Fourier series (integration of pi)

[DanPF]

Homework Statement

Hi

First of all this a textbook question from Stroud Advanced Engineering Mathematics, solution is given, but no steps are shown.

A question I just can't seem to solve at the moment, as below.

A function f(x) is defined by $$f(x) = \pi - x:0 < x < \pi$$
$$f(x + 2\pi) = f(x)$$

Express the function as a half range cosie series

Homework Equations

Im at a loss of how to start this question, I've been doing fine when they haven't included pi in the function itself.

I know as its wanted as a cosine series I'll have to make it resemble an even function and thus my a0 and an terms can be

$$2/\pi \int^{\pi}_{0}\pi - x dx$$$$\stackrel{2}{\pi}\int^{\pi}_{0}\pi - x\:cosnx:dx$$

So my main question is really how I remove pi. I've tried taking it out as a constant which doesn't work, so I have a feeling I am not realising something

The final solution according to the book is

$$\pi/2 + 4/\pi\left\{cosx + 1/9:cos3x + 1/25:cos5x+ ...\right\}$$

I keep getting $$-\pi^2$$ for a0

Thank you for reading and any help

 

[LCKurtz]

$$\stackrel{2}{\pi}\int^{\pi}_{0}\pi - x\:cosnx:dx$$

Since you haven't actually shown what you did, I will make a wild guess that your problem is caused by leaving the parentheses out:

$$\stackrel{2}{\pi}\int^{\pi}_{0}(\pi - x)\:cosnx\ dx$$

You don't treat $\pi$ any differently than you would treat a number like 5.

 

[DanPF]

Hi

Thanks for the reply, sorry I didnt put down more of my working, I was having trouble using the Latex. I have solved the problem as I was forgetting the a0/2 term, and didnt correctly integrate the -pi term.