Struve function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Struve functions Hα(x), are solutions y(x) of the non-homogeneous <a href="/facts/Bessel%27s_differential_equation/hSyvFPqC">Bessel's differential equation</a>:

x
          
            2
          
        
        
          
            
              
                d
                
                  2
                
              
              y
            
            
              d
              
                x
                
                  2
                
              
            
          
        
        +
        x
        
          
            
              d
              y
            
            
              d
              x
            
          
        
        +
        
          (
          
            
              x
              
                2
              
            
            −
            
              α
              
                2
              
            
          
          )
        
        y
        =
        
          
            
              4
              
                
                  (
                  
                    
                      x
                      2
                    
                  
                  )
                
                
                  α
                  +
                  1
                
              
            
            
              
                
                  π
                
              
              Γ
              
                (
                
                  α
                  +
                  
                    
                      1
                      2
                    
                  
                
                )
              
            
          
        
      
    
    {\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+\left(x^{2}-\alpha ^{2}\right)y={\frac {4\left({\frac {x}{2}}\right)^{\alpha +1}}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}}

introduced by <a href="/facts/Hermann_Struve/PeItKQoT">Hermann Struve</a> (1882). The <a href="/facts/Complex_number/2w7mqI7e">complex number</a> α is the order of the Struve function, and is often an integer.
And further defined its second-kind version 
 
 
 
 
 
 K
 
 
 α
 
 
 (
 x
 )
 
 
 {\displaystyle \mathbf {K} _{\alpha }(x)}
 
 as 
 
 
 
 
 
 K
 
 
 α
 
 
 (
 x
 )
 =
 
 
 H
 
 
 α
 
 
 (
 x
 )
 −
 
 Y
 
 α
 
 
 (
 x
 )
 
 
 {\displaystyle \mathbf {K} _{\alpha }(x)=\mathbf {H} _{\alpha }(x)-Y_{\alpha }(x)}
 
.
The modified Struve functions Lα(x) are equal to −ie−iαπ / 2Hα(ix) and are solutions y(x) of the non-homogeneous <a href="/facts/Bessel%27s_differential_equation/hSyvFPqC">Bessel's differential equation</a>:

x
          
            2
          
        
        
          
            
              
                d
                
                  2
                
              
              y
            
            
              d
              
                x
                
                  2
                
              
            
          
        
        +
        x
        
          
            
              d
              y
            
            
              d
              x
            
          
        
        −
        
          (
          
            
              x
              
                2
              
            
            +
            
              α
              
                2
              
            
          
          )
        
        y
        =
        
          
            
              4
              
                
                  (
                  
                    
                      x
                      2
                    
                  
                  )
                
                
                  α
                  +
                  1
                
              
            
            
              
                
                  π
                
              
              Γ
              
                (
                
                  α
                  +
                  
                    
                      1
                      2
                    
                  
                
                )
              
            
          
        
      
    
    {\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}-\left(x^{2}+\alpha ^{2}\right)y={\frac {4\left({\frac {x}{2}}\right)^{\alpha +1}}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}}

And further defined its second-kind version 
 
 
 
 
 
 M
 
 
 α
 
 
 (
 x
 )
 
 
 {\displaystyle \mathbf {M} _{\alpha }(x)}
 
 as 
 
 
 
 
 
 M
 
 
 α
 
 
 (
 x
 )
 =
 
 
 L
 
 
 α
 
 
 (
 x
 )
 −
 
 I
 
 α
 
 
 (
 x
 )
 
 
 {\displaystyle \mathbf {M} _{\alpha }(x)=\mathbf {L} _{\alpha }(x)-I_{\alpha }(x)}
 
.

Struve function open-in-new

Struve function