page_title: Complex analysis
section_title: Complex functions
hierarchical_section_title: Complex analysis / Complex functions
context_page_description: Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.
  As a differentiable function of a complex variable is equal to the sum of its Taylor series, complex analysis is particularly concerned with analytic functions of a complex variable.
context_section_description: A complex function is a function from complex numbers to complex numbers. In other words, it is a function that has a subset of the complex numbers as a domain and the complex numbers as a codomain. Complex functions are generally supposed to have a domain that contains a nonempty open subset of the complex plane.
  For any complex function, the values  from the domain and their images  in the range may be separated into real and imaginary parts:
  where  are all real-valued.  
  In other words, a complex function  may be decomposed into 
   and 
  i.e., into two real-valued functions (, ) of two real variables (, ).
  Similarly, any complex-valued function f on an arbitrary set X can be considered as an ordered pair of two real-valued functions: (Re f, Im f) or, alternatively, as a vector-valued function from X into 
  Some properties of complex-valued functions (such as continuity) are nothing more than the corresponding properties of vector valued functions of two real variables. Other concepts of complex analysis, such as differentiability are direct generalizations of the similar concepts for real functions, but may have very different properties. In particular, every differentiable complex function is analytic (see next section), and two differentiable functions that are equal in a neighborhood of a point are equal on the intersection of their domain (if the domains are connected). The latter property is the basis of the principle of analytic continuation which allows extending every real analytic function in a unique way for getting a complex analytic function whose domain is the whole complex plane with a finite number of curve arcs removed. Many basic and special complex functions are defined in this way, including exponential functions, logarithmic functions, and trigonometric functions.
caption: An exponential function An of a discrete (integer) variable n, similar to geometric progression