Maximum modulus principle

Last updated
A plot of the modulus of
cos
[?]
(
z
)
{\displaystyle \cos(z)}
(in red) for
z
{\displaystyle 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). Maximum modulus principle.png
A plot of the modulus of (in red) for 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 is a holomorphic function, then the modulus cannot exhibit a strict maximum that is strictly within the domain of .

Contents

In other words, either is locally a constant function, or, for any point inside the domain of there exist other points arbitrarily close to at which takes larger values.

Formal statement

Let be a holomorphic function on some connected open subset of the complex plane and taking complex values. If is a point in such that

for all in some neighborhood of , then is constant on .

This statement can be viewed as a special case of the open mapping theorem, which states that a nonconstant holomorphic function maps open sets to open sets: If attains a local maximum at , then the image of a sufficiently small open neighborhood of cannot be open, so is constant.

Suppose that is a bounded nonempty connected open subset of . Let be the closure of . Suppose that is a continuous function that is holomorphic on . Then attains a maximum at some point of the boundary of .

This follows from the first version as follows. Since is compact and nonempty, the continuous function attains a maximum at some point of . If is not on the boundary, then the maximum modulus principle implies that is constant, so also attains the same maximum at any point of the boundary.

Minimum modulus principle

For a holomorphic function on a connected open set of , if is a point in such that

for all in some neighborhood of , then is constant on .

Proof: Apply the maximum modulus principle to .

Sketches of proofs

Using the maximum principle for harmonic functions

One can use the equality

for complex natural logarithms to deduce that is a harmonic function. Since is a local maximum for this function also, it follows from the maximum principle that is constant. Then, using the Cauchy–Riemann equations we show that = 0, and thus that is constant as well. Similar reasoning shows that can only have a local minimum (which necessarily has value 0) at an isolated zero of .

Using Gauss's mean value theorem

Another proof works by using Gauss's mean value theorem to "force" all points within overlapping open disks to assume the same value as the maximum. The disks are laid such that their centers form a polygonal path from the value where is maximized to any other point in the domain, while being totally contained within the domain. Thus the existence of a maximum value implies that all the values in the domain are the same, thus is constant.

Using Cauchy's Integral Formula [1]

As is open, there exists (a closed ball centered at with radius ) such that . We then define the boundary of the closed ball with positive orientation as . Invoking Cauchy's integral formula, we obtain

For all , , so . This also holds for all balls of radius less than centered at . Therefore, for all .

Now consider the constant function for all . Then one can construct a sequence of distinct points located in where the holomorphic function vanishes. As is closed, the sequence converges to some point in . This means vanishes everywhere in which implies for all .

Physical interpretation

A physical interpretation of this principle comes from the heat equation. That is, since is harmonic, it is thus the steady state of a heat flow on the region . Suppose a strict maximum was attained on the interior of , the heat at this maximum would be dispersing to the points around it, which would contradict the assumption that this represents the steady state of a system.

Applications

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

Related Research Articles

<span class="mw-page-title-main">Complex analysis</span> Branch of mathematics studying functions of a complex variable

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 helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.

<span class="mw-page-title-main">Riemann mapping theorem</span> Mathematical theorem

In complex analysis, the Riemann mapping theorem states that if is a non-empty simply connected open subset of the complex number plane which is not all of , then there exists a biholomorphic mapping from onto the open unit disk

<span class="mw-page-title-main">Harmonic function</span> Functions in mathematics

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function where U is an open subset of that satisfies Laplace's equation, that is,

<span class="mw-page-title-main">Cauchy's integral theorem</span> Theorem in complex analysis

In mathematics, the Cauchy integral theorem in complex analysis, named after Augustin-Louis Cauchy, is an important statement about line integrals for holomorphic functions in the complex plane. Essentially, it says that if is holomorphic in a simply connected domain Ω, then for any simply closed contour in Ω, that contour integral is zero.

<span class="mw-page-title-main">Analytic function</span> Type of function in mathematics

In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions.

<span class="mw-page-title-main">Complex conjugate</span> Fundamental operation on complex numbers

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if and are real numbers then the complex conjugate of is The complex conjugate of is often denoted as or .

In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its derivative is continuous and non-zero at the point. The theorem also gives a formula for the derivative of the inverse function. In multivariable calculus, this theorem can be generalized to any continuously differentiable, vector-valued function whose Jacobian determinant is nonzero at a point in its domain, giving a formula for the Jacobian matrix of the inverse. There are also versions of the inverse function theorem for complex holomorphic functions, for differentiable maps between manifolds, for differentiable functions between Banach spaces, and so forth.

In complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function for which there exists a positive number such that for all is constant. Equivalently, non-constant holomorphic functions on have unbounded images.

The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space, that is, n-tuples of complex numbers. The name of the field dealing with the properties of these functions is called several complex variables, which the Mathematics Subject Classification has as a top-level heading.

In mathematics, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform. It is named after Raymond Paley (1907–1933) and Norbert Wiener (1894–1964) who, in 1934, introduced various versions of the theorem. The original theorems did not use the language of distributions, and instead applied to square-integrable functions. The first such theorem using distributions was due to Laurent Schwartz. These theorems heavily rely on the triangle inequality.

In complex analysis, the Phragmén–Lindelöf principle, first formulated by Lars Edvard Phragmén (1863–1937) and Ernst Leonard Lindelöf (1870–1946) in 1908, is a technique which employs an auxiliary, parameterized function to prove the boundedness of a holomorphic function on an unbounded domain when an additional condition constraining the growth of on is given. It is a generalization of the maximum modulus principle, which is only applicable to bounded domains.

<span class="mw-page-title-main">Schwarz lemma</span> Statement in complex analysis

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 theorem, which it helps to prove. It is, however, one of the simplest results capturing the rigidity of holomorphic functions.

In mathematics, Carathéodory's theorem is a theorem in complex analysis, named after Constantin Carathéodory, which extends the Riemann mapping theorem. The theorem, published by Carathéodory in 1913, states that any conformal mapping sending the unit disk to some region in the complex plane bounded by a Jordan curve extends continuously to a homeomorphism from the unit circle onto the Jordan curve. The result is one of Carathéodory's results on prime ends and the boundary behaviour of univalent holomorphic functions.

In mathematics, plurisubharmonic functions form an important class of functions used in complex analysis. On a Kähler manifold, plurisubharmonic functions form a subset of the subharmonic functions. However, unlike subharmonic functions plurisubharmonic functions can be defined in full generality on complex analytic spaces.

In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory.

<span class="mw-page-title-main">Complex logarithm</span> Logarithm of a complex number

In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related:

Geometric function theory is the study of geometric properties of analytic functions. A fundamental result in the theory is the Riemann mapping theorem.

<span class="mw-page-title-main">Antiderivative (complex analysis)</span> Concept in complex analysis

In complex analysis, a branch of mathematics, the antiderivative, or primitive, of a complex-valued function g is a function whose complex derivative is g. More precisely, given an open set in the complex plane and a function the antiderivative of is a function that satisfies .

In mathematics, Lindelöf's theorem is a result in complex analysis named after the Finnish mathematician Ernst Leonard Lindelöf. It states that a holomorphic function on a half-strip in the complex plane that is bounded on the boundary of the strip and does not grow "too fast" in the unbounded direction of the strip must remain bounded on the whole strip. The result is useful in the study of the Riemann zeta function, and is a special case of the Phragmén–Lindelöf principle. Also, see Hadamard three-lines theorem.

In complex analysis, given initial data consisting of points in the complex unit disc and target data consisting of points in , the Nevanlinna–Pick interpolation problem is to find a holomorphic function that interpolates the data, that is for all ,

References

  1. Conway, John B. (1978). Axler, S.; Gehring, F.W.; Ribet, K.A. (eds.). Functions of One Complex Variable I (2 ed.). New York: Springer Science+Business Media, Inc. ISBN   978-1-4612-6314-2.