# The complex Fourier Series and its relation to the Fourier Transform¶

In two recent articles we have talked about the Fourier Series and an application in harmonic analysis of instrument sounds in terms of their Fourier coefficients. In this article, we will analyze the relation between the Fourier Series and the Fourier Transform.

## The Fourier Series as sums of sines and cosines¶

To recap, the Fourier series of a signal x(t) with period P is given by

\begin{align}x(t)=\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos(2\pi nt/P)+b_n\sin(2\pi nt/P)\end{align}

where the coefficients are given by

\begin{align}a_n&=\frac{2}{P}\int_{-\frac{P}{2}}^\frac{P}{2}x(t)\cos(2\pi nt/P)dt\\b_n&=\frac{2}{P}\int_{-\frac{P}{2}}^\frac{P}{2}x(t)\sin(2\pi nt/P)dt\end{align}.

As we see, the Fourier series is a sum of sines and cosines with different amplitudes. Let us first look at the sum of a sine and cosine with different amplitudes:

### Sum of a sine and cosine with equal frequency¶

{% includeTrinket “a9b2abf8de” %}

T = 3 # time duration to look at

P = 1 # signal period

t = …