[<< wikibooks] Advanced Mathematics for Engineers and Scientists/Details and Applications of Fourier Series
== Details and Applications of Fourier Series ==
In the study of PDEs (and much elsewhere), a Fourier series (or more generally, trigonometric expansion) often needs to be constructed.


=== Preliminaries ===
Suppose that a function f(x) may be expressed in the following way:

  
    
      
        f
        (
        x
        )
        =
        
          
            
              A
              
                0
              
            
            2
          
        
        +
        
          ∑
          
            n
            =
            1
          
          
            ∞
          
        
        
          (
          
            
              A
              
                n
              
            
            cos
            ⁡
            
              (
              
                
                  
                    
                      n
                      π
                    
                    L
                  
                
                x
              
              )
            
            +
            
              B
              
                n
              
            
            sin
            ⁡
            
              (
              
                
                  
                    
                      n
                      π
                    
                    L
                  
                
                x
              
              )
            
          
          )
        
      
    
    {\displaystyle f(x)={\frac {A_{0}}{2}}+\sum _{n=1}^{\infty }\left(A_{n}\cos \left({\frac {n\pi }{L}}x\right)+B_{n}\sin \left({\frac {n\pi }{L}}x\right)\right)}
  It can be shown (not too difficult, but beyond this text) that the above expansion will converge to f(x), except at discontinuities, if the following conditions hold:

f(x) = f(x + 2L), i.e.  f(x) has period 2L .
f(x), f'(x), and f''(x) are piecewise continuous on the interval -L ≤ x ≤ L.
The pieces that make up f(x), f'(x), and f''(x) are continuous over closed subintervals.The first requirement is most significant; the last two requirements can, to an extent, be partly eased off in most cases without any trouble. An interesting thing happens at discontinuities. Suppose that f(x) is discontinuous at x = a; the expansion will converge to the following value:

  
    
      
        
          
            1
            2
          
        
        
          (
          
            
              lim
              
                x
                →
                
                  a
                  
                    −
                  
                
              
            
            f
            (
            x
            )
            +
            
              lim
              
                x
                →
                
                  a
                  
                    +
                  
                
              
            
            f
            (
            x
            )
          
          )
        
        
      
    
    {\displaystyle {\frac {1}{2}}\left(\lim _{x\rightarrow a^{-}}f(x)+\lim _{x\rightarrow a^{+}}f(x)\right)\,}
  So the expansion converges to the average of the values to the left and the right of the discontinuity. This, and the fact that it converges in the first place, is very convenient. The Fourier series looks unfriendly but it's honestly working for you.
The information needed to express f(x) as a Fourier series are the sequences An and Bn. This is done using orthogonality, which for the sinusoids may be derived easily using a few identities. The following are some useful orthogonality relations, with m and n restricted to integers:

  
    
      
        
          ∫
          
            0
          
          
            L
          
        
        
          
            2
            L
          
        
        sin
        ⁡
        
          (
          
            
              
                
                  m
                  π
                
                L
              
            
            x
          
          )
        
        sin
        ⁡
        
          (
          
            
              
                
                  n
                  π
                
                L
              
            
            x
          
          )
        
        d
        x
        =
        
          δ
          
            m
            ,
            n
          
        
      
    
    {\displaystyle \int _{0}^{L}{\frac {2}{L}}\sin \left({\frac {m\pi }{L}}x\right)\sin \left({\frac {n\pi }{L}}x\right)dx=\delta _{m,n}}
  
  
    
      
        
          ∫
          
            0
          
          
            L
          
        
        
          
            2
            L
          
        
        cos
        ⁡
        
          (
          
            
              
                
                  m
                  π
                
                L
              
            
            x
          
          )
        
        cos
        ⁡
        
          (
          
            
              
                
                  n
                  π
                
                L
              
            
            x
          
          )
        
        d
        x
        =
        
          δ
          
            m
            ,
            n
          
        
      
    
    {\displaystyle \int _{0}^{L}{\frac {2}{L}}\cos \left({\frac {m\pi }{L}}x\right)\cos \left({\frac {n\pi }{L}}x\right)dx=\delta _{m,n}}
  
  
    
      
        
          ∫
          
            −
            L
          
          
            L
          
        
        
          
            1
            L
          
        
        sin
        ⁡
        
          (
          
            
              
                
                  m
                  π
                
                L
              
            
            x
          
          )
        
        sin
        ⁡
        
          (
          
            
              
                
                  n
                  π
                
                L
              
            
            x
          
          )
        
        d
        x
        =
        
          δ
          
            m
            ,
            n
          
        
      
    
    {\displaystyle \int _{-L}^{L}{\frac {1}{L}}\sin \left({\frac {m\pi }{L}}x\right)\sin \left({\frac {n\pi }{L}}x\right)dx=\delta _{m,n}}
  
  
    
      
        
          ∫
          
            −
            L
          
          
            L
          
        
        
          
            1
            L
          
        
        cos
        ⁡
        
          (
          
            
              
                
                  m
                  π
                
                L
              
            
            x
          
          )
        
        cos
        ⁡
        
          (
          
            
              
                
                  n
                  π
                
                L
              
            
            x
          
          )
        
        d
        x
        =
        
          δ
          
            m
            ,
            n
          
        
      
    
    {\displaystyle \int _{-L}^{L}{\frac {1}{L}}\cos \left({\frac {m\pi }{L}}x\right)\cos \left({\frac {n\pi }{L}}x\right)dx=\delta _{m,n}}
  
  
    
      
        
          ∫
          
            −
            L
          
          
            L
          
        
        sin
        ⁡
        
          (
          
            
              
                
                  m
                  π
                
                L
              
            
            x
          
          )
        
        cos
        ⁡
        
          (
          
            
              
                
                  n
                  π
                
                L
              
            
            x
          
          )
        
        d
        x
        =
        0
      
    
    {\displaystyle \int _{-L}^{L}\sin \left({\frac {m\pi }{L}}x\right)\cos \left({\frac {n\pi }{L}}x\right)dx=0}
  δm,n is called the Kronecker delta, defined by:

  
    
      
        
          δ
          
            m
            ,
            n
          
        
        =
        
          
            {
            
              
                
                  1
                  ,
                
                
                  m
                  =
                  n
                
              
              
                
                  0
                  ,
                
                
                  m
                  ≠
                  n
                
              
            
            
          
        
      
    
    {\displaystyle \delta _{m,n}={\begin{cases}1,&m=n\\0,&m\neq n\end{cases}}}
  The Kronecker delta may be thought of as a discrete version of the Dirac delta "function". Relevant to this topic is its sifting property:

  
    
      
        
          ∑
          
            n
            =
            −
            ∞
          
          
            ∞
          
        
        
          A
          
            n
          
        
        
          δ
          
            m
            ,
            n
          
        
        =
        
          A
          
            m
          
        
        
      
    
    {\displaystyle \sum _{n=-\infty }^{\infty }A_{n}\delta _{m,n}=A_{m}\,}
  


=== Derivation of the Fourier Series ===
We're now ready to find An and Bn.

  
    
      
        f
        (
        x
        )
        =
        
          
            
              A
              
                0
              
            
            2
          
        
        +
        
          ∑
          
            n
            =
            1
          
          
            ∞
          
        
        
          (
          
            
              A
              
                n
              
            
            cos
            ⁡
            
              (
              
                
                  
                    
                      n
                      π
                    
                    L
                  
                
                x
              
              )
            
            +
            
              B
              
                n
              
            
            sin
            ⁡
            
              (
              
                
                  
                    
                      n
                      π
                    
                    L
                  
                
                x
              
              )
            
          
          )
        
      
    
    {\displaystyle f(x)={\frac {A_{0}}{2}}+\sum _{n=1}^{\infty }\left(A_{n}\cos \left({\frac {n\pi }{L}}x\right)+B_{n}\sin \left({\frac {n\pi }{L}}x\right)\right)}
  
  
    
      
        
          
            1
            L
          
        
        cos
        ⁡
        
          (
          
            
              
                
                  m
                  π
                
                L
              
            
            x
          
          )
        
        ⋅
        f
        (
        x
        )
        =
        
          
            1
            L
          
        
        cos
        ⁡
        
          (
          
            
              
                
                  m
                  π
                
                L
              
            
            x
          
          )
        
        ⋅
        
          (
          
            
              
                
                  A
                  
                    0
                  
                
                2
              
            
            +
            
              ∑
              
                n
                =
                1
              
              
                ∞
              
            
            
              (
              
                
                  A
                  
                    n
                  
                
                cos
                ⁡
                
                  (
                  
                    
                      
                        
                          n
                          π
                        
                        L
                      
                    
                    x
                  
                  )
                
                
                  B
                  
                    n
                  
                
                sin
                ⁡
                
                  (
                  
                    
                      
                        
                          n
                          π
                        
                        L
                      
                    
                    x
                  
                  )
                
              
              )
            
          
          )
        
        
      
    
    {\displaystyle {\frac {1}{L}}\cos \left({\frac {m\pi }{L}}x\right)\cdot f(x)={\frac {1}{L}}\cos \left({\frac {m\pi }{L}}x\right)\cdot \left({\frac {A_{0}}{2}}+\sum _{n=1}^{\infty }\left(A_{n}\cos \left({\frac {n\pi }{L}}x\right)B_{n}\sin \left({\frac {n\pi }{L}}x\right)\right)\right)\,}
  
  
    
      
        
          ∫
          
            −
            L
          
          
            L
          
        
        
          
            1
            L
          
        
        cos
        ⁡
        
          (
          
            
              
                
                  m
                  π
                
                L
              
            
            x
          
          )
        
        f
        (
        x
        )
        d
        x
        =
        
          ∫
          
            −
            L
          
          
            L
          
        
        
          
            
              A
              
                0
              
            
            
              2
              L
            
          
        
        cos
        ⁡
        
          (
          
            
              
                
                  m
                  π
                
                L
              
            
            x
          
          )
        
        d
        x
        +
        
          ∑
          
            n
            =
            1
          
          
            ∞
          
        
        
          (
          
            
              A
              
                n
              
            
            
              δ
              
                m
                ,
                n
              
            
            +
            
              
                
                  B
                  
                    n
                  
                
                L
              
            
            ⋅
            0
          
          )
        
      
    
    {\displaystyle \int _{-L}^{L}{\frac {1}{L}}\cos \left({\frac {m\pi }{L}}x\right)f(x)dx=\int _{-L}^{L}{\frac {A_{0}}{2L}}\cos \left({\frac {m\pi }{L}}x\right)dx+\sum _{n=1}^{\infty }\left(A_{n}\delta _{m,n}+{\frac {B_{n}}{L}}\cdot 0\right)}
  This is supposed to hold for an arbitrary integer m. If m = 0, note that the sum doesn't allow n = 0 and so the sum would be zero since in no case does m = n. This leads to:

  
    
      
        
          ∫
          
            −
            L
          
          
            L
          
        
        
          
            1
            L
          
        
        cos
        ⁡
        
          (
          
            
              
                
                  0
                  ⋅
                  π
                
                L
              
            
            x
          
          )
        
        f
        (
        x
        )
        d
        x
        =
        
          ∫
          
            −
            L
          
          
            L
          
        
        
          
            
              A
              
                0
              
            
            
              2
              L
            
          
        
        cos
        ⁡
        
          (
          
            
              
                
                  0
                  ⋅
                  π
                
                L
              
            
            x
          
          )
        
        d
        x
      
    
    {\displaystyle \int _{-L}^{L}{\frac {1}{L}}\cos \left({\frac {0\cdot \pi }{L}}x\right)f(x)dx=\int _{-L}^{L}{\frac {A_{0}}{2L}}\cos \left({\frac {0\cdot \pi }{L}}x\right)dx}
  
  
    
      
        
          ∫
          
            −
            L
          
          
            L
          
        
        
          
            1
            L
          
        
        f
        (
        x
        )
        d
        x
        =
        
          A
          
            0
          
        
        
          ∫
          
            −
            L
          
          
            L
          
        
        
          
            
              d
              x
            
            
              2
              L
            
          
        
      
    
    {\displaystyle \int _{-L}^{L}{\frac {1}{L}}f(x)dx=A_{0}\int _{-L}^{L}{\frac {dx}{2L}}}
  
  
    
      
        
          ∫
          
            −
            L
          
          
            L
          
        
        
          
            1
            L
          
        
        f
        (
        x
        )
        d
        x
        =
        
          A
          
            0
          
        
      
    
    {\displaystyle \int _{-L}^{L}{\frac {1}{L}}f(x)dx=A_{0}}
  This secures A0. Now suppose that m > 0. Since m and n are now in the same domain, the Kronecker delta will do its sifting:

  
    
      
        
          ∫
          
            −
            L
          
          
            L
          
        
        
          
            1
            L
          
        
        cos
        ⁡
        
          (
          
            
              
                
                  m
                  π
                
                L
              
            
            x
          
          )
        
        f
        (
        x
        )
        d
        x
        =
        
          ∫
          
            −
            L
          
          
            L
          
        
        
          
            
              A
              
                0
              
            
            
              2
              L
            
          
        
        cos
        ⁡
        
          (
          
            
              
                
                  m
                  π
                
                L
              
            
            x
          
          )
        
        d
        x
        +
        
          ∑
          
            n
            =
            1
          
          
            ∞
          
        
        
          (
          
            
              A
              
                n
              
            
            
              δ
              
                m
                ,
                n
              
            
          
          )
        
      
    
    {\displaystyle \int _{-L}^{L}{\frac {1}{L}}\cos \left({\frac {m\pi }{L}}x\right)f(x)dx=\int _{-L}^{L}{\frac {A_{0}}{2L}}\cos \left({\frac {m\pi }{L}}x\right)dx+\sum _{n=1}^{\infty }\left(A_{n}\delta _{m,n}\right)}
  
  
    
      
        
          ∫
          
            −
            L
          
          
            L
          
        
        
          
            1
            L
          
        
        cos
        ⁡
        
          (
          
            
              
                
                  m
                  π
                
                L
              
            
            x
          
          )
        
        f
        (
        x
        )
        d
        x
        =
        
          
            
              
                A
                
                  0
                
              
              sin
              ⁡
              (
              m
              π
              )
            
            
              m
              π
            
          
        
        +
        
          A
          
            m
          
        
      
    
    {\displaystyle \int _{-L}^{L}{\frac {1}{L}}\cos \left({\frac {m\pi }{L}}x\right)f(x)dx={\frac {A_{0}\sin(m\pi )}{m\pi }}+A_{m}}
  
  
    
      
        
          ∫
          
            −
            L
          
          
            L
          
        
        
          
            1
            L
          
        
        cos
        ⁡
        
          (
          
            
              
                
                  m
                  π
                
                L
              
            
            x
          
          )
        
        f
        (
        x
        )
        d
        x
        =
        0
        +
        
          A
          
            m
          
        
      
    
    {\displaystyle \int _{-L}^{L}{\frac {1}{L}}\cos \left({\frac {m\pi }{L}}x\right)f(x)dx=0+A_{m}}
  
  
    
      
        
          A
          
            n
          
        
        =
        
          ∫
          
            −
            L
          
          
            L
          
        
        
          
            1
            L
          
        
        cos
        ⁡
        
          (
          
            
              
                
                  n
                  π
                
                L
              
            
            x
          
          )
        
        f
        (
        x
        )
        d
        x
      
    
    {\displaystyle A_{n}=\int _{-L}^{L}{\frac {1}{L}}\cos \left({\frac {n\pi }{L}}x\right)f(x)dx}
  In the second to the last step, sin(mπ) = 0 for integer m. In the last step, m was replaced with n. This defines An for n > 0. For the case n = 0,

  
    
      
        
          A
          
            0
          
        
        =
        
          ∫
          
            −
            L
          
          
            L
          
        
        
          
            1
            L
          
        
        cos
        ⁡
        
          (
          
            
              
                
                  0
                  ⋅
                  π
                
                L
              
            
            x
          
          )
        
        f
        (
        x
        )
        d
        x
      
    
    {\displaystyle A_{0}=\int _{-L}^{L}{\frac {1}{L}}\cos \left({\frac {0\cdot \pi }{L}}x\right)f(x)dx}
  
  
    
      
        
          A
          
            0
          
        
        =
        
          ∫
          
            −
            L
          
          
            L
          
        
        
          
            1
            L
          
        
        f
        (
        x
        )
        d
        x
      
    
    {\displaystyle A_{0}=\int _{-L}^{L}{\frac {1}{L}}f(x)dx}
  Which happens to match the previous development (now you know why it's A0/2 and not just A0). So the sequence An is now completely defined for any value of n of interest:

  
    
      
        
          A
          
            n
          
        
        =
        
          ∫
          
            −
            L
          
          
            L
          
        
        
          
            1
            L
          
        
        cos
        ⁡
        
          (
          
            
              
                
                  n
                  π
                
                L
              
            
            x
          
          )
        
        f
        (
        x
        )
        d
        x
      
    
    {\displaystyle A_{n}=\int _{-L}^{L}{\frac {1}{L}}\cos \left({\frac {n\pi }{L}}x\right)f(x)dx}
  To get Bn, nearly the same routine is used.

  
    
      
        f
        (
        x
        )
        =
        
          
            
              A
              
                0
              
            
            2
          
        
        +
        
          ∑
          
            n
            =
            1
          
          
            ∞
          
        
        
          (
          
            
              A
              
                n
              
            
            cos
            ⁡
            
              (
              
                
                  
                    
                      n
                      π
                    
                    L
                  
                
                x
              
              )
            
            +
            
              B
              
                n
              
            
            sin
            ⁡
            
              (
              
                
                  
                    
                      n
                      π
                    
                    L
                  
                
                x
              
              )
            
          
          )
        
      
    
    {\displaystyle f(x)={\frac {A_{0}}{2}}+\sum _{n=1}^{\infty }\left(A_{n}\cos \left({\frac {n\pi }{L}}x\right)+B_{n}\sin \left({\frac {n\pi }{L}}x\right)\right)}
  
  
    
      
        
          
            1
            L
          
        
        sin
        ⁡
        
          (
          
            
              
                
                  m
                  π
                
                L
              
            
            x
          
          )
        
        ⋅
        f
        (
        x
        )
        =
        
          
            1
            L
          
        
        sin
        ⁡
        
          (
          
            
              
                
                  m
                  π
                
                L
              
            
            x
          
          )
        
        ⋅
        
          (
          
            
              
                
                  A
                  
                    0
                  
                
                2
              
            
            +
            
              ∑
              
                n
                =
                1
              
              
                ∞
              
            
            
              (
              
                
                  A
                  
                    n
                  
                
                cos
                ⁡
                
                  (
                  
                    
                      
                        
                          n
                          π
                        
                        L
                      
                    
                    x
                  
                  )
                
                
                  B
                  
                    n
                  
                
                sin
                ⁡
                
                  (
                  
                    
                      
                        
                          n
                          π
                        
                        L
                      
                    
                    x
                  
                  )
                
              
              )
            
          
          )
        
        
      
    
    {\displaystyle {\frac {1}{L}}\sin \left({\frac {m\pi }{L}}x\right)\cdot f(x)={\frac {1}{L}}\sin \left({\frac {m\pi }{L}}x\right)\cdot \left({\frac {A_{0}}{2}}+\sum _{n=1}^{\infty }\left(A_{n}\cos \left({\frac {n\pi }{L}}x\right)B_{n}\sin \left({\frac {n\pi }{L}}x\right)\right)\right)\,}
  
  
    
      
        
          ∫
          
            −
            L
          
          
            L
          
        
        
          
            1
            L
          
        
        sin
        ⁡
        
          (
          
            
              
                
                  m
                  π
                
                L
              
            
            x
          
          )
        
        f
        (
        x
        )
        d
        x
        =
        
          
            
              A
              
                0
              
            
            
              2
              L
            
          
        
        ⋅
        0
        +
        
          ∑
          
            n
            =
            1
          
          
            ∞
          
        
        
          (
          
            
              
                
                  A
                  
                    n
                  
                
                L
              
            
            ⋅
            0
            +
            
              B
              
                n
              
            
            
              δ
              
                m
                ,
                n
              
            
          
          )
        
      
    
    {\displaystyle \int _{-L}^{L}{\frac {1}{L}}\sin \left({\frac {m\pi }{L}}x\right)f(x)dx={\frac {A_{0}}{2L}}\cdot 0+\sum _{n=1}^{\infty }\left({\frac {A_{n}}{L}}\cdot 0+B_{n}\delta _{m,n}\right)}
  
  
    
      
        
          ∫
          
            −
            L
          
          
            L
          
        
        
          
            1
            L
          
        
        sin
        ⁡
        
          (
          
            
              
                
                  m
                  π
                
                L
              
            
            x
          
          )
        
        f
        (
        x
        )
        d
        x
        =
        
          B
          
            m
          
        
      
    
    {\displaystyle \int _{-L}^{L}{\frac {1}{L}}\sin \left({\frac {m\pi }{L}}x\right)f(x)dx=B_{m}}
  
  
    
      
        
          B
          
            n
          
        
        =
        
          ∫
          
            −
            L
          
          
            L
          
        
        
          
            1
            L
          
        
        sin
        ⁡
        
          (
          
            
              
                
                  n
                  π
                
                L
              
            
            x
          
          )
        
        f
        (
        x
        )
        d
        x
      
    
    {\displaystyle B_{n}=\int _{-L}^{L}{\frac {1}{L}}\sin \left({\frac {n\pi }{L}}x\right)f(x)dx}
  The Fourier series expansion of f(x) is now complete. To have it all in one place:

  
    
      
        f
        (
        x
        )
        =
        
          
            
              A
              
                0
              
            
            2
          
        
        +
        
          ∑
          
            n
            =
            1
          
          
            ∞
          
        
        
          (
          
            
              A
              
                n
              
            
            cos
            ⁡
            
              (
              
                
                  
                    
                      n
                      π
                    
                    L
                  
                
                x
              
              )
            
            +
            
              B
              
                n
              
            
            sin
            ⁡
            
              (
              
                
                  
                    
                      n
                      π
                    
                    L
                  
                
                x
              
              )
            
          
          )
        
      
    
    {\displaystyle f(x)={\frac {A_{0}}{2}}+\sum _{n=1}^{\infty }\left(A_{n}\cos \left({\frac {n\pi }{L}}x\right)+B_{n}\sin \left({\frac {n\pi }{L}}x\right)\right)}
  
  
    
      
        
          A
          
            n
          
        
        =
        
          ∫
          
            −
            L
          
          
            L
          
        
        
          
            1
            L
          
        
        cos
        ⁡
        
          (
          
            
              
                
                  n
                  π
                
                L
              
            
            x
          
          )
        
        f
        (
        x
        )
        d
        x
      
    
    {\displaystyle A_{n}=\int _{-L}^{L}{\frac {1}{L}}\cos \left({\frac {n\pi }{L}}x\right)f(x)dx}
  
  
    
      
        
          B
          
            n
          
        
        =
        
          ∫
          
            −
            L
          
          
            L
          
        
        
          
            1
            L
          
        
        sin
        ⁡
        
          (
          
            
              
                
                  n
                  π
                
                L
              
            
            x
          
          )
        
        f
        (
        x
        )
        d
        x
      
    
    {\displaystyle B_{n}=\int _{-L}^{L}{\frac {1}{L}}\sin \left({\frac {n\pi }{L}}x\right)f(x)dx}
  It's finally time for an example. Let's derive the Fourier series representation of a square wave, pictured at the right:
This "wave" may be quantified as f(x):

  
    
      
        f
        (
        x
        )
        =
        
          
            {
            
              
                
                  1
                  ,
                
                
                  −
                  1
                  <
                  x
                  <
                  0
                
              
              
                
                  −
                  1
                  ,
                
                
                  0
                  <
                  x
                  <
                  1
                
              
              
                
                  f
                  (
                  x
                  +
                  2
                  )
                
              
            
            
          
        
      
    
    {\displaystyle f(x)={\begin{cases}1,&-1 0, the solution has nothing in common with a Fourier series.
What's trying to be emphasized is flexibility. Knowledge of Fourier series makes it much easier to solve problems. In the parallel plate problem, knowing what a Fourier sine series is motivates the construction of the sum of un. In the end it's the problem that dictates what needs to be done. For the separable IBVPs, expansions will be a recurring nightmare theme and it is most important to be familiar and comfortable with orthogonality and its application to making sense out of infinite sums. Many functions have orthogonality properties, including Bessel functions, Legendre polynomials, and others.
The keyword is orthogonality. If an orthogonality relation exists for a given situation, then a series solution is easily possible. As an example, the diffusion equation used in the previous chapter can, with sufficiently ugly BCs, require a trigonometric series solution that is not a Fourier series (non-integer, not evenly spaced frequencies of the sinusoids). Sturm-Liouville theory rescues us in such cases, providing the right orthogonality relation.