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Maximum modulus principle

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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 .

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:

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.