complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

, the open mapping theorem states that if ''U'' is a

domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...

of the

complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...

C and ''f'' : ''U'' → C is a non-constant

holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...

, then ''f'' is an

open map In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, a ...

(i.e. it sends open subsets of ''U'' to open subsets of C, and we have

invariance of domain Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space \R^n. It states: :If U is an open subset of \R^n and f : U \rarr \R^n is an injective continuous map, then V := f(U) is open in \R^n and f is a homeomorph ...

.). The open mapping theorem points to the sharp difference between holomorphy and real-differentiability. On the

real line In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...

, for example, the differentiable function ''f''(''x'') = ''x''² is not an open map, as the image of the

open interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...

(−1, 1) is the half-open interval [0, 1). The theorem for example implies that a non-constant

cannot map an open disk ''onto'' a portion of any line embedded in the complex plane. Images of holomorphic functions can be of real dimension zero (if constant) or two (if non-constant) but never of dimension 1.

Proof

Assume ''f'' : ''U'' → C is a non-constant holomorphic function and ''U'' is a

of the complex plane. We have to show that every

point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...

in ''f''(''U'') is an

interior point In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of t ...

of ''f''(''U''), i.e. that every point in ''f''(''U'') has a neighborhood (open disk) which is also in ''f''(''U''). Consider an arbitrary ''w''₀ in ''f''(''U''). Then there exists a point ''z''₀ in ''U'' such that ''w''₀ = ''f''(''z''₀). Since ''U'' is open, we can find ''d'' > 0 such that the closed disk ''B'' around ''z''₀ with radius ''d'' is fully contained in ''U''. Consider the function ''g''(''z'') = ''f''(''z'')−''w''₀. Note that ''z''₀ is a

root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...

of the function. We know that ''g''(''z'') is not constant and holomorphic. The roots of ''g'' are isolated by the

identity theorem In real analysis and complex analysis, branches of mathematics, the identity theorem for analytic functions states: given functions ''f'' and ''g'' analytic on a domain ''D'' (open and connected subset of \mathbb or \mathbb), if ''f'' = ''g'' on so ...

, and by further decreasing the radius of the image disk ''d'', we can assure that ''g''(''z'') has only a single root in ''B'' (although this single root may have multiplicity greater than 1). The boundary of ''B'' is a circle and hence a

compact set In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i. ...

, on which , ''g''(''z''), is a positive

continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...

, so the

extreme value theorem In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed interval ,b/math>, then f must attain a maximum and a minimum, each at least once. That is, there exist numbers c and d in ,b/math> su ...

guarantees the existence of a positive minimum ''e'', that is, ''e'' is the minimum of , ''g''(''z''), for ''z'' on the boundary of ''B'' and ''e'' > 0. Denote by ''D'' the open disk around ''w''₀ with

radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...

''e''. By

Rouché's theorem Rouché's theorem, named after Eugène Rouché, states that for any two complex-valued functions and holomorphic inside some region K with closed contour \partial K, if on \partial K, then and have the same number of zeros inside K, wher ...

, the function ''g''(''z'') = ''f''(''z'')−''w''₀ will have the same number of roots (counted with multiplicity) in ''B'' as ''h''(''z''):=''f''(''z'')−''w₁'' for any ''w₁'' in ''D''. This is because ''h''(''z'') = ''g''(''z'') + (''w''₀ - ''w''₁), and for ''z'' on the boundary of ''B'', , ''g''(''z''), ≥ ''e'' > , ''w''₀ - ''w''₁, . Thus, for every ''w''₁ in ''D'', there exists at least one ''z''₁ in ''B'' such that ''f''(''z''₁) = ''w₁''. This means that the disk ''D'' is contained in ''f''(''B''). The image of the ball ''B'', ''f''(''B'') is a subset of the image of ''U'', ''f''(''U''). Thus ''w''₀ is an interior point of ''f''(''U''). Since ''w''₀ was arbitrary in ''f''(''U'') we know that ''f''(''U'') is open. Since ''U'' was arbitrary, the function ''f'' is open.

Applications

Maximum modulus principle In mathematics, the maximum modulus principle in complex analysis states that if ''f'' is a holomorphic function, then the modulus , ''f'' , cannot exhibit a strict local maximum that is properly within the domain of ''f''. In other words, eit ...

Schwarz lemma In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than deeper theorems, such as the Riemann mapping ...

References

* {{citation, first=Walter, last=Rudin, authorlink=Walter Rudin, title=Real & Complex Analysis, publisher=McGraw-Hill, year=1966, isbn=0-07-054234-1 Theorems in complex analysis Articles containing proofs

Proof

Applications

See also

References