Bessel function

Bessel functions, named after <a href="/facts/Friedrich_Bessel/82K1lJ7t">Friedrich Bessel</a> who was the first to systematically study them in 1824, are canonical solutions y(x) of Bessel's <a href="/facts/Differential_equation/9MGbE3Ca">differential equation</a>

x
          
            2
          
        
        
          
            
              
                d
                
                  2
                
              
              y
            
            
              d
              
                x
                
                  2
                
              
            
          
        
        +
        x
        
          
            
              d
              y
            
            
              d
              x
            
          
        
        +
        
          (
          
            
              x
              
                2
              
            
            −
            
              α
              
                2
              
            
          
          )
        
        y
        =
        0
      
    
    {\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+\left(x^{2}-\alpha ^{2}\right)y=0}

for an arbitrary <a href="/facts/Complex_number/2w7mqI7e">complex number</a> 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
, which represents the order of the Bessel function. Although 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 and 
 
 
 
 −
 α
 
 
 {\displaystyle -\alpha }
 
 produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly <a href="/facts/Smooth_function/mffL4ch2">smooth functions</a> of 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
.
The most important cases are when 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 is an <a href="/facts/Integer/PUwwrawx">integer</a> or <a href="/facts/Half-integer/oJ29mf8G">half-integer</a>. Bessel functions for integer 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 are also known as cylinder functions or the <a href="/facts/Cylindrical_harmonics/9PHMAYx0">cylindrical harmonics</a> because they appear in the solution to <a href="/facts/Laplace%2527s_equation/8n1YkmFS">Laplace's equation</a> in <a href="/facts/Cylindrical_coordinates/1Vc6lhxD">cylindrical coordinates</a>. Spherical Bessel functions with half-integer 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 are obtained when solving the <a href="/facts/Helmholtz_equation/SWfaXrTB">Helmholtz equation</a> in <a href="/facts/Spherical_coordinates/5cVnNof3">spherical coordinates</a>.

Bessel function open-in-new

Bessel function