Isolated singularity

<h2 id="examples">Examples</h2>
<ul><li>The function 
  
    
      
        
          
            1
            z
          
        
      
    
    {\displaystyle {\frac {1}{z}}}
  
 has 0 as an isolated singularity.</li>
<li>The <a href="/facts/Cosecant/czej7ur0">cosecant</a> function 
  
    
      
        csc
        ⁡
        
          (
          
            π
            z
          
          )
        
      
    
    {\displaystyle \csc \left(\pi z\right)}
  
 has every <a href="/facts/Integer/PUwwrawx">integer</a> as an isolated singularity.</li></ul>
<h2 id="nonisolated-singularities">Nonisolated singularities</h2>
<p>Other than isolated singularities, complex functions of one variable may exhibit other singular behavior. Namely, two kinds of nonisolated singularities exist:
</p>
<ul><li>Cluster points, i.e. <a href="/facts/Limit_points/95MykoGU">limit points</a> of isolated singularities: if they are all poles, despite admitting <a href="/facts/Laurent_series/sSMevFU8">Laurent series</a> expansions on each of them, no such expansion is possible at its limit.</li>
<li>Natural boundaries, i.e. any non-isolated set (e.g. a curve) around which functions cannot be <a href="/facts/Analytic_continuation/aBCuEgvt">analytically continued</a> (or outside them if they are closed curves in the <a href="/facts/Riemann_sphere/4XgBzplC">Riemann sphere</a>).</li></ul>
<h3>Examples</h3>

<ul><li>The function 
  
    
      
        tan
        ⁡
        
          (
          
            
              1
              z
            
          
          )
        
      
    
    {\textstyle \tan \left({\frac {1}{z}}\right)}
  
 is <a href="/facts/Meromorphic/LeNBd6Sh">meromorphic</a> on 
  
    
      
        
          C
        
        ∖
        {
        0
        }
      
    
    {\displaystyle \mathbb {C} \setminus \{0\}}
  
, with simple poles at 
  
    
      
        
          z
          
            n
          
        
        =
        
          
            (
            
              
                
                  π
                  2
                
              
              +
              n
              π
            
            )
          
          
            −
            1
          
        
      
    
    {\textstyle z_{n}=\left({\frac {\pi }{2}}+n\pi \right)^{-1}}
  
, for every 
  
    
      
        n
        ∈
        
          
            N
          
          
            0
          
        
      
    
    {\displaystyle n\in \mathbb {N} _{0}}
  
. Since 
  
    
      
        
          z
          
            n
          
        
        →
        0
      
    
    {\displaystyle z_{n}\rightarrow 0}
  
, every punctured disk centered at 
  
    
      
        0
      
    
    {\displaystyle 0}
  
 has an infinite number of singularities within, so no Laurent expansion is available for 
  
    
      
        tan
        ⁡
        
          (
          
            
              1
              z
            
          
          )
        
      
    
    {\textstyle \tan \left({\frac {1}{z}}\right)}
  
 around 
  
    
      
        0
      
    
    {\displaystyle 0}
  
, which is in fact a cluster point of its poles.</li>
<li>The function 
  
    
      
        csc
        ⁡
        
          (
          
            
              π
              z
            
          
          )
        
      
    
    {\textstyle \csc \left({\frac {\pi }{z}}\right)}
  
 has a singularity at 0 which is <i>not</i> isolated, since there are additional singularities at the <a href="/facts/Multiplicative_inverse/yczdYyCw">reciprocal</a> of every <a href="/facts/Integer/PUwwrawx">integer</a>, which are located arbitrarily close to 0 (though the singularities at these reciprocals are themselves isolated).</li>
<li>The function defined via the <a href="/facts/Maclaurin_series/M0aTuftH">Maclaurin series</a> 
  
    
      
        
          ∑
          
            n
            =
            0
          
          
            ∞
          
        
        
          z
          
            
              2
              
                n
              
            
          
        
      
    
    {\textstyle \sum _{n=0}^{\infty }z^{2^{n}}}
  
 converges inside the open unit disk centred at 
  
    
      
        0
      
    
    {\displaystyle 0}
  
 and has the unit circle as its natural boundary.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="/facts/Lars_Ahlfors/aukXlC2t">Ahlfors, L.</a>, <i>Complex Analysis, 3 ed.</i> (McGraw-Hill, 1979).</li>
<li><a href="/facts/Walter_Rudin/9mxRAECY">Rudin, W.</a>, <i>Real and Complex Analysis, 3 ed.</i> (McGraw-Hill, 1986).</li>
<li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/Singularity.html">"Singularity"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul>

Isolated singularity open-in-new

Isolated singularity