Chiral anomaly: Difference between revisions
Jump to navigation
Jump to search
en>TakuyaMurata In physics |
en>Monkbot |
||
| Line 1: | Line 1: | ||
{{About|Liouville's theorem in complex analysis||Liouville's theorem (disambiguation)}} | |||
In [[complex analysis]], '''Liouville's theorem''', named after [[Joseph Liouville]], states that every [[bounded function|bounded]] [[entire function]] must be constant. That is, every [[holomorphic]] function ''f'' for which there exists a positive number ''M'' such that |''f''(''z'')| ≤ ''M'' for all ''z'' in '''C''' is constant. | |||
The theorem is considerably improved by [[Picard theorem|Picard's little theorem]], which says that every entire function whose image omits at least two complex numbers must be constant. | |||
== Proof == | |||
The theorem follows from the fact that [[Proof that holomorphic functions are analytic|holomorphic functions are analytic]]. If ''f'' is an entire function, it can be represented by its [[Taylor series]] about 0: | |||
: <math>f(z) = \sum_{k=0}^\infty a_k z^k</math> | |||
where (by [[Cauchy's integral formula]]) | |||
:<math> | |||
a_k = \frac{f^{(k)}(0)}{k!} = {1 \over 2 \pi i} \oint_{C_r} | |||
\frac{f( \zeta )}{\zeta^{k+1}}\,d\zeta | |||
</math> | |||
and ''C''<sub>''r''</sub> is the circle about 0 of radius ''r'' > 0. Suppose ''f'' is bounded: i.e. there exists a constant ''M'' such that |''f''(''z'')| ≤ ''M'' for all ''z''. We can estimate directly | |||
:<math> | |||
| a_k | | |||
\le \frac{1}{2 \pi} \oint_{C_r} \frac{ | f ( \zeta ) | }{ | \zeta |^{k+1} } \, |d\zeta| | |||
\le \frac{1}{2 \pi} \oint_{C_r} \frac{ M }{ r^{k+1} } \, |d\zeta| | |||
= \frac{M}{2 \pi r^{k+1}} \oint_{C_r} |d\zeta| | |||
= \frac{M}{2 \pi r^{k+1}} 2 \pi r | |||
= \frac{M}{r^k}, | |||
</math> | |||
where in the second inequality we have used the fact that |''z''|=''r'' on the circle ''C''<sub>''r''</sub>. But the choice of ''r'' in the above is an arbitrary positive number. Therefore, letting ''r'' tend to infinity (we let ''r'' tend to infinity since f is analytic on the entire plane) gives ''a''<sub>''k''</sub> = 0 for all ''k'' ≥ 1. Thus ''f''(''z'') = ''a''<sub>0</sub> and this proves the theorem. | |||
== Corollaries == | |||
===Fundamental theorem of algebra=== | |||
There is a short [[Fundamental theorem of algebra#Complex-analytic proofs|proof of the fundamental theorem of algebra]] based upon Liouville's theorem. | |||
===No entire function dominates another entire function=== | |||
A consequence of the theorem is that "genuinely different" entire functions cannot dominate each other, i.e. if ''f'' and ''g'' are entire, and |''f''| ≤ |''g''| everywhere, then ''f'' = α·''g'' for some complex number α. To show this, consider the function ''h'' = ''f''/''g''. It is enough to prove that ''h'' can be extended to an entire function, in which case the result follows by Liouville's theorem. The holomorphy of ''h'' is clear except at points in ''g''<sup>−1</sup>(0). But since ''h'' is bounded, any singularities must be removable. Thus ''h'' can be extended to an entire bounded function which by Liouville's theorem implies it is constant. | |||
===If ''f'' is less than or equal to a scalar times its input, then it is linear=== | |||
Suppose that ''f'' is entire and |''f''(''z'')| is less than or equal to ''M''|''z''|, for ''M'' a positive real number. We can apply Cauchy's integral formula; we have that | |||
:<math>|f'(z)|=\frac{1}{2\pi}\left|\oint_{C_r }\frac{f(\zeta)}{(\zeta-z)^2}d\zeta\right|\leq \frac{1}{2\pi} \oint_{C_r} \frac{\left| f(\zeta) \right|}{\left| (\zeta-z)^2\right|} \left|d\zeta\right|\leq \frac{1}{2\pi} \oint_{C_r} \frac{M\left| \zeta \right|}{\left| (\zeta-z)^2\right|} \left|d\zeta\right|=\frac{MI}{2\pi}</math> | |||
where ''I'' is the value of the remaining integral. This shows that ''f''' is bounded and entire, so it must be constant, by Liouville's theorem. Integrating then shows that ''f'' is [[Affine transformation|affine]] and then, by referring back to the original inequality, we have that the constant term is zero. | |||
===Non-constant elliptic functions cannot be defined on C=== | |||
The theorem can also be used to deduce that the domain of a non-constant [[elliptic function]] ''f'' cannot be '''C'''. Suppose it was. Then, if ''a'' and ''b'' are two periods of ''f'' such that <sup>''a''</sup>⁄<sub>''b''</sub> is not real, consider the [[parallelogram]] ''P'' whose [[Vertex (geometry)|vertices]] are 0, ''a'', ''b'' and ''a'' + ''b''. Then the image of ''f'' is equal to ''f''(''P''). Since ''f'' is [[continuous functions|continuous]] and ''P'' is [[Compact space|compact]], ''f''(''P'') is also compact and, therefore, it is bounded. So, ''f'' is constant. | |||
The fact that the domain of a non-constant elliptic function ''f'' can not be '''C''' is what Liouville actually proved, in 1847, using the theory of elliptic functions.<ref>{{Citation|last = Liouville|first = Joseph|author-link = Joseph Liouville|publication-date = 1879|year = 1847|title = Leçons sur les fonctions doublement périodiques|periodical = [[Crelle's journal|Journal für die Reine und Angewandte Mathematik]]|volume = 88|pages = 277–310|issn = 0075-4102|url = http://gdz.sub.uni-goettingen.de/no_cache/en/dms/load/img/?IDDOC=266004}}</ref> In fact, it was [[Augustin Louis Cauchy|Cauchy]] who proved Liouville's theorem.<ref>{{Citation|last = Cauchy|first = Augustin-Louis|authorlink = Augustin Louis Cauchy|year = 1844|publication-date = 1882|contribution = Mémoires sur les fonctions complémentaires|contribution-url = http://visualiseur.bnf.fr/StatutConsulter?N=VERESS5-1212867208163&B=1&E=PDF&O=NUMM-90188|title = Œuvres complètes d'Augustin Cauchy|series = 1|volume = 8|place = Paris|publisher = Gauthiers-Villars}}</ref><ref>{{Citation|last = Lützen|first = Jesper|year = 1990|title = Joseph Liouville 1809–1882: Master of Pure and Applied Mathematics|series = Studies in the History of Mathematics and Physical Sciences|volume = 15|publisher = Springer-Verlag|isbn = 3-540-97180-7}}</ref> | |||
===Entire functions have dense images=== | |||
If ''f'' is a non-constant entire function, then its image is [[Dense set|dense]] in '''C'''. This might seem to be a much stronger result than Liouville's theorem, but it is actually an easy corollary. If the image of ''f'' is not dense, then there is a complex number ''w'' and a real number ''r'' > 0 such that the open disk centered at ''w'' with radius ''r'' has no element of the image of ''f''. Define ''g''(''z'') = 1/(''f''(''z'') − ''w''). Then ''g'' is a bounded entire function, since | |||
:<math>(\forall z\in\mathbb{C}):|g(z)|=\frac1{|f(z)-w|}<\frac1r\cdot</math> | |||
So, ''g'' is constant, and therefore ''f'' is constant. | |||
== Remarks == | |||
Let '''C''' ∪ {∞} be the one point compactification of the complex plane '''C'''. In place of holomorphic functions defined on regions in '''C''', one can consider regions in '''C''' ∪ {∞}. Viewed this way, the only possible singularity for entire functions, defined on '''C''' ⊂ '''C''' ∪ {∞}, is the point ∞. If an entire function ''f'' is bounded in a neighborhood of ∞, then ∞ is a [[removable singularity]] of ''f'', i.e. ''f'' cannot blow up or behave erratically at ∞. In light of the power series expansion, it is not surprising that Liouville's theorem holds. | |||
Similarly, if an entire function has a [[Pole (complex analysis)|pole]] at ∞, i.e. blows up like ''z<sup>n</sup>'' in some neighborhood of ∞, then ''f'' is a polynomial. This extended version of Liouville's theorem can be more precisely stated: if |''f''(''z'')| ≤ ''M''.|''z<sup>n</sup>''| for |''z''| sufficiently large, then ''f'' is a polynomial of degree at most ''n''. This can be proved as follows. Again take the Taylor series representation of ''f'', | |||
:<math> f(z) = \sum_{k=0}^\infty a_k z^k.</math> | |||
The argument used during the proof using Cauchy estimates shows that | |||
:<math>(\forall k\in\mathbb{N}):|a_k|\leqslant Mr^{n-k}.</math> | |||
So, if ''k'' > ''n'', | |||
:<math>|a_k|\leqslant\lim_{r\rightarrow+\infty}Mr^{n-k}=0.</math> | |||
Therefore, ''a<sub>k</sub>'' = 0. | |||
==See also== | |||
* [[Mittag-Leffler's theorem]] | |||
==References== | |||
<references/> | |||
==External links== | |||
*{{planetmath reference|title=Liouville's theorem|id=1145}} | |||
* {{MathWorld | urlname= LiouvillesBoundednessTheorem | title= Liouville’s Boundedness Theorem}} | |||
* [http://math.fullerton.edu/mathews/c2003/LiouvilleMoreraGaussMod.html Module for Liouville’s Theorem by John H. Mathews] | |||
[[Category:Theorems in complex analysis]] | |||
[[Category:Articles containing proofs]] | |||
[[Category:Analytic functions|holomorphic functions]] | |||
Revision as of 15:49, 3 February 2014
29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church. In complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function f for which there exists a positive number M such that |f(z)| ≤ M for all z in C is constant.
The theorem is considerably improved by Picard's little theorem, which says that every entire function whose image omits at least two complex numbers must be constant.
Proof
The theorem follows from the fact that holomorphic functions are analytic. If f is an entire function, it can be represented by its Taylor series about 0:
where (by Cauchy's integral formula)
and Cr is the circle about 0 of radius r > 0. Suppose f is bounded: i.e. there exists a constant M such that |f(z)| ≤ M for all z. We can estimate directly
where in the second inequality we have used the fact that |z|=r on the circle Cr. But the choice of r in the above is an arbitrary positive number. Therefore, letting r tend to infinity (we let r tend to infinity since f is analytic on the entire plane) gives ak = 0 for all k ≥ 1. Thus f(z) = a0 and this proves the theorem.
Corollaries
Fundamental theorem of algebra
There is a short proof of the fundamental theorem of algebra based upon Liouville's theorem.
No entire function dominates another entire function
A consequence of the theorem is that "genuinely different" entire functions cannot dominate each other, i.e. if f and g are entire, and |f| ≤ |g| everywhere, then f = α·g for some complex number α. To show this, consider the function h = f/g. It is enough to prove that h can be extended to an entire function, in which case the result follows by Liouville's theorem. The holomorphy of h is clear except at points in g−1(0). But since h is bounded, any singularities must be removable. Thus h can be extended to an entire bounded function which by Liouville's theorem implies it is constant.
If f is less than or equal to a scalar times its input, then it is linear
Suppose that f is entire and |f(z)| is less than or equal to M|z|, for M a positive real number. We can apply Cauchy's integral formula; we have that
where I is the value of the remaining integral. This shows that f' is bounded and entire, so it must be constant, by Liouville's theorem. Integrating then shows that f is affine and then, by referring back to the original inequality, we have that the constant term is zero.
Non-constant elliptic functions cannot be defined on C
The theorem can also be used to deduce that the domain of a non-constant elliptic function f cannot be C. Suppose it was. Then, if a and b are two periods of f such that a⁄b is not real, consider the parallelogram P whose vertices are 0, a, b and a + b. Then the image of f is equal to f(P). Since f is continuous and P is compact, f(P) is also compact and, therefore, it is bounded. So, f is constant.
The fact that the domain of a non-constant elliptic function f can not be C is what Liouville actually proved, in 1847, using the theory of elliptic functions.[1] In fact, it was Cauchy who proved Liouville's theorem.[2][3]
Entire functions have dense images
If f is a non-constant entire function, then its image is dense in C. This might seem to be a much stronger result than Liouville's theorem, but it is actually an easy corollary. If the image of f is not dense, then there is a complex number w and a real number r > 0 such that the open disk centered at w with radius r has no element of the image of f. Define g(z) = 1/(f(z) − w). Then g is a bounded entire function, since
So, g is constant, and therefore f is constant.
Remarks
Let C ∪ {∞} be the one point compactification of the complex plane C. In place of holomorphic functions defined on regions in C, one can consider regions in C ∪ {∞}. Viewed this way, the only possible singularity for entire functions, defined on C ⊂ C ∪ {∞}, is the point ∞. If an entire function f is bounded in a neighborhood of ∞, then ∞ is a removable singularity of f, i.e. f cannot blow up or behave erratically at ∞. In light of the power series expansion, it is not surprising that Liouville's theorem holds.
Similarly, if an entire function has a pole at ∞, i.e. blows up like zn in some neighborhood of ∞, then f is a polynomial. This extended version of Liouville's theorem can be more precisely stated: if |f(z)| ≤ M.|zn| for |z| sufficiently large, then f is a polynomial of degree at most n. This can be proved as follows. Again take the Taylor series representation of f,
The argument used during the proof using Cauchy estimates shows that
So, if k > n,
Therefore, ak = 0.
See also
References
- ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
External links
- Template:Planetmath reference
I had like 17 domains hosted on single account, and never had any special troubles. If you are not happy with the service you will get your money back with in 45 days, that's guaranteed. But the Search Engine utility inside the Hostgator account furnished an instant score for my launched website. Fantastico is unable to install WordPress in a directory which already have any file i.e to install WordPress using Fantastico the destination directory must be empty and it should not have any previous installation files. When you share great information, others will take note. Once your hosting is purchased, you will need to setup your domain name to point to your hosting. Money Back: All accounts of Hostgator come with a 45 day money back guarantee. If you have any queries relating to where by and how to use Hostgator Discount Coupon, you can make contact with us at our site. If you are starting up a website or don't have too much website traffic coming your way, a shared plan is more than enough. Condition you want to take advantage of the worldwide web you prerequisite a HostGator web page, -1 of the most trusted and unfailing web suppliers on the world wide web today. Since, single server is shared by 700 to 800 websites, you cannot expect much speed.
Hostgator tutorials on how to install Wordpress need not be complicated, especially when you will be dealing with a web hosting service that is friendly for novice webmasters and a blogging platform that is as intuitive as riding a bike. After that you can get Hostgator to host your domain and use the wordpress to do the blogging. Once you start site flipping, trust me you will not be able to stop. I cut my webmaster teeth on Control Panel many years ago, but since had left for other hosting companies with more commercial (cough, cough) interfaces. If you don't like it, you can chalk it up to experience and go on. First, find a good starter template design. When I signed up, I did a search for current "HostGator codes" on the web, which enabled me to receive a one-word entry for a discount. Your posts, comments, and pictures will all be imported into your new WordPress blog.- Module for Liouville’s Theorem by John H. Mathews