Analytic function

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an analytic function is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> that is locally given by a <a href="/facts/Convergent_series/DTok170z">convergent</a> <a href="/facts/Power_series/vYh6sTps">power series</a>. There exist both real analytic functions and complex analytic functions. Functions of each type are <a href="/facts/Smooth_function/mffL4ch2">infinitely differentiable</a>, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. 
</p><p>A function is analytic if and only if for every 
  
    
      
        
          x
          
            0
          
        
      
    
    {\displaystyle x_{0}}
  
 in its <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a>, its <a href="/facts/Taylor_series/M0aTuftH">Taylor series</a> about 
  
    
      
        
          x
          
            0
          
        
      
    
    {\displaystyle x_{0}}
  
 converges to the function in some <a href="/facts/Neighborhood_(topology)/XHKgrpkp">neighborhood</a> of 
  
    
      
        
          x
          
            0
          
        
      
    
    {\displaystyle x_{0}}
  
. This is stronger than merely being <a href="/facts/Smoothness/mffL4ch2">infinitely differentiable</a> at 
  
    
      
        
          x
          
            0
          
        
      
    
    {\displaystyle x_{0}}
  
, and therefore having a well-defined Taylor series; the <a href="/facts/Fabius_function/EAKqfvoj">Fabius function</a> provides an example of a function that is infinitely differentiable but not analytic.
</p>

Analytic function open-in-new

Analytic function