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[[Image:Maximum modulus principle.png|right|thumb|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).]]
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
In [[mathematics]], the '''maximum modulus principle''' in [[complex analysis]] states that if ''f'' is a [[holomorphic function]], then the [[absolute value|modulus]] <math>|f|</math> cannot exhibit a true [[local maximum]] that is properly within the [[domain (mathematics)|domain]] of ''f''.  


In other words, either ''f'' is a [[constant function]], or, for any point ''z''<sub>0</sub> inside the domain of ''f'' there exist other points arbitrarily close to ''z''<sub>0</sub> at which |''f'' | takes larger values.  
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
* Only registered users will be able to execute this rendering mode.
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


==Formal statement==
Registered users will be able to choose between the following three rendering modes:  
Let ''f'' be a function holomorphic on some [[connected set|connected]] [[open set|open]] [[subset]] ''D'' of the [[complex plane]] <math>\mathbb{C}</math>  and taking complex values. If ''z''<sub>0</sub> is a point in ''D'' such that
:<math>|f(z_0)|\ge |f(z)|</math>
for all ''z'' in a [[neighborhood (topology)|neighborhood]] of ''z''<sub>0</sub>, then the function ''f'' is constant on ''D''.


By switching to the [[Multiplicative_inverse|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 (topology)|boundary]] of ''D'', and non-zero at all points, then |''f'' (z)| takes its minimum value on the boundary of ''D''.
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


Alternatively, the maximum modulus principle can be viewed as a special case of the [[open mapping theorem (complex analysis)|open mapping theorem]], which states that a nonconstant holomorphic function maps open sets to open sets. If |''f''| attains a local maximum at ''z'', then the image of a sufficiently small open neighborhood of ''z'' cannot be open. Therefore, ''f'' is constant.
<!--'''PNG''' (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


==Sketches of proofs==
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


===Using the maximum principle for harmonic functions===
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
One can use the equality
:log ''f''(''z'') = ln |''f''(''z'')| + i arg ''f''(''z'') 
for complex [[natural logarithm]]s to deduce that ln |''f''(''z'')| is a [[harmonic function]]. Since ''z''<sub>0</sub> 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.


===Using Gauss's mean value theorem===
==Demos==
Another proof works by using Gauss's mean value theorem to "force" all points within overlapping open disks to assume the same value. The disks are laid such that their centers form a polygonal path from the value where ''f''(''z'') 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 ''f''(''z'') is constant.


==Physical Interpretation==
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


A physical interpretation of this principle comes from the [[heat equation]]. That is, since log |''f''(''z'')| is harmonic, it is thus the steady state of a heat flow on the region ''D''. Suppose a strict maximum was attained on the interior of ''D'', 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 ==
* accessibility:
The maximum modulus principle has many uses in complex analysis, and may be used to prove the following:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
* The [[fundamental theorem of algebra]].
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
* [[Schwarz's lemma]], a result which in turn has many generalisations and applications in complex analysis.
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
* The [[Phragmén–Lindelöf principle]], an extension to unbounded domains.
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
* The [[Borel–Carathéodory theorem]], which bounds an analytic function in terms of its real part.
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
* 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.
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


==References==
==Test pages ==
* {{cite book |first=E. C. |last=Titchmarsh |authorlink=E. C. Titchmarsh |title=The Theory of Functions |edition=2nd |year=1939 |publisher=Oxford University Press }} ''(See chapter 5.)''
* {{springer|author=E.D. Solomentsev|title=Maximum-modulus principle|id=m/m063110}}


== External links ==
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
* {{MathWorld | urlname= MaximumModulusPrinciple | title= Maximum Modulus Principle}}
*[[Displaystyle]]
* [http://math.fullerton.edu/mathews/c2003/LiouvilleMoreraGaussMod.html The Maximum Modulus Principle by John H. Mathews]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


[[Category:Mathematical principles]]
*[[Inputtypes|Inputtypes (private Wikis only)]]
[[Category:Theorems in complex analysis]]
*[[Url2Image|Url2Image (private Wikis only)]]
 
==Bug reporting==
[[de:Maximumprinzip (Mathematik)]]
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .

Latest revision as of 22:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML

E=mc2


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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