FS: Difference between revisions

Revision as of 02:43, 4 November 2013

28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance. In mathematics, a rational function is any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers, they may be taken in any field K. In this case, one speaks of a rational function and a rational fraction over K. The values of the variables may be taken in any field L containing K. Then the domain of the function is the set of the values of the variables for which the denominator is not zero and the codomain is L.

By modifying the definition to use equivalence classes the set of rational functions becomes a field.

Definitions

A function $f (x)$ is called a rational function if and only if it can be written in the form

f (x) = \frac{P (x)}{Q (x)}

where $P$ and $Q$ are polynomials in $x$ and $Q$ is not the zero polynomial. The domain of $f$ is the set of all points $x$ for which the denominator $Q (x)$ is not zero.

However, if $P$ and $Q$ have a non constant polynomial greatest common divisor $R$ , then setting $P = P_{1} R$ and $Q = Q_{1} R$ produces a rational function

f_{1} (x) = \frac{P_{1} (x)}{Q_{1} (x)},

which may have a larger domain than $f (x)$ , and is equal to $f (x)$ on the domain of $f (x) .$ It is a common usage to identify $f (x)$ and $f_{1} (x)$ , that is to extend "by continuity" the domain of $f (x)$ to that of $f_{1} (x) .$ Indeed, one can define a rational fraction as an equivalence class of fractions of polynomials, where two fractions A(x)/B(x) and C(x)/D(x) are considered equivalent if A(x)D(x)=B(x)C(x). In this case $\frac{P (x)}{Q (x)}$ is equivalent to $\frac{P_{1} (x)}{Q_{1} (x)}$ .

Examples

People are nonetheless rushing to buy properties in Singapore. A new residential condominium mission in Bedok had a long queue three days before its official launch and bought more than 350 units on the very first day. Nonetheless, there is a ‘development' that had emerged in 2011 that the majority have possibly overlooked.

In August, more than a hundred individuals across 16 states reportedly had been sickened with salmonella after consuming mangos In September, the Meals and Drug Administration detained mango imports from a Mexican packing house after the fruits had been linked to the sickenings Save Cash On Your Online Purchasing Make saving money when purchasing on-line a breeze with Bodge's favourite app, Invisible Hands. This helpful program does the price-checking analysis for you, and will warn you when a unique site has a better value - or let you realize that you have found the most effective deal. Watch Posted by Edison Foo September 29, 2013 HEARTH SALE!!! From only $9xx,000! We hope this listing will help launch your search for new properties in Singapore. It's unlawful to purchase ice cream after 6 pm, in Newark.

for a borrower with one or more excellent residential property loans3; and No SSD will be payable by the seller if the property is bought more than three years after it was bought. Non-extension of the Jan 2009 Funds assistance measures for the property market when the measures expire. Permitting up to 2 years of property tax deferral for land accepted for improvement. Lively secondary marketplace for easy exit of property investments. Curiosity on mortgage fee is tax deductible when the property is rented out. We've progressive property tax rates for residential properties- the higher the value of the residential property, the upper the tax charge. Why do I've to pay property tax on my house when I'm an onwer-occupier and not deriving rental earnings? Palacio Cluster Landed

After obtaining a Sale License (topic to authorities circumstances meant to guard individuals shopping for property in Singapore), he might www.rmmonline.com proceed to sell units in his growth. Funding in the property market of Singapore is likely one of the few investment choices the place utilizing the bank's cash could not be any easier. The power of expats, to make a down payment, leverage the capital and consequently improve total return on investment, is excessive in Singapore. PRs who own a HDB flat must sell their flat inside six months of buying a non-public residential property in Singapore. EVERLASTING residents (PRs) now face unprecedented limits on their skill to purchase property in Singapore. New Condo in 2013 March Foreigners are eligible for Singapore greenback mortgage loan. housing grant

Skilled property brokers in Singapore will assist the client and defend their interests throughout the acquisition, and will help safe the provide at the best possible price. The agents may even be sure that all paperwork are in order and that the vendor is the rightful proprietor of the property. Most of the property brokers share their databases with each other. Tuesday's measures come after the Monetary Authority of Singapore in June tightened rules on property loans, and closed loopholes that allowed some householders to circumvent loan restrictions and avoid paying levies when buying extra property. Contact us for a FREE housing loan analysis and to study concerning the newest mortgage to worth on Singapore housing loans. J Gateway Condominium (Bought Out) Oct, 2013

6. Completion of automotive park, roads and drains serving the housing project 5% 7. Notice of Vacant Possession (Upon obtaining Temporary Occupation Permit) The estimates given by SingaporeHousingLoan.SG's calculator are not quotations and are usually not binding in any way on SingaporeHousingLoan.sg or the consumer. The method of calculation does not exactly mirror that of which is being used by the individual monetary establishment and the data provided should be used as a information only. SingaporeHousingLoan.SG's calculator does not pre-qualify you for any housing loan from any monetary institution. Undercover Report Finds Unlawful Rat Meat Sold In London Market Unpasteurized Tempeh Linked To Salmonella Outbreak That Sickens 60 Hottest EC Launch in Sengkang!

BuyRentSingaporeProperty.com is dedicated in providing the newest residential launches, Developer's sales, new launches, resale, HIGH tasks and rental of properties in Singapore. Then there are new launch property condos in the mid-tier range. These are sometimes in the vicinity of the city fringe, in areas corresponding to Newton and Novena Tanjong Rhu. These new launch projects are popular due to their proximity to town heart, the place the enterprise and entertainment centers and all the amenities they provide, but they're usually cheaper than the City core region tasks. Why submit cheques to register for new property launches in Singapore? Riversails Apartment @ Higher Serangoon LakeVille @ Jurong Lakeside Condo by MCL Ferra Condo @ Leonie Hill

The rational function $f (x) = \frac{x^{3} - 2 x}{2 (x^{2} - 5)}$ is not defined at $x^{2} = 5 \Leftrightarrow x = \pm \sqrt{5}$ . It is asymptotic to $\frac{x}{2}$ as x approaches infinity.

The rational function $f (x) = \frac{x^{2} + 2}{x^{2} + 1}$ is defined for all real numbers, but not for all complex numbers, since if x were a square root of $- 1$ (i.e. the imaginary unit or its negative), then formal evaluation would lead to division by zero: $f (i) = \frac{i^{2} + 2}{i^{2} + 1} = \frac{- 1 + 2}{- 1 + 1} = \frac{1}{0}$ , which is undefined.

A constant function such as f(x) = π is a rational function since constants are polynomials. Note that the function itself is rational, even though the value of f(x) is irrational for all x.

Every polynomial function $f (x) = P (x)$ is a rational function with $Q (x) = 1$ . A function that cannot be written in this form, such as $f (x) = \sin (x)$ , is not a rational function. The adjective "irrational" is not generally used for functions.

The rational function $f (x) = \frac{x}{x}$ is equal to 1 for all x except 0, where there is a removable singularity. The sum, product, or quotient (excepting division by the zero polynomial) of two rational functions is itself a rational function. However, the process of reduction to standard form may inadvertently result in the removal of such singularities unless care is taken. Using the definition of rational functions as equivalence classes gets around this, since x/x is equivalent to 1/1.

Taylor series

The coefficients of a Taylor series of any rational function satisfy a linear recurrence relation, which can be found by setting the rational function equal to its Taylor series and collecting like terms.

For example,

\frac{1}{x^{2} - x + 2} = \sum_{k = 0}^{\infty} a_{k} x^{k} .

Multiplying through by the denominator and distributing,

1 = (x^{2} - x + 2) \sum_{k = 0}^{\infty} a_{k} x^{k}

1 = \sum_{k = 0}^{\infty} a_{k} x^{k + 2} - \sum_{k = 0}^{\infty} a_{k} x^{k + 1} + 2 \sum_{k = 0}^{\infty} a_{k} x^{k} .

After adjusting the indices of the sums to get the same powers of x, we get

1 = \sum_{k = 2}^{\infty} a_{k - 2} x^{k} - \sum_{k = 1}^{\infty} a_{k - 1} x^{k} + 2 \sum_{k = 0}^{\infty} a_{k} x^{k} .

Combining like terms gives

1 = 2 a_{0} + (2 a_{1} - a_{0}) x + \sum_{k = 2}^{\infty} (a_{k - 2} - a_{k - 1} + 2 a_{k}) x^{k} .

Since this holds true for all x in the radius of convergence of the original Taylor series, we can compute as follows. Since the constant term on the left must equal the constant term on the right it follows that

a_{0} = \frac{1}{2} .

Then, since there are no powers of x on the left, all of the coefficients on the right must be zero, from which it follows that

a_{1} = \frac{1}{4}

a_{k} = \frac{1}{2} (a_{k - 1} - a_{k - 2}) f o r k \geq 2 .

Conversely, any sequence that satisfies a linear recurrence determines a rational function when used as the coefficients of a Taylor series. This is useful in solving such recurrences, since by using partial fraction decomposition we can write any rational function as a sum of factors of the form 1 / (ax + b) and expand these as geometric series, giving an explicit formula for the Taylor coefficients; this is the method of generating functions.

Abstract algebra and geometric notion

In abstract algebra the concept of a polynomial is extended to include formal expressions in which the coefficients of the polynomial can be taken from any field. In this setting given a field F and some indeterminate X, a rational expression is any element of the field of fractions of the polynomial ring F[X]. Any rational expression can be written as the quotient of two polynomials P/Q with Q ≠ 0, although this representation isn't unique. P/Q is equivalent to R/S, for polynomials P, Q, R, and S, when PS = QR. However since F[X] is a unique factorization domain, there is a unique representation for any rational expression P/Q with P and Q polynomials of lowest degree and Q chosen to be monic. This is similar to how a fraction of integers can always be written uniquely in lowest terms by canceling out common factors.

The field of rational expressions is denoted F(X). This field is said to be generated (as a field) over F by (a transcendental element) X, because F(X) does not contain any proper subfield containing both F and the element X.

Complex rational functions

In complex analysis, a rational function

f (z) = \frac{P (z)}{Q (z)}

is the ratio of two polynomials with complex coefficients, where Q is not the zero polynomial and P and Q have no common factor (this avoids f taking the indeterminate value 0/0). The domain and range of f are usually taken to be the Riemann sphere, which avoids any need for special treatment at the poles of the function (where Q(z) is 0).

The degree of a rational function is the maximum of the degrees of its constituent polynomials P and Q. If the degree of f is d, then the equation

f (z) = w

has d distinct solutions in z except for certain values of w, called critical values, where two or more solutions coincide. The function f can therefore be thought of as a d-fold covering of the w-sphere by the z-sphere.

Rational functions with degree 1 are called Möbius transformations and form the automorphisms group of the Riemann sphere. Rational functions are representative examples of meromorphic functions.

Notion of a rational function on an algebraic variety

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.

Like polynomials, rational expressions can also be generalized to n indeterminates X₁,..., X_n, by taking the field of fractions of F[X₁,..., X_n], which is denoted by F(X₁,..., X_n).

An extended version of the abstract idea of rational function is used in algebraic geometry. There the function field of an algebraic variety V is formed as the field of fractions of the coordinate ring of V (more accurately said, of a Zariski-dense affine open set in V). Its elements f are considered as regular functions in the sense of algebraic geometry on non-empty open sets U, and also may be seen as morphisms to the projective line.

Applications

These objects are first encountered in school algebra. In more advanced mathematics they play an important role in ring theory, especially in the construction of field extensions. They also provide an example of a nonarchimedean field (see Archimedean property).

Rational functions are used in numerical analysis for interpolation and approximation of functions, for example the Padé approximations introduced by Henri Padé. Approximations in terms of rational functions are well suited for computer algebra systems and other numerical software. Like polynomials, they can be evaluated straightforwardly, and at the same time they express more diverse behavior than polynomials.

Rational functions are used to approximate or model more complex equations in science and engineering including (i) fields and forces in physics, (ii) spectroscopy in analytical chemistry, (iii) enzyme kinetics in biochemistry, (iv) electronic circuitry, (v) aerodynamics, (vi) medicine concentrations in vivo, (vii) wave functions for atoms and molecules, (viii) optics and photography to improve image resolution, and (ix) acoustics and sound.

References

Other Sports Official Kull from Drumheller, has hobbies such as telescopes, property developers in singapore and crocheting. Identified some interesting places having spent 4 months at Saloum Delta.

my web-site http://himerka.com/
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

Dynamic visualization of rational functions with JSXGraph

@@ Line 1: / Line 1: @@
-== right Jordan Basketball Shoes Australia ==
+{{For|the use in automata theory|Finite state transducer}}
+In [[mathematics]], a '''rational function''' is any [[function (mathematics)|function]] which can be defined by a [[rational fraction]], ''i.e.'' an [[algebraic fraction]] such that both the numerator and the denominator are [[polynomial]]s. The [[coefficient]]s of the polynomials need not be [[rational number]]s, they may be taken in any [[field (mathematics)|field]] ''K''. In this case, one speaks of a rational function and a rational fraction ''over K''. The values of the [[variable (mathematics)|variable]]s may be taken in any field ''L'' containing ''K''. Then the [[domain (function)|domain]] of the function is the set of the values of the variables for which the denominator is not zero and the [[codomain]] is ''L''.
-Depo provera pregnancy symptoms ? . I have been having several the signs of pregnancy lately, however Im beginning to get confused. I just recently stopped take. Pediatric growth charts could be a valuable tool to parents. It can benefit you know where you child is as far as physical growth goes. Its not all child grows the same, so taking into account your families tendencies towards weight and height must be considered. <br><br>CSS's biggest success would be a case of insider trading of the public company employee who was feeding sensitive information concerning an impending suit and eventual buyout with a larger competitor to a group of shady Shanghai investors. For the information the worker received 5,000 rmb each month. The young man was nabbed by CSS prior to the stock play took place and three shady Shanghai investors fled China to prevent prosecution. <br><br>(All kids are into that, right?) Generally people know it's hard for moms to get to the computer. But smartphones are not any easier. We'd love to reach that email or voicemail [http://www.maryboroughbridgeclub.com/documents/brifiles.asp?id=49-Jordan-Basketball-Shoes-Australia Jordan Basketball Shoes Australia] or text. The shop has over 300 sunglasses displayed. Normally, all sunglasses are $50 and under, but from Friday to Sunday, PSP will cut yet another 25% off almost everything in the store, excluding perfumes. PSP offers brands such as XLoop, Biohazard, Locs, Snooki, DG and Hello Kitty. <br><br>Some experts speculate that youngsters may regress before learning a brand new skill to give their brain a rest before making a big leap forward. If you notice your child stutters more when she's tired, excited, or upset. I mostly get grunts and whines from her right now and points. <br><br>Of course, I find out about it the day before the deadline, so my pimpage may be worth squat, but woonelly. That is SOME contest.A graceful curtsey to Qadesh [http://www.maryboroughbridgeclub.com/documents/brifiles.asp?id=137-Air-Jordan-6 Air Jordan 6] for the info.I don mind the rules, but I can participate. And Simon Schuster still hasn come completely clean about the ForeverRites issues.Najida, it has to be [http://www.drlindycrocker.com/AWStats/header.asp?n=164-New-Balance-Store-Au New Balance Store Au] the whole MS. <br><br>Based on WPRI, Ruggiero can expect to bring home between $60$100 in tips per night. Like most waitresses, Ruggiero's income relies heavily on tips. She earns $2.89 each hour in salary, which is well below Rhode Island's minimum wage of $7.40. <br><br>Try Skelion is a website dedicated to alternative religions in general. More specifically, ancient and forgotten religions would be the focal point. Considered to be the perfect library for ancient religion, Try Skelion has a tremendous amount of information regarding moon rituals. <br><br>The 53yearold suffers from herniated disks in his lumbar spine that impinge on his nerves. Surgery will decrease the [http://www.jrhc.com.au/js/style.asp?u=61-Ugg-Boots-For-Sale-Melbourne Ugg Boots For Sale Melbourne] pain, but that would also put him out of commission. And right now, Dicky Eklund can't be out of commission. If you happen to take a product that is already available and wish to bring in a different audience then you will have to change the message to draw in the correct crowd. Younger people might be far more comfortable shopping online than seniors. Targeting older people you would have to get into print media rather than online resources..<ul>
+By modifying the definition to use [[equivalence class]]es the set of rational functions becomes a field.
-   <li>[http://israeliz.net/activity/p/40215/ http://israeliz.net/activity/p/40215/]</li>
-   <li>[http://166.111.7.59/luo/forum.php?mod=viewthread&tid=2617191&extra= http://166.111.7.59/luo/forum.php?mod=viewthread&tid=2617191&extra=]</li>
-   <li>[http://forum.tlmacau.com/viewthread.php?tid=224125&extra=page%3D1&frombbs=1 http://forum.tlmacau.com/viewthread.php?tid=224125&extra=page%3D1&frombbs=1]</li>
-   <li>[http://huanxun9999.com/news/html/?70188.html http://huanxun9999.com/news/html/?70188.html]</li>
- </ul>
-== .. Nike Dunk Heels Ebay ==
+==Definitions==
+A function <math>f(x)</math> is called a rational function if and only if it can be written in the form
-A healthy pet is a delight to the household. Pet owners pride themselves on the shining coat or even the boundless energy of their pet. However a lot of people put the dental care of their furry friends on the back burner. A virtual assistant called Google Now will now be able to deliver reminders to get the milk when a individual is in a grocery friend or call certain friends when visiting certain cities. Google Now has been programmed to understand more spoken questions so it can be even more helpful. We've got the technology is being expanded to work on Chrome Web browsers to it can be accessed on pcs, extending its reach beyond smartphones and tablets. <br><br>Gather is a site that compiles news articles and also provides an outlet for members to post their own commentaries as well. Users can discuss the provided content or they are able to create their own. The site aims to supply a unique perspective on the latest goingons. <br><br>Women's team continues to excel in the World Cup in Germany, ChaCha, the fir free realtime questionandanswer service, has seen a dramatic increase [http://www.gympiebridgeclub.com/results/unprocessed/processed.asp?n=72-Nike-Dunk-Heels-Ebay Nike Dunk Heels Ebay] in questions relating [http://www.ceremonieswithsoul.com.au/install/footer.asp?id=29-Le-Pliage-Tote-Australia Le Pliage Tote Australia] to soccer in general and the World Cup tournament particularly. Over the past two weeks, ChaCha questions about the World Cup jumped up by 76%, while questions about the team's breakout star, goalie Hope Solo, increased by 734%. Team, and ChaCha has seen increased interest in her background, family life, and career. <br><br>This is because APD is defined mainly by behaviors (Factor 2 antisocial behaviors) [http://www.jrhc.com.au/js/style.asp?u=73-Ugg-Boots-Cheap-Sydney Ugg Boots Cheap Sydney] and doesn't tap the affective/interpersonal dimensions (Factor 1 core psychopathic features, narcissism) of psychopathy. Further, criminals and APDs tend to "age out" of crime; psychopaths do not, and therefore are at high risk of recidivism. Psychopaths love to intellectualize in treatment with their halfbaked understanding of rules. <br><br>At least have the PS4 subforum ready to rock. I'm sure we could have it loaded ourselves if the option was there for us. Good idea mate.. Some NGOs that found themselves with unexpected amounts of money to spend responded by extending the time horizons or scope of the programs. Only Mdecins sans Frontires was quick to admit that they had enough donations for that tsunami and request [http://www.maryboroughbridgeclub.com/documents/brifiles.asp?id=160-Jordan-Shoes-Online-Australia Jordan Shoes Online Australia] that additional funds help people elsewhere, moving which initially drew criticism from other NGOs. (MSF posted a for Haiti a week ago).. <br><br>We use industry standard safeguards to protect the confidentiality of personal information we collect from people to this Web site. Personal information collected is obtainable only by designated staff or agents of our organization who are also bound to this statement. It is important to recognize that "perfect security" does not exist on the Internet.<ul>
+:<math> f(x) = \frac{P(x)}{Q(x)} </math>
-   <li>[http://www.xinfengfa168.com/news/html/?53357.html http://www.xinfengfa168.com/news/html/?53357.html]</li>
+where <math>P\,</math> and <math>Q\,</math> are [[polynomial]]s in <math>x\,</math> and <math>Q\,</math> is not the [[zero polynomial]].  The [[domain of a function|domain]] of <math>f\,</math> is the set of all points <math>x\,</math> for which the denominator <math>Q(x)\,</math> is not zero.
-   <li>[http://www.miyuehui.com/forum.php?mod=viewthread&tid=35614 http://www.miyuehui.com/forum.php?mod=viewthread&tid=35614]</li>
+However, if <math>\textstyle P</math> and <math>\textstyle Q</math> have a non constant [[polynomial greatest common divisor]] <math>\textstyle R</math>, then setting <math>\textstyle P=P_1R</math> and <math>\textstyle Q=Q_1R</math> produces a rational function
+:<math> f_1(x) = \frac{P_1(x)}{Q_1(x)}, </math>
-   <li>[http://demon-inferno.net/forum/viewtopic.php?f=10&t=256178 http://demon-inferno.net/forum/viewtopic.php?f=10&t=256178]</li>
+which may have a larger domain than <math> f(x)</math>, and is equal to <math> f(x)</math> on the domain of <math> f(x).</math> It is a common usage to identify <math> f(x)</math> and <math> f_1(x)</math>, that is to extend "by continuity" the domain of <math> f(x)</math> to that of <math> f_1(x).</math> Indeed, one can define a rational fraction as an [[equivalence class]] of fractions of polynomials, where two fractions ''A''(''x'')/''B''(''x'') and ''C''(''x'')/''D''(''x'') are considered equivalent if ''A''(''x'')''D''(''x'')=''B''(''x'')''C''(''x''). In this case <math>\frac{P(x)}{Q(x)}</math> is equivalent to <math>\frac{P_1(x)}{Q_1(x)}</math>.
-   <li>[http://gjddbbs.51wwz.com/forum.php?mod=viewthread&tid=3132 http://gjddbbs.51wwz.com/forum.php?mod=viewthread&tid=3132]</li>
+==Examples==
+{{multiple image
- </ul>
+ | header = Examples of rational functions
+ | align = right
+ | direction = vertical
+ | width = 300
+ | image1 = RationalDegree3.svg
+ | alt1 = Rational function of degree 3
+ | caption1 = Rational function of degree 3: <math>y = \frac{x^3-2x}{2(x^2-5)}</math>
+ | image2 = RationalDegree2byXedi.svg
+ | alt2 = Rational function of degree 2
+ | caption2 = Rational function of degree 2: <math>y = \frac{x^2-3x-2}{x^2-4}</math>
+}}
+The rational function <math>f(x) = \frac{x^3-2x}{2(x^2-5)}</math> is not defined at <math>x^2=5 \Leftrightarrow x=\pm \sqrt{5}</math>. It is asymptotic to <math>\frac{x}{2}</math> as ''x'' approaches infinity.
+The rational function <math>f(x) = \frac{x^2 + 2}{x^2 + 1}</math> is defined for all [[real number]]s, but not for all [[complex number]]s, since if ''x'' were a square root of <math>-1</math> (i.e. the [[imaginary unit]] or its negative), then formal evaluation would lead to division by zero: <math>f(i) = \frac{i^2 + 2}{i^2 + 1} = \frac{-1 + 2}{-1 + 1} = \frac{1}{0}</math>, which is undefined.
+A [[constant function]] such as ''f''(''x'') = π is a rational function since constants are polynomials.  Note that the function itself is rational, even though the [[value (mathematics)|value]] of ''f''(''x'') is irrational for all ''x''.
+Every [[polynomial function]] <math>f(x) = P(x)</math> is a rational function with <math>Q(x) = 1</math>.  A function that cannot be written in this form, such as <math>f(x) = \sin(x)</math>, is not a rational function. The adjective "irrational" is not generally used for functions.
+The rational function <math>f(x) = \frac{x}{x}</math> is equal to 1 for all ''x'' except 0, where there is a [[removable singularity]].
+The sum, product, or quotient (excepting division by the zero polynomial) of two rational functions is itself a rational function. However, the process of reduction to standard form may inadvertently result in the removal of such singularities unless care is taken. Using the definition of rational functions as equivalence classes gets around this, since ''x''/''x'' is equivalent to 1/1.
+==Taylor series==
+The coefficients of a [[Taylor series]] of any rational function satisfy a [[Recurrence relation|linear recurrence relation]], which can be found by setting the rational function equal to its Taylor series and collecting [[like terms]].
+For example,
+:<math>\frac{1}{x^2 - x + 2} = \sum_{k=0}^{\infty} a_k x^k.</math>
+Multiplying through by the denominator and distributing,
+:<math>1 = (x^2 - x + 2) \sum_{k=0}^{\infty} a_k x^k</math>
+:<math>1 = \sum_{k=0}^{\infty} a_k x^{k+2} - \sum_{k=0}^{\infty} a_k x^{k+1} + 2\sum_{k=0}^{\infty} a_k x^k.</math>
+After adjusting the indices of the sums to get the same powers of ''x'', we get
+:<math>1 = \sum_{k=2}^{\infty} a_{k-2} x^k - \sum_{k=1}^{\infty} a_{k-1} x^k + 2\sum_{k=0}^{\infty} a_k x^k.</math>
+Combining like terms gives
+:<math>1 = 2a_0 + (2a_1 - a_0)x + \sum_{k=2}^{\infty} (a_{k-2} - a_{k-1} + 2a_k) x^k.</math>
+Since this holds true for all ''x'' in the radius of convergence of the original Taylor series, we can compute as follows.  Since the [[constant term]] on the left must equal the constant term on the right it follows that
+:<math>a_0 = \frac{1}{2}.</math>
+Then, since there are no powers of ''x'' on the left, all of the [[coefficient]]s on the right must be zero, from which it follows that
+:<math>a_1 = \frac{1}{4}</math>
+:<math>a_{k} = \frac{1}{2} (a_{k-1} - a_{k-2})\quad for\ k \ge 2.</math>
+Conversely, any sequence that satisfies a linear recurrence determines a rational function when used as the coefficients of a Taylor series. This is useful in solving such recurrences, since by using [[partial fraction|partial fraction decomposition]] we can write any rational function as a sum of factors of the form ''1 / (ax + b)'' and expand these as [[geometric series]], giving an explicit formula for the Taylor coefficients; this is the method of [[generating functions]].
+== Abstract algebra and geometric notion == <!-- Rational expression redirects here -->
+In [[abstract algebra]] the concept of a polynomial is extended to include formal expressions in which the coefficients of the polynomial can be taken from any [[field (mathematics)|field]].  In this setting given a field ''F'' and some indeterminate ''X'', a '''rational expression''' is any element of the [[field of fractions]] of the [[polynomial ring]] ''F''[''X''].  Any rational expression can be written as the quotient of two polynomials ''P''/''Q'' with ''Q'' ≠ 0, although this representation isn't unique.  ''P''/''Q'' is equivalent to ''R''/''S'', for polynomials ''P'', ''Q'', ''R'', and ''S'', when ''PS'' = ''QR''.  However since ''F''[''X''] is a [[unique factorization domain]], there is a [[irreducible fraction|unique representation]] for any rational expression ''P''/''Q'' with ''P'' and ''Q'' polynomials of lowest degree and ''Q'' chosen to be [[monic polynomial|monic]].  This is similar to how a [[Fraction (mathematics)|fraction]] of integers can always be written uniquely in lowest terms by canceling out common factors.
+The field of rational expressions is denoted ''F''(''X''). This field is said to be generated (as a field) over ''F'' by (a [[transcendental element]]) ''X'', because ''F''(''X'') does not contain any proper subfield containing both ''F'' and the element ''X''.
+===Complex rational functions===
+In [[complex analysis]], a rational function
+:<math>f(z) = \frac{P(z)}{Q(z)}</math>
+is the ratio of two polynomials with complex coefficients, where ''Q'' is not the zero polynomial and ''P'' and ''Q'' have no common factor (this avoids ''f'' taking the indeterminate value 0/0). The domain and range of ''f'' are usually taken to be the [[Riemann sphere]], which avoids any need for special treatment at the [[pole (complex analysis)|poles]] of the function (where ''Q''(''z'') is 0).
+The ''degree'' of a rational function is the maximum of the [[degree of a polynomial|degrees]] of its constituent polynomials ''P'' and ''Q''. If the degree of ''f'' is ''d'', then the equation
+:<math>f(z) = w \,</math>
+has ''d'' distinct solutions in ''z'' except for certain values of ''w'', called ''critical values'', where two or more solutions coincide. The function ''f'' can therefore be thought of as a ''d''-fold [[covering map|covering]] of the ''w''-sphere by the ''z''-sphere.
+Rational functions with degree 1 are called ''[[Möbius transformation]]s'' and form the [[automorphism]]s [[group (mathematics) | group]] of the [[Riemann sphere]]. Rational functions are representative examples of [[meromorphic function]]s.
+===Notion of a rational function on an algebraic variety===
+{{main | Function field of an algebraic variety}}
+Like [[Polynomial ring#The polynomial ring in several variables|polynomials]], rational expressions can also be generalized to ''n'' indeterminates ''X''<sub>1</sub>,..., ''X''<sub>''n''</sub>, by taking the field of fractions of ''F''[''X''<sub>1</sub>,..., ''X''<sub>''n''</sub>], which is denoted by ''F''(''X''<sub>1</sub>,..., ''X''<sub>''n''</sub>).
+An extended version of the abstract idea of rational function is used in algebraic geometry. There the [[function field of an algebraic variety]] ''V'' is formed as the field of fractions of the [[coordinate ring]] of ''V'' (more accurately said, of a Zariski-dense affine open set in ''V''). Its elements ''f'' are considered as regular functions in the sense of algebraic geometry on non-empty open sets ''U'', and also may be seen as morphisms to the [[projective line]].
+==Applications==
+These objects are first encountered in school algebra.  In more advanced mathematics they play an important role in [[ring theory]], especially in the construction of [[field extension]]s. They also provide an example of a ''nonarchimedean field'' (see [[Archimedean property]]).
+Rational functions are used in [[numerical analysis]] for [[interpolation]] and [[approximation]] of functions, for example the [[Padé approximation]]s introduced by [[Henri Padé]]. Approximations in terms of rational functions are well suited for [[computer algebra system]]s and other numerical [[software]]. Like polynomials, they can be evaluated straightforwardly, and at the same time they express more diverse behavior than polynomials. <!-- Care must be taken, however, since small errors in denominators close to zero can cause large errors in evaluation. -->
+Rational functions are used to approximate or model more complex equations in science and engineering including (i) fields and forces in physics, (ii) spectroscopy in analytical chemistry, (iii) enzyme kinetics in biochemistry, (iv) electronic circuitry, (v) aerodynamics, (vi) medicine concentrations in vivo, (vii) wave functions for atoms and molecules,
+(viii) optics and photography to improve image resolution, and (ix) acoustics and sound.
+==References==
+*{{springer|id=Rational_function&oldid=17805|title=Rational function}}
+*{{Citation |last1=Press|first1=WH|last2=Teukolsky|first2=SA|last3=Vetterling|first3=WT|last4=Flannery|first4=BP|year=2007|title=Numerical Recipes: The Art of Scientific Computing|edition=3rd|publisher=Cambridge University Press| publication-place=New York|isbn=978-0-521-88068-8|chapter=Section 3.4. Rational Function Interpolation and Extrapolation|chapter-url=http://apps.nrbook.com/empanel/index.html?pg=124}}
+==See also==
+*[[Partial fraction decomposition]]
+*[[Partial fractions in integration]]
+*[[Function field of an algebraic variety]]
+==External links==
+* [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Rational_functions Dynamic visualization of rational functions with JSXGraph]
+[[Category:Algebraic varieties]]
+[[Category:Morphisms of schemes]]
+[[Category:Meromorphic functions]]

FS: Difference between revisions

Revision as of 02:43, 4 November 2013

Contents

Definitions

Examples

Taylor series

Abstract algebra and geometric notion

Complex rational functions

Notion of a rational function on an algebraic variety

Applications

References

See also

External links

Navigation menu

FS: Difference between revisions

Revision as of 02:43, 4 November 2013

Definitions

Examples

Taylor series

Abstract algebra and geometric notion

Complex rational functions

Notion of a rational function on an algebraic variety

Applications

References

See also

External links

Navigation menu

Search