Frobenius method

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the method of Frobenius, named after <a href="/facts/Ferdinand_Georg_Frobenius/4cXXQyu5">Ferdinand Georg Frobenius</a>, is a way to find an <a href="/facts/Infinite_series/yDnlsgIe">infinite series</a> solution for a linear second-order <a href="/facts/Ordinary_differential_equation/ygKGQ8kD">ordinary differential equation</a> of the form

z
          
            2
          
        
        
          u
          ″
        
        +
        p
        (
        z
        )
        z
        
          u
          ′
        
        +
        q
        (
        z
        )
        u
        =
        0
      
    
    {\displaystyle z^{2}u''+p(z)zu'+q(z)u=0}

with 
 
 
 
 
 u
 ′
 
 ≡
 
 
 
 d
 u
 
 
 d
 z
 
 
 
 
 
 {\textstyle u'\equiv {\frac {du}{dz}}}
 
 and 
 
 
 
 
 u
 ″
 
 ≡
 
 
 
 
 d
 
 2
 
 
 u
 
 
 d
 
 z
 
 2
 
 
 
 
 
 
 
 {\textstyle u''\equiv {\frac {d^{2}u}{dz^{2}}}}
 
.
in the vicinity of the <a href="/facts/Regular_singular_point/exnS3T3l">regular singular point</a> 
 
 
 
 z
 =
 0
 
 
 {\displaystyle z=0}
 
. 
One can divide by 
 
 
 
 
 z
 
 2
 
 
 
 
 {\displaystyle z^{2}}
 
 to obtain a differential equation of the form

u
          ″
        
        +
        
          
            
              p
              (
              z
              )
            
            z
          
        
        
          u
          ′
        
        +
        
          
            
              q
              (
              z
              )
            
            
              z
              
                2
              
            
          
        
        u
        =
        0
      
    
    {\displaystyle u''+{\frac {p(z)}{z}}u'+{\frac {q(z)}{z^{2}}}u=0}

which will not be solvable with regular <a href="/facts/Power_series_solution_of_differential_equations/1w2StG6o">power series methods</a> if either p(z)/z or q(z)/z2 is not <a href="/facts/Analytic_function/JhL3z8FP">analytic</a> at z = 0. The Frobenius method enables one to create a power series solution to such a differential equation, provided that p(z) and q(z) are themselves analytic at 0 or, being analytic elsewhere, both their limits at 0 exist (and are finite).

Frobenius method open-in-new

Frobenius method