Fourier Cosine Series

[RobertPaulson]

Hi, I'm going through past papers for revision and I was trying to do this question but have no idea where the 2 before the integral of a_0 comes from.
If anyone could help that would be great.

Also because this function is even, I don't understand why it says "n odd" underneath sigma

 

[RobertPaulson]

Ok, I realise I misread the half-range formulae to find where the 2 comes from. I would still like to know about the "n odd" though if that's possible.
Thanks

 

[galactus]

I am used to using an and bn.

If f(x) is even, all the \(\displaystyle b_{n}\)'s are 0, and the \(\displaystyle a_{n}\)'s are integrals of even functions.

\(\displaystyle \text{If f(x) even}=\left\{ \begin{array}{rcl}a_{n}=\frac{2}{L}\int_{0}^{L}f(x)cos(\frac{n{\pi}x}{L})dx\\b_{n}=0\end{array}\right\)

For even or odd functions, the coefficient formulas for \(\displaystyle a_{n}, \;\ b_{n}\) simplify. Suppose f(x) is odd.

Since \(\displaystyle sin(\frac{n{\pi}x}{L})\) is odd, \(\displaystyle f(x)sin(\frac{n{\pi}x}{L})\) is even and \(\displaystyle f(x)cos(\frac{n{\pi}x}{L})\) is odd.

Then \(\displaystyle a_{n}\) is the integral, over a symmetric interval (-L,L), of an odd function, namely \(\displaystyle f(x)cos(\frac{n{\pi}x}{L})\).

\(\displaystyle a_{n}\) is therefore 0. But, \(\displaystyle b_{n}\) is the integral of an even function over a symmetric interval and is therefore twice the 0

to L integral.

Does that explanation help?.

 

[RobertPaulson]

Yeah man that's awesome, well put. Having re-read the question again, I'm having trouble seeing how f(x) = x is an even function which is what they're implying in the answer right? If you could enlighten me that would be superb.
Thanks for your input so far

 

[galactus]

Duh, I overlooked that as well. f(x)=x is an odd function. Because f(-x)=-f(x).

With respect to the origin, of course. Odd functions would be \(\displaystyle x, x^{3},\;\ x^{5},.........\)

Evens would be \(\displaystyle x^{2}, \;\ x^{4}, \;\ x^{6},..................\)

See. With respect to the above, if the exponents are odd, then the function is odd. If the exponents are even, then the function is even.

Remember, an even times an even or an odd times an odd gives an even function.

an even times an odd gives an odd.

For \(\displaystyle f(x)=x \;\ for \;\ 0<x<1\), with period 2:

Even: \(\displaystyle f_{c}(x)=\frac{1}{2}-\frac{4}{{\pi}^{2}}\left(cos({\pi}x)+\frac{1}{9}cos(3{\pi}x)+...............\right)\)

Odd:\(\displaystyle f_{s}(x)=\frac{2}{\pi}\left(sin({\pi}x)-\frac{1}{2}sin(2{\pi}x)+\frac{1}{3}sin(3{\pi}x)+.............\right)\)

Here's an animated graph:

 

[RobertPaulson]

Thanks so much for your help, I love this site you guys are amazing

 

[galactus]

Thanks so much for your help, I love this site you guys are amazing

Yes, I know. We can't help it. We're just cuddly, lovable, little fuzzballs.