The exponential form of Fourier series of a continuous-time periodic signal x(t) is given by,
$$\mathrm{x(t)= \sum_{n=-\infty}^{\infty}C_{n}\:e^{jn\omega_{0}t}}$$
The set of coefficients $[C_{n}]$ is called the set of the Fourier series coefficients or the spectral coefficients of signal x(t).
$$\mathrm{C_{n}=\frac{1}{T}\int_{-T/2}^{T/2}x(t)\:e^{-jn\omega_{0}t}dt}$$
The complex coefficients $[C_{n}]$ measure the portion of the signal x(t),that is at each harmonic of the fundamental component.
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The coefficient $[C_{0}]$ is the DC component of the function x(t), which is the average value of the signal over one period, i.e.,
$$\mathrm{C_{n}=\frac{1}{T}\int_{-T/2}^{T/2}x(t)dt}$$
The Fourier series coefficients of the function x(t) are discrete in nature and hence we obtain a discrete spectrum.
The Fourier transform of a continuous-time non-periodic signal x(t) is defined as,
$$\mathrm{X(\omega)=\int_{-\infty }^{\infty}x(t)e^{-j\omega t}dt}$$
The inverse Fourier transform is defined as,
$$\mathrm{x(t)=\frac{1}{2\pi}\int_{-\infty }^{\infty}X(\omega)e^{j\omega t}d\omega}$$
x(t) and X(ω) form a Fourier transform pair, which is represented as,
$$\mathrm{x(t)\overset{FT}{\leftrightarrow}X(\omega)}$$
The equation of inverse Fourier transform$[i.e.,x(t)=\frac{1}{2\pi}\int_{-\infty }^{\infty}X(\omega)e^{j\omega t}d\omega]$ plays a role for non-periodic signals similar to the equation of Fourier series $[i.e.,x(t)=\sum_{n=-\infty}^{\infty}= C_{n}\:e^{jn\omega_{o}t}]$ for periodic signals. Because both the equations represent the linear combination of complex exponentials.
For periodic signals, the spectral coefficients have amplitudes Cn and occur at discrete set of harmonically related frequencies $(n\omega_{0})$, where,$(n=0,\pm 1,\pm 2...)$ These amplitudes are given by,
$$\mathrm{C_{n}=\frac{1}{T}\int_{-T/2}^{T/2}x(t)\:e^{-jn\omega_{0}}dt}$$
For non-periodic signals, the complex exponentials occur at continuous frequencies with magnitude $[X(\omega)d\omega/2\pi]$.
The Fourier transform $X(\omega)$ of an aperiodic signal x(t) is called the spectrum of the signal x(t).