 converges/diverges.

Yes. (Although I'm pretty sure it diverges, but that's only because I have a vague recollection of what a_n actually ends up being for this series).

I confess I don't know how you're supposed to "fix" the method. Term by term integration is going to be a lot safer than term by term integration, but I'm not sure it's valid (the original series has a discontinuity, so the fourier series isn't going to be terribly well behaved). And you then have to worry about arbitrary constants of integration.

Reply 5

hohoho11

Well if you can't do it, then I don't have a hope in hell!

So you're basically saying my lecturer has set a question that is wrong to begin with then? She's a new lecturer and she keeps setting stupid questions like this that don't involve using the standard method, to make the course more 'exciting'. Pfffffft.

Reply 6

DFranklin

hohoho11

Well if you can't do it, then I don't have a hope in hell!

I could probably bodge something out, but it wouldn't be "right". I suspect the solution your lecturer has in mind isn't "right" either (in the sense that it won't be rigourously justified), but the problem is my "not right" solution isn't necessarily going to match what they wanted.

So you're basically saying my lecturer has set a question that is wrong to begin with then?

No - I don't think I said that.

Its perfectly reasonable for her to ask the question and ask you what's wrong with it. It's also perfectly reasonable for you to say "the argument is invalid if the differentiated series doesn't converge to the derivative of e^x, and there's no reason to think it does" (*).

Basically I don't want to even try to second guess what your lecturer had in mind here. Now the fact I can't guess what she was thinking does imply it's not a great question, but I don't have the context of what she's been lecturing in your course. For all I know she's spent the last 3 lectures going over stuff that makes it very obvious how you're supposed to proceed here,

She's a new lecturer and she keeps setting stupid questions like this that don't involve using the standard method, to make the course more 'exciting'.

Sounds like she's a bit overenthusiastic. To be honest, she probably knows the subject area too well. (Worst lectures I ever attended were given by someone with a Fields medal in the subject).

(*) Actually, you want to take a little care, along the lines of "if the series converged, the argument would be valid. But the argument gives a nonsense result. So the argument isn't valid. So the series can't converge".