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In [[mathematics]], '''Liouville's theorem''', originally formulated by [[Joseph Liouville]] in the 1830s and 1840s, places an important restriction on [[antiderivative]]s that can be expressed as elementary functions. | |||
The antiderivatives of certain [[elementary function]]s cannot themselves be expressed as elementary functions. A standard example of such a function is <math>e^{-x^2},</math> whose antiderivative is (with a multiplier of a constant) the [[error function]], familiar from [[statistics]]. Other examples include the functions <math> \frac{ \sin ( x ) }{ x }</math> and <math>x^x.</math> | |||
Liouville's theorem states that elementary antiderivatives, if they exist, must be in the same [[differential field]] as the function, plus possibly a finite number of logarithms. | |||
== Definitions == | |||
For any differential field ''F'', there is a subfield | |||
: Con(''F'') = {''f'' in ''F'' | ''Df'' = 0}, | |||
called the [[Mathematical constant|constant]]s of ''F''. Given two differential fields ''F'' and ''G'', ''G'' is called a '''logarithmic extension''' of ''F'' if ''G'' is a [[field extension|simple transcendental extension]] of ''F'' (i.e. ''G'' = ''F''(''t'') for some [[Transcendental element|transcendental]] ''t'') such that | |||
: ''Dt'' = ''Ds''/''s'' for some ''s'' in ''F''. | |||
This has the form of a [[logarithmic derivative]]. Intuitively, one may think of ''t'' as the [[logarithm]] of some element ''s'' of ''F'', in which case, this condition is analogous to the ordinary [[chain rule]]. But it must be remembered that ''F'' is not necessarily equipped with a unique logarithm; one might adjoin many "logarithm-like" extensions to ''F''. Similarly, an '''exponential extension''' is a simple transcendental extension that satisfies | |||
: ''Dt'' = ''t'' ''Ds''. | |||
With the above caveat in mind, this element may be thought of as an exponential of an element ''s'' of ''F''. Finally, ''G'' is called an '''elementary differential extension''' of ''F'' if there is a finite chain of subfields from ''F'' to ''G'' where each extension in the chain is either algebraic, logarithmic, or exponential. | |||
== Basic theorem == | |||
Suppose ''F'' and ''G'' are differential fields, with Con(''F'') = Con(''G''), and that ''G'' is an elementary differential extension of ''F''. Let ''a'' be in ''F'', ''y'' in G, and suppose ''Dy'' = ''a'' (in words, suppose that ''G'' contains an antiderivative of ''a''). Then there exist ''c''<sub>1</sub>, ..., ''c''<sub>''n''</sub> in Con(''F''), ''u''<sub>1</sub>, ..., ''u''<sub>''n''</sub>, ''v'' in ''F'' such that | |||
:<math>a = c_1\frac{Du_1}{u_1}+\dotsb+c_n\frac{Du_n}{u_n} + Dv.</math> | |||
In other words, the only functions that have "elementary antiderivatives" (i.e. antiderivatives living in, at worst, an elementary differential extension of ''F'') are those with this form prescribed by the theorem. Thus, on an intuitive level, the theorem states that the only elementary antiderivatives are the "simple" functions plus a finite number of logarithms of "simple" functions. | |||
A proof of Liouville's theorem can be found in section 12.4 of Geddes, et al. | |||
== Examples == | |||
As an example, the field '''C'''(''x'') of [[rational function]]s in a single variable has a derivation given by the standard [[derivative]] with respect to that variable. The constants of this field are just the [[complex number]]s '''C'''. | |||
The function <math> \frac{1}{x} </math>, which exists in '''C'''(''x''), does not have an antiderivative in '''C'''(''x''). Its antiderivatives ln ''x'' + ''C'' do, however, exist in the logarithmic extension '''C'''(''x'', ln ''x''). | |||
Likewise, the function <math>\frac{1}{x^2+1}</math> does not have an antiderivative in '''C'''(''x''). Its antiderivatives tan<sup>−1</sup>(''x'') + ''C'' do not seem to satisfy the requirements of the theorem, since they are not (apparently) sums of rational functions and logarithms of rational functions. However, a calculation with [[Euler's formula]] shows that in fact the antiderivatives can be written in the required manner (as logarithms of rational functions). | |||
:<math> | |||
\begin{align} | |||
e^{i \theta} & = \cos \theta + i \sin \theta \\ | |||
e^{-i \theta} & = \cos \theta - i \sin \theta \\ | |||
e^{2i \theta} & = \frac{e^{i \theta}}{e^{-i \theta}} = \frac{\cos \theta + i \sin \theta}{\cos \theta - i \sin \theta} \\ | |||
& = \frac{1 + i \tan \theta}{1 - i \tan \theta} \\[8pt] | |||
2i \theta & = \ln \frac{1 + i \tan \theta}{1 - i \tan \theta} \\[8pt] | |||
2i \tan^{-1} x & = \ln \frac{1 + ix}{1 - ix} \\[8pt] | |||
\tan^{-1} x & = - \frac{1}{2} i \ln \frac{1+ix}{1-ix} | |||
\end{align} | |||
</math> | |||
== Relationship with differential Galois theory == | |||
Liouville's theorem is sometimes presented as a theorem in [[differential Galois theory]], but this is not strictly true. The theorem can be proved without any use of Galois theory. Furthermore, the Galois group of a simple antiderivative is either trivial (if no field extension is required to express it), or is simply the additive group of the constants (corresponding to the constant of integration). Thus, an antiderivative's differential Galois group does not encode enough information to determine if it can be expressed using elementary functions, the major condition of Liouville's theorem. | |||
<!-- | |||
== Example of theorem == | |||
Suppose we want to know whether a function of the form f*e<sup>g</sup> has an elementary antiderivative, with ''f'' and ''g'' in ''C''(''x'') --> | |||
==References== | |||
*{{Citation | last1=Bertrand | first1=D. | title=Review of "Lectures on differential Galois theory" | url=http://www.ams.org/bull/1996-33-02/S0273-0979-96-00652-0/S0273-0979-96-00652-0.pdf | year=1996 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=33 | issue=2}} | |||
* {{cite book | author=Geddes, Czapor, Labahn| title=Algorithms for Computer Algebra | publisher=Kluwer Academic Publishers | year=1992 | isbn=0-7923-9259-0}} | |||
*{{Citation | last1=Magid | first1=Andy R. | title=Lectures on differential Galois theory | url=http://books.google.com/books?id=cJ9vByhPqQ8C | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=University Lecture Series | isbn=978-0-8218-7004-4 | mr=1301076 | year=1994 | volume=7}} | |||
*{{Citation | last1=Magid | first1=Andy R. | title=Differential Galois theory | url=http://www.ams.org/notices/199909/fea-magid.pdf | mr=1710665 | year=1999 | journal=[[Notices of the American Mathematical Society]] | issn=0002-9920 | volume=46 | issue=9 | pages=1041–1049}} | |||
*{{Citation | last1=van der Put | first1=Marius | last2=Singer | first2=Michael F. | title=Galois theory of linear differential equations | url=http://www4.ncsu.edu/~singer/ms_papers.html | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | isbn=978-3-540-44228-8 | mr=1960772 | year=2003 | volume=328}} | |||
== See also == | |||
*[[Risch algorithm]] | |||
[[Category:Field theory]] | |||
[[Category:Differential algebra]] | |||
[[Category:Differential equations]] | |||
[[Category:Theorems in algebra]] | |||
Revision as of 15:18, 22 September 2013
In mathematics, Liouville's theorem, originally formulated by Joseph Liouville in the 1830s and 1840s, places an important restriction on antiderivatives that can be expressed as elementary functions.
The antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions. A standard example of such a function is whose antiderivative is (with a multiplier of a constant) the error function, familiar from statistics. Other examples include the functions and
Liouville's theorem states that elementary antiderivatives, if they exist, must be in the same differential field as the function, plus possibly a finite number of logarithms.
Definitions
For any differential field F, there is a subfield
- Con(F) = {f in F | Df = 0},
called the constants of F. Given two differential fields F and G, G is called a logarithmic extension of F if G is a simple transcendental extension of F (i.e. G = F(t) for some transcendental t) such that
- Dt = Ds/s for some s in F.
This has the form of a logarithmic derivative. Intuitively, one may think of t as the logarithm of some element s of F, in which case, this condition is analogous to the ordinary chain rule. But it must be remembered that F is not necessarily equipped with a unique logarithm; one might adjoin many "logarithm-like" extensions to F. Similarly, an exponential extension is a simple transcendental extension that satisfies
- Dt = t Ds.
With the above caveat in mind, this element may be thought of as an exponential of an element s of F. Finally, G is called an elementary differential extension of F if there is a finite chain of subfields from F to G where each extension in the chain is either algebraic, logarithmic, or exponential.
Basic theorem
Suppose F and G are differential fields, with Con(F) = Con(G), and that G is an elementary differential extension of F. Let a be in F, y in G, and suppose Dy = a (in words, suppose that G contains an antiderivative of a). Then there exist c1, ..., cn in Con(F), u1, ..., un, v in F such that
In other words, the only functions that have "elementary antiderivatives" (i.e. antiderivatives living in, at worst, an elementary differential extension of F) are those with this form prescribed by the theorem. Thus, on an intuitive level, the theorem states that the only elementary antiderivatives are the "simple" functions plus a finite number of logarithms of "simple" functions.
A proof of Liouville's theorem can be found in section 12.4 of Geddes, et al.
Examples
As an example, the field C(x) of rational functions in a single variable has a derivation given by the standard derivative with respect to that variable. The constants of this field are just the complex numbers C.
The function , which exists in C(x), does not have an antiderivative in C(x). Its antiderivatives ln x + C do, however, exist in the logarithmic extension C(x, ln x).
Likewise, the function does not have an antiderivative in C(x). Its antiderivatives tan−1(x) + C do not seem to satisfy the requirements of the theorem, since they are not (apparently) sums of rational functions and logarithms of rational functions. However, a calculation with Euler's formula shows that in fact the antiderivatives can be written in the required manner (as logarithms of rational functions).
![{\displaystyle {\begin{aligned}e^{i\theta }&=\cos \theta +i\sin \theta \\e^{-i\theta }&=\cos \theta -i\sin \theta \\e^{2i\theta }&={\frac {e^{i\theta }}{e^{-i\theta }}}={\frac {\cos \theta +i\sin \theta }{\cos \theta -i\sin \theta }}\\&={\frac {1+i\tan \theta }{1-i\tan \theta }}\\[8pt]2i\theta &=\ln {\frac {1+i\tan \theta }{1-i\tan \theta }}\\[8pt]2i\tan ^{-1}x&=\ln {\frac {1+ix}{1-ix}}\\[8pt]\tan ^{-1}x&=-{\frac {1}{2}}i\ln {\frac {1+ix}{1-ix}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ede876ac5a132103868f3baec8f74f01ec4e412b)
Relationship with differential Galois theory
Liouville's theorem is sometimes presented as a theorem in differential Galois theory, but this is not strictly true. The theorem can be proved without any use of Galois theory. Furthermore, the Galois group of a simple antiderivative is either trivial (if no field extension is required to express it), or is simply the additive group of the constants (corresponding to the constant of integration). Thus, an antiderivative's differential Galois group does not encode enough information to determine if it can be expressed using elementary functions, the major condition of Liouville's theorem.
References
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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.
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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
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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