Analyticity of holomorphic functions

<p>In <a href="/facts/Complex_analysis/hgCoQlH0">complex analysis</a>, a <a href="/facts/Complex_number/2w7mqI7e">complex</a>-valued <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> 
  
    
      
        f
      
    
    {\displaystyle f}
  
 of a complex variable 
  
    
      
        z
      
    
    {\displaystyle z}
  
:
</p>
<ul><li>is said to be <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic</a> at a point 
  
    
      
        a
      
    
    {\displaystyle a}
  
 if it is <a href="/facts/Differentiable_function/jQqTgLk1">differentiable</a> at every point within some <a href="/facts/Open_disk/DgMVr97H">open disk</a> centered at 
  
    
      
        a
      
    
    {\displaystyle a}
  
, and</li>
<li>is said to be <a href="/facts/Analytic_function/JhL3z8FP">analytic</a> at 
  
    
      
        a
      
    
    {\displaystyle a}
  
 if in some open disk centered at 
  
    
      
        a
      
    
    {\displaystyle a}
  
 it can be expanded as a <a href="/facts/Convergent_series/DTok170z">convergent</a> <a href="/facts/Power_series/vYh6sTps">power series</a> 
  
    
      
        f
        (
        z
        )
        =
        
          ∑
          
            n
            =
            0
          
          
            ∞
          
        
        
          c
          
            n
          
        
        (
        z
        −
        a
        
          )
          
            n
          
        
      
    
    {\displaystyle f(z)=\sum _{n=0}^{\infty }c_{n}(z-a)^{n}}
  
 (this implies that the <a href="/facts/Radius_of_convergence/4Un0ANfP">radius of convergence</a> is positive).</li></ul>
<p>One of the most important theorems of complex analysis is that holomorphic functions are analytic and vice versa.  Among the corollaries of this theorem are
</p>
<ul><li>the <a href="/facts/Identity_theorem/jEu4TXXn">identity theorem</a> that two holomorphic functions that agree at every point of an <a href="/facts/Infinite_set/9rTN0xGH">infinite set</a> 
  
    
      
        S
      
    
    {\displaystyle S}
  
 with an <a href="/facts/Accumulation_point/95MykoGU">accumulation point</a> inside the <a href="/facts/Intersection/hVW6QFWD">intersection</a> of their <a href="/facts/Domain_of_a_function/FQvoeUpX">domains</a> also agree everywhere in every connected open <a href="/facts/Subset/SJGERmbA">subset</a> of their domains that contains the set 
  
    
      
        S
      
    
    {\displaystyle S}
  
, and</li>
<li>the fact that, since power series are <a href="/facts/Infinitely_differentiable/mffL4ch2">infinitely differentiable</a>, so are holomorphic functions (this is in contrast to the case of real differentiable functions), and</li>
<li>the fact that the radius of convergence is always the <a href="/facts/Distance/hbXlaEFk">distance</a> from the center 
  
    
      
        a
      
    
    {\displaystyle a}
  
 to the nearest non-removable <a href="/facts/Mathematical_singularity/Y5sMQY4I">singularity</a>; if there are no singularities (i.e., if 
  
    
      
        f
      
    
    {\displaystyle f}
  
 is an <a href="/facts/Entire_function/XNluKzu7">entire function</a>), then the radius of convergence is infinite.  Strictly speaking, this is not a corollary of the theorem but rather a by-product of the proof.</li>
<li>no <a href="/facts/Bump_function/ExdBtMHN">bump function</a> on the complex plane can be entire.  In particular, on any <a href="/facts/Connected_set/5CUTxfoA">connected</a> open subset of the complex plane, there can be no bump function defined on that set which is holomorphic on the set.  This has important ramifications for the study of <a href="/facts/Complex_manifold/DlvPnpzm">complex manifolds</a>, as it precludes the use of <a href="/facts/Partitions_of_unity/4byX17aR">partitions of unity</a>.  In contrast the partition of unity is a tool which can be used on any real manifold.</li></ul>

Analyticity of holomorphic functions open-in-new

Analyticity of holomorphic functions