In
complex analysis, a subfield of
mathematics, a lacunary value or gap of a
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
-valued
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
defined on a subset of the
complex plane is a complex number which is not in the
image of the function.
[.]
More specifically, given a subset ''X'' of the complex plane C and a function ''f'' : ''X'' → C, a complex number ''z'' is called a ''lacunary value'' of ''f'' if ''z'' ∉ image(''f'').
Note, for example, that 0 is the only lacunary value of the complex
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
. The two
Picard theorem
In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard.
The theorems
Little Picard Theorem: If a function f: \mathbb \to\mathbb ...
s limit the number of possible lacunary values of certain types of
holomorphic functions.
References
Complex analysis
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