A plot of the modulus of cos(z) (in red) for z in the unit disk centered at the origin (shown in blue). As predicted by the theorem, the maximum of the modulus cannot be inside of the disk (so the highest value on the red surface is somewhere along its edge).

In mathematics, the maximum modulus principle in complex analysis states that if f is a holomorphic function, then the modulus | f | cannot exhibit a true local maximum that is properly within the domain of f. Image File history File links Size of this preview: 569 × 599 pixelsFull resolution (675 × 711 pixel, file size: 30 KB, MIME type: image/png) function main() % the number of data points. ... Image File history File links Size of this preview: 569 × 599 pixelsFull resolution (675 × 711 pixel, file size: 30 KB, MIME type: image/png) function main() % the number of data points. ... In mathematics, the open unit disk around P (where P is a given point in the plane), is the set of points whose distance from P is less than 1: The closed unit disk around P is the set of points whose distance from P is less than or equal... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... A graph illustrating local min/max and global min/max points In mathematics, a point x* is a local maximum of a function f if there exists some &#949; > 0 such that f(x*) &#8805; f(x) for all x with |x-x*| < &#949;. Stated less formally, a local maximum... In mathematics, the domain of a function is the set of all input values to the function. ...

In other words, either f is a constant function, or, for any point z₀ inside the domain of f there exist other points arbitrarily close to z₀ at which |f | takes larger values. In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ...

Formal statement

Let f be a function holomorphic on some connected open subset D of the complex plane C and taking complex values. If z₀ is a point in D such that In topology and related branches of mathematics, a connected space is a topological space which cannot be written as the disjoint union of two or more nonempty spaces. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... “Superset” redirects here. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...

for all z in a neighborhood of z₀, then the function f is constant on D. This is a glossary of some terms used in the branch of mathematics known as topology. ...

Sketch of the proof

One uses the equality

log f(z) = log |f(z)| + i arg f(z)

for complex natural logarithms to deduce that log |f(z)| is a harmonic function. Since z₀ is a local maximum for this function also, it follows from the maximum principle that |f(z)| is constant. Then, using the Cauchy-Riemann equations we show that f'(z)=0, and thus that f(z) is constant as well. The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2. ... In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R (where U is an open subset of Rn) which satisfies Laplaces equation, i. ... In mathematics, the maximum principle in harmonic analysis states that if f is a harmonic function, then f cannot exhibit a true local maximum within the domain of definition of f. ... In mathematics, the Cauchy-Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, are two partial differential equations which provide a necessary but not sufficient condition for a function to be holomorphic. ...

By switching to the reciprocal, we can get the minimum modulus principle. It states that if f is holomorphic within a bounded domain D, continuous up to the boundary of D, and non-zero at all points, then the modulus |f (z)| takes its minimum value on the boundary of D. The reciprocal function: y = 1/x. ... In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure of S, not belonging to the interior of...

Alternatively, the maximum modulus principle can be viewed as a special case of the open mapping theorem, which states that a holomorphic function maps open sets to open sets. If |f| attains a local maximum at a, then clearly the direct image of sufficiently small open neighborhoods of a cannot be open. Therefore, f is constant. In mathematics, there are two theorems with the name open mapping theorem. In functional analysis, the open mapping theorem, also known as the Banach-Schauder theorem, is a fundamental result which states: if A : X → Y is a surjective continuous linear operator between Banach spaces X and Y, then A...

Applications

The maximum modulus principle has many uses in complex analysis, and may be used to prove the following:

• The fundamental theorem of algebra, as may be seen in the classic text "Introduction to Complex Analysis", by Nevanlinna and Paatero.
• Schwarz's lemma, a result which in turn has many generalisations and applications in complex analysis.
• The Phragmén-Lindelöf principle, an extension to unbounded domains.

In mathematics, the fundamental theorem of algebra states that every complex polynomial in one variable and of degree  â‰¥  has some complex root. ... In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions defined on the open unit disk. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

References

• E.C. Titchmarsh, The Theory of Functions (2nd Ed) (1939) Oxford University Press. (See chapter 5.)
• E.D. Solomentsev (2001), "Maximum-modulus principle", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 1-55608-010-7

Edward Charles (Ted) Titchmarsh (born 1 June 1899 in Newbury died 18 January 1963 at Oxford) was a leading British mathematician. ... The Encyclopaedia of Mathematics is a large reference work in mathematics. ...

External links

• Eric W. Weisstein, Maximum Modulus Principle at MathWorld.
• The Maximum Modulus Principle by John H. Mathews