Fourier Series

Chapter 1  Introduction
Trigonometric Fourier Series
Every periodic function f(t) may be approximated as a Trigonometric Fourier Series

Fig. 1-1
Trigonometric Fourier Series
Fig. 1-1a Formula with the sigma sign
Fig. 1-1b Formula without the sigma sign. It’s more clear for me and i will use this notation later. The consecutive a1,a2,…b1,b2 amplitudes and especially 1ω,2ω…pulsatances  are more visible here. Note that the consecutive pulsatances are multiplies.
Fig. 1-1c f(t) periodical function example with the period.
The a/m formulas are exactly for n=∞ only. It’s an approximation for finite mostly. Why mostly and not always? The Fig. 2-3 in the next chapter is  an answer.

Complex Fourier Series
It’s simpler but less clear in the first moment. There is an analogy with the the Solar System. The Earth circles around the Sun, the Moon circles around the Earth… etc. More–> see Chapter 4

Fig. 1-2
f(t) 
function as a Complex Fourier Series
The formulas are simpler transparently than Fig. 1-1. Unfortunately the used coefficients are complex numbers, not real.
So all  the
 Chapter 3 is about complex numbers.

Fig. 1-3
Fourier Series coefficients as complex numbers.
Note that the Fig. 1-2  right hand formula is a complicated complex numbers sum but the left is a real f(t) function!

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