section_path: Complex analysis/Holomorphic functions
section_name: Holomorphic functions
paragraph_text: Holomorphic functions
  
  A function f: C {\displaystyle f:\mathbb {C} } → C {\displaystyle \mathbb {C} } is called holomorphic or complex differentiable at a point z 0 {\displaystyle z_{0}} if the limit
  
  exists (in which case it is denoted by f ′ (z 0) {\displaystyle f'(z_{0})}). This mimics the definition for real differentiable functions, except that all quantities are complex numbers. Loosely speaking, the freedom of approaching z 0 {\displaystyle z_{0}} in different directions imposes a much stronger condition than being (real) differentiable. For example, the function
  
  is differentiable as a function R 2 → R 2 {\displaystyle \mathbb {R} ^{2}\to \mathbb {R} ^{2}}, but is not complex differentiable.A real differentiable function is complex differentiable if and only if it satisfies the Cauchy–Riemann equations, which are sometimes abbreviated as
  
  Complex analysis shows some features not apparent in real analysis. For example, the identity theorem asserts that two holomorphic functions f and g agree if they agree on an arbitrarily small open subset of C {\displaystyle \mathbb {C} }. Meromorphic functions, functions that can locally be written as f (z)/(z − z) with a holomorphic function f, still share some of the features of holomorphic functions. Other functions have essential singularities, such as sin(1/ z) at z = 0.