Other definition for fourier series

[Jhenrique]

Is correct to define Fourier series like:

$$f(t)=\sum_{k=0}^{\infty}a_k \cos \left (\frac{2 \pi k t}{T} \right ) + b_k \sin \left (\frac{2 \pi k t}{T} \right )$$

Where ak and bk:

$$a_k=\frac{1}{T} \int_{-T}^{+T} f(t) \cos \left (\frac{2 \pi k t}{T} \right ) dt$$

$$b_k=\frac{1}{T} \int_{-T}^{+T} f(t) \sin \left (\frac{2 \pi k t}{T} \right ) dt$$

?

 

[Xiuh]

No. You are counting the period twice.
If this were true, could you expand something like $\cos \left (\frac{\pi t}{T} \right )$?

 

[Jhenrique]

I don't understand your answer

 

[Jhenrique]

I take this topic to introduce another question: in wikipedia, I found others difinitions to Fourier series:

$$f(t)=A_0+\sum_{n=1}^{\infty } A_n cos\left ( \frac{2 \pi n t}{T}-\phi_n \right )$$
where:

$A_0 = \frac{1}{2}a_0$
$A_n = \sqrt{a_{n}^{2}+b_{n}^{2}}$
$\phi_n = tan^{-1}\left ( \frac{b_n}{a_n} \right )$


and:

$$f(t)=\gamma_0+2\sum_{n=1}^{\infty } \gamma_n cos\left ( \frac{2 \pi n t}{T}+\varphi_n \right )$$
where:

$\gamma_0 = c_0$
$\gamma_n = abs(c_n)$
$\varphi_n = arg(c_n)$



I'd like to know if $\varphi_n$ is or isn't equal to $\phi_n$ ?