Trigonometric series

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, trigonometric series are a special class of <a href="/facts/Orthogonality_(mathematics)/eKmrl6oQ">orthogonal</a> <a href="/facts/Series_(mathematics)/yDnlsgIe">series</a> of the form

A
          
            0
          
        
        +
        
          ∑
          
            n
            =
            1
          
          
            ∞
          
        
        
          A
          
            n
          
        
        cos
        ⁡
        
          (
          n
          x
          )
        
        +
        
          B
          
            n
          
        
        sin
        ⁡
        
          (
          n
          x
          )
        
        ,
      
    
    {\displaystyle A_{0}+\sum _{n=1}^{\infty }A_{n}\cos {(nx)}+B_{n}\sin {(nx)},}

where 
 
 
 
 x
 
 
 {\displaystyle x}
 
 is the variable and 
 
 
 
 {
 
 A
 
 n
 
 
 }
 
 
 {\displaystyle \{A_{n}\}}
 
 and 
 
 
 
 {
 
 B
 
 n
 
 
 }
 
 
 {\displaystyle \{B_{n}\}}
 
 are <a href="/facts/Coefficient/5Hwq2Wuq">coefficients</a>. It is an infinite version of a <a href="/facts/Trigonometric_polynomial/K2FEr7Ic">trigonometric polynomial</a>.
A trigonometric series is called the <a href="/facts/Fourier_series/s3fsxPAT">Fourier series</a> of the <a href="/facts/Integrable_function/V7lyV997">integrable function</a> 
 
 
 
 f
 
 
 {\textstyle f}
 
 if the coefficients have the form:

A
          
            n
          
        
        =
        
          
            1
            π
          
        
        
          ∫
          
            0
          
          
            2
            π
          
        
        
        f
        (
        x
        )
        cos
        ⁡
        
          (
          n
          x
          )
        
        
        d
        x
      
    
    {\displaystyle A_{n}={\frac {1}{\pi }}\int _{0}^{2\pi }\!f(x)\cos {(nx)}\,dx}

B
          
            n
          
        
        =
        
          
            1
            π
          
        
        
          
            ∫
            
              0
            
            
              2
              π
            
          
          
          f
          (
          x
          )
          sin
          ⁡
          
            (
            n
            x
            )
          
          
          d
          x
        
      
    
    {\displaystyle B_{n}={\frac {1}{\pi }}\displaystyle \int _{0}^{2\pi }\!f(x)\sin {(nx)}\,dx}

Trigonometric series open-in-new

Trigonometric series