# Why this equation “Fourier series” is important?

I am a student majoring in electrical engineering.

There is three equations about Fourier series.

\begin{align} x(t)&=\sum_{n=-\infty}^{\infty}X_n e^{j2\pi nf_0t} &&&& (1)\\ X_n&=\frac1{T_0}\int_{t_0}^{t_0+T_0}x(t)e^{-j2\pi nf_0t}dt &&&& (2)\\ x(t)&=X_0 + \sum_{n=1}^{\infty} A_n \cos(2\pi nf_0t) + \sum_{n=1}^{\infty}B_n \sin(2\pi nf_0t) &&&& (3)\\ &~~~~~~~~~~~~,where~~~~A_n = \frac2{T_0} \int_{t_0}^{t_0+T_0} x(t) \cos(2\pi nf_0t)dt\\ &~~~~~~~~~~~~~~~~~~~~~~~~~~~~B_n = \frac2{T_0} \int_{t_0}^{t_0+T_0} x(t) \sin(2\pi nf_0t)dt\\ \end{align}

I know equations $(1)$ and $(2)$ are used when changing domain from time to frequency, and vice versa.

I want to know where the equation $(3)$ is used.

I regard equation $(3)$ is weird because $x(t)$ is expressed as a function of $x(t)$ itself.

Please let me know why equation $(3)$ is also important. Thank you.

• FYI: $x(t)$ is represented in terms of $x(t)$ in equations $(1)$ and $(2)$ as well. – JimmyK4542 Dec 24 '15 at 9:19
• But when I have $X_n$ data, I can get $x(t)$ using the equation $(1)$. In the similar way, I can gain $X_n$ from $x(t)$ using the equation $(2)$. However, it is weird that I can gain $x(t)$ from $x(t)$ using the equation $(3)$. – Danny_Kim Dec 24 '15 at 9:30

From a theoretical standpoint, Equation 3 shows you how to decompose $x(t)$ into orthogonal functions (sines and cosines), which can help you draw conclusions about $x(t)$ which aren't obvious from the original function.