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Mathematica 2015

No.1 Trigonometric notation for Fourier series The Fourier series corresponding to f(x) is given by:  0 2 +βˆ‘( cos  𝐿 +  sin  𝐿 ) ∞ =1 In this expression,  and 

are called the Fourier coefficients given by:

∫ ( ) πœ‹π‘₯ 𝐿 𝐿 βˆ’πΏ ∫ ( )𝑖 πœ‹π‘₯ 𝐿 𝐿 βˆ’πΏ

= 1 𝐿 = 1 𝐿





{

, n ∈ N

(1)





No.2 Complex notation for Fourier series Since Euler’s identity says that:

 π‘–πœƒ = πœƒ + 𝑖𝑖πœƒ  βˆ’π‘–πœƒ = πœƒ βˆ’ 𝑖𝑖πœƒ We can then express πœƒ and 𝑖πœƒ in terms of complex exponential:

 π‘–πœƒ +  βˆ’π‘–πœƒ 2

πœƒ =

𝑖πœƒ =  π‘–πœƒ βˆ’  βˆ’π‘–πœƒ 2𝑖 After some mathematical manipulations, which are not detailed here, the Fourier series can then be written as:

∞

 π‘–πœ‹π‘₯/𝐿

( ) = βˆ‘

βˆ’βˆž

∫ ( ) π‘–πœ‹π‘₯/𝐿 𝐿 βˆ’πΏ

= 1





Where

2𝐿

ο‚·

Real application examples:

After knowing the background knowledge, let’s have a look at the following two examples and go further into Fourier series. Example 1:

First, let’s consider a saw tooth wave with a function f(x),which is defined as below: ( ) = | |, βˆ’ ≀ < , 𝑖 2 So 2L = 2Ο€ β†’ L = Ο€ According to (1), we find that  is:

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