Böhmer積分
在數學中，Böhmer 積分是一種特殊的積分

  
    
      
        
          C
          (
          x
          ,
          α
          )
          =
          
            ∫
            
              x
            
            
              ∞
            
          
          
            t
            
              α
              −
              1
            
          
          cos
          ⁡
          (t)
          
          d
          t
        
      
    
    {\displaystyle \displaystyle C (x,\alpha)=\int _{x}^{\infty} t^{\alpha -1}\cos (t)\,dt}
  

  
    
      
        
          S
          (
          x
          ,
          α
          )
          =
          
            ∫
            
              x
            
            
              ∞
            
          
          
            t
            
              α
              −
              1
            
          
          sin
          ⁡
          (t)
          
          d
          t
        
      
    
    {\displaystyle \displaystyle S (x,\alpha)=\int _{x}^{\infty} t^{\alpha -1}\sin (t)\,dt}
  因此，菲涅耳積分可以用 Böhmer 積分表示為

  
    
      
        S
        ⁡
        (y)
        =
        
          
            1
            2
          
        
        −
        
          
            1
            
              2
              π
            
          
        
        ⋅
        S
        ⁡
        
          (
          
            
              
                1
                2
              
            
            ,
            
              y
              
                2
              
            
          
          )
        
      
    
    {\displaystyle \operatorname {S} (y)={\frac {1}{2}}-{\frac {1}{\sqrt {2\pi}}}\cdot \operatorname {S} \left ({\frac {1}{2}},y^{2}\right)}
  

  
    
      
        C
        ⁡
        (y)
        =
        
          
            1
            2
          
        
        −
        
          
            1
            
              2
              π
            
          
        
        ⋅
        C
        ⁡
        
          (
          
            
              
                1
                2
              
            
            ,
            
              y
              
                2
              
            
          
          )
        
      
    
    {\displaystyle \operatorname {C} (y)={\frac {1}{2}}-{\frac {1}{\sqrt {2\pi}}}\cdot \operatorname {C} \left ({\frac {1}{2}},y^{2}\right)}
  正弦積分和餘弦積分也可以用 Böhmer 積分表示

  
    
      
        Si
        ⁡
        (x)
        =
        
          
            π
            2
          
        
        −
        S
        ⁡
        (
        x
        ,
        0
        )
      
    
    {\displaystyle \operatorname {Si} (x)={\frac {\pi}{2}}-\operatorname {S} (x,0)}
  

  
    
      
        Ci
        ⁡
        (x)
        =
        
          
            π
            2
          
        
        −
        C
        ⁡
        (
        x
        ,
        0
        )
      
    
    {\displaystyle \operatorname {Ci} (x)={\frac {\pi}{2}}-\operatorname {C} (x,0)}