Removable singularity

In <a href="/facts/Complex_analysis/hgCoQlH0">complex analysis</a>, a removable singularity of a <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic function</a> is a point at which the function is <a href="/facts/Undefined_(mathematics)/Bc1Q3px4">undefined</a>, but it is possible to redefine the function at that point in such a way that the resulting function is <a href="/facts/Analytic_function/JhL3z8FP">regular</a> in a <a href="/facts/Neighbourhood_(mathematics)/XHKgrpkp">neighbourhood</a> of that point.
For instance, the (unnormalized) <a href="/facts/Sinc_function/lSvDtHy0">sinc function</a>, as defined by

sinc
        
        (
        z
        )
        =
        
          
            
              sin
              ⁡
              z
            
            z
          
        
      
    
    {\displaystyle {\text{sinc}}(z)={\frac {\sin z}{z}}}

has a singularity at z = 0. This singularity can be removed by defining 
 
 
 
 
 sinc
 
 (
 0
 )
 :=
 1
 ,
 
 
 {\displaystyle {\text{sinc}}(0):=1,}
 
 which is the <a href="/facts/Limit_of_a_function/HFbQ69e5">limit</a> of sinc as z tends to 0. The resulting function is holomorphic. In this case the problem was caused by sinc being given an <a href="/facts/Indeterminate_form/5Y3JWKjS">indeterminate form</a>. Taking a <a href="/facts/Power_series/vYh6sTps">power series</a> expansion for 
 
 
 
 
 
 
 sin
 ⁡
 (
 z
 )
 
 z
 
 
 
 
 {\textstyle {\frac {\sin(z)}{z}}}
 
 around the singular point shows that

sinc
        
        (
        z
        )
        =
        
          
            1
            z
          
        
        
          (
          
            
              ∑
              
                k
                =
                0
              
              
                ∞
              
            
            
              
                
                  (
                  −
                  1
                  
                    )
                    
                      k
                    
                  
                  
                    z
                    
                      2
                      k
                      +
                      1
                    
                  
                
                
                  (
                  2
                  k
                  +
                  1
                  )
                  !
                
              
            
          
          )
        
        =
        
          ∑
          
            k
            =
            0
          
          
            ∞
          
        
        
          
            
              (
              −
              1
              
                )
                
                  k
                
              
              
                z
                
                  2
                  k
                
              
            
            
              (
              2
              k
              +
              1
              )
              !
            
          
        
        =
        1
        −
        
          
            
              z
              
                2
              
            
            
              3
              !
            
          
        
        +
        
          
            
              z
              
                4
              
            
            
              5
              !
            
          
        
        −
        
          
            
              z
              
                6
              
            
            
              7
              !
            
          
        
        +
        ⋯
        .
      
    
    {\displaystyle {\text{sinc}}(z)={\frac {1}{z}}\left(\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{(2k+1)!}}\right)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{(2k+1)!}}=1-{\frac {z^{2}}{3!}}+{\frac {z^{4}}{5!}}-{\frac {z^{6}}{7!}}+\cdots .}

Formally, if 
 
 
 
 U
 ⊂
 
 C
 
 
 
 {\displaystyle U\subset \mathbb {C} }
 
 is an <a href="/facts/Open_subset/QfTCIqMu">open subset</a> of the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a> 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
, 
 
 
 
 a
 ∈
 U
 
 
 {\displaystyle a\in U}
 
 a point of 
 
 
 
 U
 
 
 {\displaystyle U}
 
, and 
 
 
 
 f
 :
 U
 ∖
 {
 a
 }
 →
 
 C
 
 
 
 {\displaystyle f:U\setminus \{a\}\rightarrow \mathbb {C} }
 
 is a <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic function</a>, then 
 
 
 
 a
 
 
 {\displaystyle a}
 
 is called a removable singularity for 
 
 
 
 f
 
 
 {\displaystyle f}
 
 if there exists a holomorphic function 
 
 
 
 g
 :
 U
 →
 
 C
 
 
 
 {\displaystyle g:U\rightarrow \mathbb {C} }
 
 which coincides with 
 
 
 
 f
 
 
 {\displaystyle f}
 
 on 
 
 
 
 U
 ∖
 {
 a
 }
 
 
 {\displaystyle U\setminus \{a\}}
 
. We say 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is holomorphically extendable over 
 
 
 
 U
 
 
 {\displaystyle U}
 
 if such a 
 
 
 
 g
 
 
 {\displaystyle g}
 
 exists.

Removable singularity open-in-new

Removable singularity