Is Author Wrong? Solving a Homework Equation

[foo9008]

Homework Statement

is the author wrong ? i was told that the f(x) = 0.5(a_0) +Σ(a_n)cos (nπx / L ) ... but , in the example(photo2) , the author ignore the L , which the author gave f(x) = 0.5(a_0) +Σ(a_n)cos (nπx ) +...

Homework Equations

The Attempt at a Solution

P/ s : i have tried to make some correction beside the working , is it correct ?[/B]

 

[BvU]

Don't know what photo 2 is, but in 154002 the author carefully uses L = 2.
And in 1550002 L is $\pi$

 

[foo9008]

Don't know what photo 2 is, but in 154002 the author carefully uses L = 2.
And in 1550002 L is $\pi$

So, the author is wrong, right? In155002, the L should be 2, right??

 

[BvU]

If 150 says $n\pi\x\over L$ and 154 says $n\pi\over 2$, doesn't that mean the author did take L = 2 ?

As for 155, I'm not so sure: does the definition in your book agree with

The http://www.math24.net/definition-of-fourier-series.html of the function f(x) is given by
$$f(x)={a_0\over 2}+\sum_{n=1}^\infty \{a_n\cos nx+b_n\sin nx\}$$
where the Fourier coefficients $a_0$, $a_n$, and $b_n$ are defined by the integrals$$
a_0={1\over \pi} \int _{−\pi}^\pi f(x)\, dx,\quad a_n={1\over \pi} \int _{−\pi}^\pi f(x)\cos nx\,dx,\quad b_n{1\over \pi} \int _{−\pi}^\pi f(x)\sin nx\,dx$$

 

[foo9008]

If 150 says $n\pi\x\over L$ and 154 says $n\pi\over 2$, doesn't that mean the author did take L = 2 ?

As for 155, I'm not so sure: does the definition in your book agree with

The http://www.math24.net/definition-of-fourier-series.html of the function f(x) is given by
$$f(x)={a_0\over 2}+\sum_{n=1}^\infty \{a_n\cos nx+b_n\sin nx\}$$
where the Fourier coefficients $a_0$, $a_n$, and $b_n$ are defined by the integrals$$
a_0={1\over \pi} \int _{−\pi}^\pi f(x)\, dx,\quad a_n={1\over \pi} \int _{−\pi}^\pi f(x)\cos nx\,dx,\quad b_n{1\over \pi} \int _{−\pi}^\pi f(x)\sin nx\,dx$$

no , as you can see it 150 , the author gave $$f(x)={a_0\over 2}+\sum_{n=1}^\infty \{a_n\cos nπx / L+b_n\sin nπx\/L}$$