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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. ## Formal statement

Let f be a function holomorphic on some connected open subset D of the complex plane C and taking complex values. If z_{0} is a point in D such that
_{0}, then the function f is constant on D.
## Sketch of the proof

_{0} 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. ## Applications

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

## External links

In other words, either f is a constant function, or, for any point z_{0} inside the domain of f there exist other points arbitrarily close to z_{0} at which |f | takes larger values.

- $|f(z\_0)|ge\; |f(z)|$

One uses the equality

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

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.

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.

- 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.

- E.C. Titchmarsh, The Theory of Functions (2nd Ed) (1939) Oxford University Press. (See chapter 5.)

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This article is licensed under the GNU Free Documentation License.

Last updated on Thursday September 25, 2008 at 04:16:13 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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