Maximum modulus principle
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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 .
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
[edit]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.
Related statement
[edit]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
[edit]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
[edit]Using the maximum principle for harmonic functions
[edit]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
[edit]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.
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
[edit]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
[edit]The maximum modulus principle has many uses in complex analysis, and may be used to prove the following:
- The fundamental theorem of algebra.
- 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.
- The Borel–Carathéodory theorem, which bounds an analytic function in terms of its real part.
- The Hadamard three-lines theorem, a result about the behaviour of bounded holomorphic functions on a line between two other parallel lines in the complex plane.
References
[edit]- ^ 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.
- Titchmarsh, E. C. (1939). The Theory of Functions (2nd ed.). Oxford University Press. (See chapter 5.)
- E. D. Solomentsev (2001) [1994], "Maximum-modulus principle", Encyclopedia of Mathematics, EMS Press
- 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.