formulasearchengine - User contributions [en]

Thermal radiation

2014-12-26T22:10:05Z

112.198.77.205: Undid revision 639713878 by 117.244.109.242 (talk) vandal

Nothing to say about myself I think. Enjoying to be a member of wmflabs.org. I just wish I am useful at all My web-site - [http://hemorrhoidtreatmentfix.com/hemorrhoid-surgery hemorrhoids surgery]

Sirius

2014-03-03T12:35:50Z

112.198.64.48: /* Red controversy */

Hi [http://Search.Un.org/search?ie=utf8&site=un_org&output=xml_no_dtd&client=UN_Website_en&num=10&lr=lang_en&proxystylesheet=UN_Website_en&oe=utf8&q=correct&Submit=Go correct]. Let me start by introducing the author, his name is Norris. Alabama has always been her residential. One of the things he loves most is ballet and they is doing make it a professional. After being out of his job for years he became a postal service worker. I'm not are able of webdesign an individual might for you to check my website: http://www.archive.org/details/miley_cyrus_playboy Feel free to surf to my homepage [http://www.archive.org/details/miley_cyrus_playboy miley cyrus playboy]

Multiplexer

2014-03-02T03:39:07Z

112.198.64.44: /* Telecommunications */

Nowadays, Nike Air Max 2011 has been introduced into the market as the third generation of Nike Air Max shoes. However, as a matter of fact, only after a long time from the introduction of these new shoes, people begin to pay certain attention to them. [http://www.futurecol.co.nz/assets/cheap-nike.asp nike running] The acknowledged model among all Nike excellent sports shoes should refer to Nike Air Max 2009 which deserves to be a typical example as a kind of great and popular running shoes. The leather and mesh adopted as material in uppers of air max 2009 make these shoes breathable and light. The heel of these shoes is made up of polyurethane and max air cushion which can be seen. Thus it is fair to say that Nike Air Max 2009 shoes are classic ones among all Nike shoes. We also need to mention Nike Air Max classic BW shoes which are considered as one of the greatest products in the world of sports shoes when it comes to the wonderful running shoes. with these shoes on feet, you will feel the excellence of their various functions. A large [http://Data.Gov.uk/data/search?q=majority majority] of people in the entire world all suppose that Nike designer Air Max shoes can be treated as one of the greatest running shoes in the sports market. Surf on the internet, and there you can learn about more details of these shoes. As to Nike designer Air Max 2011 shoes; they are wonderful sports sneakers, too. They are sure to make the wearers shoes better performance when doing sports and bring them fashion, comfort as well as much capacity. I also would like to spare some time on the internet to seek for some comments about designer Nike Air Max 2011. According to all the reviews, wearing air max 2011 shoes, your feet would enjoy great comfort, and in fact, these shoes have won much favor and praise from most people. Then I was driven by these positive comments and also attempted to buy a pair of air max shoes to have a try. These shoes turn to be cheap and great, and this makes me very satisfied. These shoes are absolutely worthy our any cost. It is known to us that the family of Nike Air Max series include many members such as Nike Air Max 87, Nike Air Max 91, Nike Air Max 95, Nike Air Max 2009, Nike Air Max LTD, Nike Air Max skyline and so on. Among all the above models of air max shoes, every one will illustrate your personality and good taste to the largest extent. As for Nike designer Air Max 91; we can infer its producing year from its name, i.e., the year of 1991. If you expect to get a pair of satisfying running shoes, then, from my point of view, you'd better buy designer Air Max 91 shoes. People who like breathable shoes can pay extra attention to the cheap air max 2011 shoes which will surly their wise choice. All these shoes are endowed with a visual landmark, great cushion and strong stability. 800x600 Normal 0 7.8 � 0 2 false false false EN-US ZH-CN X-NONE MicrosoftInternetExplorer4 /* Style Definitions */ table.MsoNormalTable mso-style-name:n

Template:Subaru vehicles

2014-03-01T09:42:41Z

112.198.196.151:

Advantages of Dwelling in a Condo or Landed property in Singapore Property New Launches & Venture Showcase In Singapore Many residential Singapore property sales contain buying property in Singapore at new launches. These are often houses underneath construction, being offered new by developers. New Launch Singapore Property, 28 Imperial Residences Coming To Geylang Lorong 26 The property market is slowing down, in response to personal property transactions in Might Cellular Apps FREE Sign Up Log in Property Brokers Suggestions Go to the City Redevelopment Authority website to examine the most recent printed figures on ‘Rentals of Private Residential Properties'. Discover a comparable unit in a nearby project that's comparatively new. Check the range of month-to-month rent to see whether or not the numbers given are too optimistic. The blame has been placed on several components, most notably the World Cup (why??) and perhaps the Government Land Sales Programme, which implies that we've got a wider (and perhaps higher) choice so there is no hurry to commit to buying a personal house so quickly. Sunnyvale Residences is a freehold house along Lorong K Telok Kurau, situated on the prime of District 15. Sunnyvale Residences comprising 30 unit Read Extra Executive Director (Strategic Planning & Communications) Property investors may be usually labeled into 2 broad categories. These investing with a deal with rental yield and people investing with a concentrate on capital gains. Seasoned traders tend to desire completed properties as it could immediately generate money circulation by way of rental collections. But when your focus is on capital positive factors, new launches tend to have a fantastic appreciation in value by the point it PRIME. When a neighborhood developer launched a [http://kran-elektro.ru/node/6105 property agents singapore] in the east in mid-2011, as an alternative of utilizing the same old industry observe of balloting for the sequence to select a flat, their advertising agent asked fascinated buyers to line up in entrance of the sales gallery. ROI = (monthly rental – loan repayment – upkeep payment – property tax) x 12 ÷ preliminary funding To live in a condominium, one has to observe sure rules & laws. Every condo association has specific checklist of guidelines, and these could be changed as properly. For example, not all condos give the permission for tenants, there are simply two options, one to maneuver in or the opposite is to sell it. There may be each other drawback. There are condo associations, which allow too many leases. Such situations are additionally too good as per investment. If more than 10% units are rented, then there is likely to be issues relating to mortgage (financing), which would in the end elevate issues on the time of reselling. It's just that you must be cautious when investing in a apartment unit and consider every possible facet when investing in a selected unit. Lush Acres EC @ Sengkang The Master Plan revealed that there will be extra plots of land used for home ownership and property launch in anticipation of the increase in inhabitants. Many analyst has revealed that they're able to notice an increase within the number of plots of land put aside for house ownership. Property in mint condition. You will be the very first proprietor of the apartment. All the things from fittings to stoves are going to be in the perfect situation. Knowing there are not any earlier occupants is also serves as a psychological peace of thoughts for some folks. More affordable when it comes to psf. Trying on the current market, it will seem that accomplished properties are transacting at a decrease psf price compared to new launch condominiums under building. The value of the condos are fairly excessive in present actual estate situation. However when you make investment in it and take wiser resolution by both renting your apartment or resell it. Either manner you will get a superb amount by making funding in the condo property. Yes, it is true that you are able to do the same with different housing properties as well, however you have to not neglect that the choice of residing an opulent life at rental is way greater and can thus be of higher worth. The Shiloh, Florida web site occupies about 200 acres (81 hectares) situated some 20 miles (32 kilometers) north of the NASA pad SpaceX has been using to launch cargo to the International Area Station The Federal Aviation Agency (FAA) not too long ago requested proposals from environmental consultants to support its study of doable impacts to the Shiloh space from development of a spaceport. The state commerce group Area Florida has applied to the FAA for a license to function business-only launch facilities there.

Natural number

2014-01-23T01:27:36Z

112.198.79.166:

{{redirect|Base e|the numbering system which uses [[e (mathematical constant)|e]] as its base|Non-integer representation#Base e}}

[[Image:Log.svg|thumb|right|300 px|Graph of the natural logarithm function. The function slowly grows to positive infinity as ''x'' increases and rapidly goes to negative infinity as ''x'' approaches 0 ("slowly" and "rapidly" as compared to any [[power law]] of ''x''); the y-axis is an [[asymptote]].]]

The '''natural logarithm''' is the [[logarithm]] to the [[base (exponentiation)|base]] ''[[e (mathematical constant)|e]]'', where ''e'' is an [[Irrational number|irrational]] and [[Transcendental number|transcendental]] constant approximately equal to 2.718281828. The natural logarithm is generally written as ln ''x'', log''e'' ''x'' or sometimes, if the base of ''e'' is implicit, as simply log ''x''.<ref>{{cite book
|title=Mathematics for physical chemistry
|edition=3rd
|first1=Robert G.
|last1=Mortimer
|publisher=Academic Press
|year=2005
|isbn=0-12-508347-5
|page=9
|url=http://books.google.com/books?id=nGoSv5tmATsC}}, [http://books.google.com/books?id=nGoSv5tmATsC&pg=PA9 Extract of page 9]
</ref> Parentheses are sometimes added for clarity, giving ln(''x''), log''e''(''x'') or log(''x''). This is done in particular when the argument to the logarithm is not a single symbol, in order to prevent ambiguity.

The natural logarithm of a number ''x'' is the [[exponentiation|power]] to which ''e'' would have to be raised to equal ''x''. For example, ln(7.389...) is 2, because ''e''2=7.389.... The natural log of ''e'' itself, ln(''e''), is 1 because ''e''1 = ''e'', while the natural logarithm of 1, ln(1), is 0, since ''e''0 = 1.

The natural logarithm can be defined for any positive [[real number]] ''a'' as the [[Integral|area under the curve]] ''y'' = 1/''x'' from 1 to ''a''. The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, leads to the term "natural." The definition can be extended to non-zero [[complex number]]s, as explained below.

The natural logarithm function, if considered as a real-valued function of a real variable, is the [[inverse function]] of the [[exponential function]], leading to the identities:
:<math>e^{\ln(x)} = x \qquad \mbox{if }x > 0\,\!</math>

:<math>\ln(e^x) = x.\,\! </math>

Like all logarithms, the natural logarithm maps multiplication into addition:

:<math> \ln(xy) = \ln(x) + \ln(y). \!\, </math>

Thus, the logarithm function is an [[isomorphism]] from the [[group (mathematics)|group]] of positive real numbers under multiplication to the group of real numbers under addition, represented as a [[function (mathematics)|function]]:
:<math>\ln : \mathbb{R}^+ \to \mathbb{R}.</math>

Logarithms can be defined to any positive base other than 1, not just ''e''. However, logarithms in other bases differ only by a constant multiplier from the natural logarithm, and are usually defined in terms of the latter. For instance, the [[binary logarithm]] is just the natural logarithm divided by ln(2), the [[natural logarithm of 2]]. Logarithms are useful for solving equations in which the unknown appears as the exponent of some other quantity. For example, logarithms are used to solve for the half-life, decay constant, or unknown time in [[exponential decay]] problems. They are important in many branches of mathematics and the sciences and are used in finance to solve problems involving [[compound interest]].

{| class=infobox width=200px
|colspan=2 align=center| Natural Logarithm
|-
|'''Representation''' ||<math>\ln x \,</math>
|-
|'''Inverse''' ||<math>e^x \,</math>
|-
|'''Derivative''' ||<math>\frac{1}{x} \,</math>
|-
|'''Indefinite Integral''' ||<math>x\ln x - x + C \,</math>
|}

{{E (mathematical constant)}}

==History==
The first mention of the natural logarithm was by [[Nicholas Mercator]] in his work ''Logarithmotechnia'' published in 1668,<ref>{{cite web |author=J J O'Connor and E F Robertson |url=http://www-history.mcs.st-and.ac.uk/HistTopics/e.html |title=The number e |publisher=The MacTutor History of Mathematics archive |date=September 2001 |accessdate=2009-02-02}}</ref> although the mathematics teacher John Speidell had already in 1619 compiled a table on the natural logarithm.<ref>{{cite book |last=Cajori |first=Florian |authorlink=Florian Cajori |title=A History of Mathematics, 5th ed |pages=152 |publisher=AMS Bookstore |year=1991 |isbn=0-8218-2102-4 |url=http://books.google.com/?id=mGJRjIC9fZgC&dq=%22Cajori%22+%22A+History+of+Mathematics%22+}}</ref> It was formerly also called hyperbolic logarithm,<ref>{{cite web |author=Flashman, Martin |url=http://users.humboldt.edu/flashman/Presentations/Estimations.html |title=Estimating Integrals using Polynomials |accessdate=2008-03-23}}</ref> as it corresponds to the area under a [[hyperbola]]. It is also sometimes referred to as the [[Napierian logarithm]], although the original meaning of this term is slightly different.

==Notational conventions==
The notations {{nowrap|"ln ''x''"}} and {{nowrap|"log''e'' ''x''"}} both refer unambiguously to the natural logarithm of ''x''.

{{nowrap|"log ''x''"}} without an explicit base may also refer to the natural logarithm. This usage is common in mathematics and some scientific contexts as well as in many [[programming language]]s.<ref>Including [[C (programming language)|C]], [[C++]], [[SAS System|SAS]], [[MATLAB]], [[Mathematica]], [[Fortran]], and [[BASIC programming language|BASIC]]</ref> {{nowrap|"log ''x''"}} is frequently used to denote the [[common logarithm|common (base 10) logarithm]], however.

==Origin of the term ''natural logarithm''==
Initially, it might seem that since the common numbering system is [[base 10]], this base would be more "natural" than base ''e''. But mathematically, the number 10 is not particularly significant. Its use culturally—as the basis for many societies’ numbering systems—likely arises from humans’ typical number of fingers.<ref>{{cite book | last=Boyers | first=Carl | title=A History of Mathematics | publisher=[[John Wiley & Sons|Wiley]] | year=1968}}</ref> Other cultures have based their [[counting systems]] on such choices as [[quinary|5]], [[octal|8]], [[duodecimal|12]], [[vigesimal|20]], and [[sexagesimal|60]].<ref>{{cite journal | last=Harris | first=John | title=Australian Aboriginal and Islander mathematics | journal=Australian Aboriginal Studies | volume=2 |pages=29–37 | year=1987 | url=http://www1.aiatsis.gov.au/exhibitions/e_access/serial/m0005975_v_a.pdf| accessdate=2008-02-12 | format=PDF}}</ref><ref>{{cite journal | last=Large| first=J.J. | title=The vigesimal system of enumeration | journal=Journal of the Polynesian Society| volume=11|issue=4 |pages=260–261 | year=1902 | url=http://www.jps.auckland.ac.nz/document/?wid=636 | accessdate=30 March 2011}}</ref><ref>{{cite journal | last=Cajori| first=Florian| authorlink = Florian Cajori | title=Sexagesimal fractions among the Babylonians | journal=American Mathematical Monthly| volume=29 |pages=8–10 | year=1922| doi=10.2307/2972914 | jstor=2972914 | issue=1 }}</ref>

log''e'' is a "natural" log because it automatically springs from, and appears so often in, mathematics. For example, consider the problem of [[derivative|differentiating]] a logarithmic function:<ref>{{cite book
|title=Calculus: An Applied Approach
|edition=8th
|first1=Ron
|last1=Larson
|publisher=Cengage Learning
|year=2007
|isbn=0-618-95825-8
|page=331
|url=http://books.google.com/books?id=rbDG7V0OV34C}}, [http://books.google.com/books?id=rbDG7V0OV34C&pg=PA331 Section 4.5, page 331]
</ref>
:<math>\frac{d}{dx}\log_b(x) = \frac{d}{dx} \left( \frac{1}{\ln(b)} \ln{x} \right) = \frac{1}{\ln(b)} \frac{d}{dx} \ln{x} = \frac{1}{x\ln(b)}. </math>

If the [[base (exponentiation)|base]] ''b'' equals ''e'', then the derivative is simply 1/''x'', and at ''x'' = 1 this derivative equals 1. Another sense in which the base-''e''-logarithm is the most natural is that it can be defined quite easily in terms of a simple integral or [[Taylor series]] and this is not true of other logarithms.

Further senses of this naturalness make no use of [[calculus]]. As an example, there are a number of simple series involving the natural logarithm. [[Pietro Mengoli]] and [[Nicholas Mercator]] called it ''logarithmus naturalis'' a few decades before [[Sir Isaac Newton|Newton]] and [[Gottfried Wilhelm Leibniz|Leibniz]] developed calculus.<ref>{{cite web | last=Ballew | first=Pat | title=Math Words, and Some Other Words, of Interest | url=http://www.pballew.net/arithme1.html#ln | accessdate=2007-09-16}}</ref>

== Definitions ==
[[Image:Log-pole-x 1.svg|thumb|ln(''a'') illustrated as the area under the curve ''f''(''x'') = 1/''x'' from 1 to ''a''. If ''a'' is less than 1, the area from ''a'' to 1 is counted as negative.]]

Formally, ln(''a'') may be defined as the [[integral]],
:<math>\ln(a)=\int_1^a \frac{1}{x}\,dx.</math>

This function is a logarithm because it satisfies the fundamental property of a logarithm:
:<math>\ln(ab)=\ln(a)+\ln(b). \,\!</math>

This can be demonstrated by splitting the integral that defines ln(''ab'') into two parts and then making the [[Integration by substitution|variable substitution]] {{nowrap|1=''x'' = ''ta''}} in the second part, as follows:
:<math>
\ln (ab)
= \int_1^{ab} \frac{1}{x} \; dx
= \int_1^a \frac{1}{x} \; dx \; + \int_a^{ab} \frac{1}{x} \; dx
=\int_1^{a} \frac{1}{x} \; dx \; + \int_1^{b} \frac{1}{at} \; d(at)
</math>
::<math>
=\int_1^{a} \frac{1}{x} \; dx \; + \int_1^{b} \frac{1}{t} \; dt
= \ln (a) + \ln (b).
</math>

The number ''[[E (mathematical constant)|e]]'' can then be defined as the unique real number ''a'' such that ln(''a'') = 1.

Alternatively, if the [[exponential function]] has been defined first, say by using an [[infinite series]], the natural logarithm may be defined as its [[inverse function]], i.e., ln is that function such that exp(ln(''x'')) = ''x''. Since the range of the exponential function on real arguments is all positive real numbers and since the exponential function is strictly increasing, this is well-defined for all positive ''x''.

== Properties ==
* <math>\ln(1) = 0 \,</math>
* <math>\ln(-1) = i \pi \,</math>
::(see [[complex logarithm]])
* <math>\ln(x) < \ln(y) \quad{\rm for}\quad 0 < x < y \,</math>
* <math>\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1 \,</math>
* <math>\ln(x^y) = y \, \ln(x) \,</math>
* <math>\frac{x-1}{x} \leq \ln(x) \leq x-1 \quad{\rm for}\quad x > 0 \,</math>
* <math>\ln{( 1+x^\alpha )} \leq \alpha x \quad{\rm for}\quad x \ge 0, \alpha \ge 1 \,</math>

:{| class="toccolours collapsible collapsed" width="90%" style="text-align:left"
!Proof
|-
|
The statement is true for <math>x=0</math>, and we now show that <math>\frac{d}{dx} \ln{( 1+x^\alpha )} \leq \frac{d}{dx} ( \alpha x ) </math> for all <math>x</math>, which completes the proof by the [[fundamental theorem of calculus#First part|fundamental theorem of calculus]]. Hence, we want to show that

:<math>\frac{d}{dx} \ln{( 1+x^\alpha )} = \frac{\alpha x^{\alpha - 1}}{1 + x^\alpha} \leq \alpha = \frac{d}{dx} ( \alpha x ) </math>

(Note that we have not yet proved that this statement is true.) If this is true, then by multiplying the middle statement by the positive quantity <math>(1+x^\alpha) / \alpha</math> and subtracting <math>x^\alpha</math> we would obtain

:<math> x^{\alpha-1} \leq x^\alpha + 1 </math>
:<math> x^{\alpha-1} (1-x) \leq 1 </math>

This statement is trivially true for <math>x \ge 1</math> since the left hand side is negative or zero. For <math>0 \le x < 1</math> it is still true since both factors on the left are less than 1 (recall that <math>\alpha \ge 1</math>). Thus this last statement is true and by repeating our steps in reverse order we find that <math>\frac{d}{dx} \ln{( 1+x^\alpha )} \leq \frac{d}{dx} ( \alpha x ) </math> for all <math>x</math>. This completes the proof.
|}

== Derivative, Taylor series ==
[[Image:LogTay.svg|300px|thumb|right|The Taylor polynomials for ln(1 + ''x'') only provide accurate approximations in the range −1 < ''x'' ≤ 1. Note that, for ''x'' > 1, the Taylor polynomials of higher degree are ''worse'' approximations.]]

The [[derivative]] of the natural logarithm is given by
:<math>\frac{d}{dx} \ln(x) = \frac{1}{x}.\,</math>

This leads to the [[Taylor series]] for ln(1 + ''x'') around 0; also known as the [[Mercator series]]
:<math>\ln(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} x^n = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots \quad{\rm for}\quad \left|x\right| \leq 1,\quad</math>
::<math>{\rm unless}\quad x = -1.</math>

([[Leonhard Euler]]<ref>[[Leonhard Euler]], Introductio in Analysin Infinitorum. Tomus Primus. Bousquet, Lausanne 1748. Exemplum 1, p. 228; quoque in: Opera Omnia, Series Prima, Opera Mathematica, Volumen Octavum, Teubner 1922</ref> nevertheless boldly applied this series to ''x= -1'',
in order to show that the harmonic series equals the (natural) logarithm of ''1/(1-1)'', that is the logarithm of infinity. Nowadays, more formally but perhaps less vividly, we prove that the harmonic series truncated at ''N'' is close to the logarithm of ''N'', when ''N'' is large).

At right is a picture of ln(1 + ''x'') and some of its [[Taylor polynomial]]s around 0. These approximations converge to the function only in the region −1 < ''x'' ≤ 1; outside of this region the higher-degree Taylor polynomials are ''worse'' approximations for the function.

Substituting ''x'' − 1 for ''x'', we obtain an alternative form for ln(x) itself, namely
:<math>\ln(x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} (x-1) ^ n</math>

:<math>\ln(x)= (x - 1) - \frac{(x-1) ^ 2}{2} + \frac{(x-1)^3}{3} - \frac{(x-1)^4}{4} \cdots</math>

:<math>{\rm for}\quad \left|x-1\right| \leq 1\quad {\rm unless}\quad x = 0 \,.</math><ref>[http://www.math2.org/math/expansion/log.htm "Logarithmic Expansions" at Math2.org]</ref>

By using the [[binomial transform#Euler transform|Euler transform]] on the Mercator series, one obtains the following, which is valid for any x with absolute value greater than 1:

:<math>\ln{x \over {x-1}} = \sum_{n=1}^\infty {1 \over {n x^n}} = {1 \over x}+ {1 \over {2x^2}} + {1 \over {3x^3}} + \cdots \,.</math>

This series is similar to a [[BBP-type formula]].

Also note that <math> x \over {x-1} </math> is its own inverse function, so to yield the natural logarithm of a certain number ''y'', simply put in <math> y \over {y-1} </math> for ''x''.

:<math>\ln{x} = \sum_{n=1}^\infty {1 \over {n}} \left( {x - 1 \over x} \right)^n = \left( {x - 1 \over x} \right) + {1 \over 2} \left( {x - 1 \over x} \right)^2 + {1 \over 3} \left( {x - 1 \over x} \right)^3 + \cdots \,</math>

:<math>{\rm for }\quad \operatorname{Re} (x) \geq \frac12 \,.</math>

== The natural logarithm in integration ==
The natural logarithm allows simple [[integral|integration]] of functions of the form ''g''(''x'') = ''f'' '(''x'')/''f''(''x''): an [[antiderivative]] of ''g''(''x'') is given by ln(|''f''(''x'')|). This is the case because of the [[chain rule]] and the following fact:

:<math>\ {d \over dx}\left( \ln \left| x \right| \right) = {1 \over x}.</math>

In other words,

:<math>\int { 1 \over x} dx = \ln|x| + C</math>

and

:<math>\int { \frac{f'(x)}{f(x)}\, dx} = \ln |f(x)| + C.</math>

Here is an example in the case of ''g''(''x'') = tan(''x''):

:<math>\int \tan (x) \,dx = \int {\sin (x) \over \cos (x)} \,dx</math>
:<math>\int \tan (x) \,dx = \int {-{d \over dx} \cos (x) \over {\cos (x)}} \,dx.</math>
Letting ''f''(''x'') = cos(''x'') and ''f'''(''x'')= – sin(''x''):
:<math>\int \tan (x) \,dx = -\ln{\left| \cos (x) \right|} + C</math>
:<math>\int \tan (x) \,dx = \ln{\left| \sec (x) \right|} + C</math>

where ''C'' is an [[arbitrary constant of integration]].

The natural logarithm can be integrated using [[integration by parts]]:

:<math>\int \ln (x) \,dx = x \ln (x) - x + C.</math>

== Numerical value ==
To calculate the numerical value of the natural logarithm of a number, the Taylor series expansion can be rewritten as:

:<math>\ln(1+x)= x \,\left( \frac{1}{1} - x\,\left(\frac{1}{2} - x \,\left(\frac{1}{3} - x \,\left(\frac{1}{4} - x \,\left(\frac{1}{5}- \cdots \right)\right)\right)\right)\right) \quad{\rm for}\quad \left|x\right|<1.\,\!</math>

To obtain a better rate of convergence, the following identity can be used.

:{|
|-
|<math>\ln(x) = \ln\left(\frac{1+y}{1-y}\right)</math>
|<math>= 2\,y\, \left( \frac{1}{1} + \frac{1}{3} y^{2} + \frac{1}{5} y^{4} + \frac{1}{7} y^{6} + \frac{1}{9} y^{8} + \cdots \right) </math>
|-
|
|<math>= 2\,y\, \left( \frac{1}{1} + y^{2} \, \left( \frac{1}{3} + y^{2} \, \left( \frac{1}{5} + y^{2} \, \left( \frac{1}{7} + y^{2} \, \left( \frac{1}{9} + \cdots \right) \right) \right)\right) \right) </math>
|}

:provided that ''y'' = (''x''−1)/(''x''+1) and Re(''x'') ≥ 0 but ''x'' ≠ 0.

For ln(''x'') where ''x'' > 1, the closer the value of ''x'' is to 1, the faster the rate of convergence. The identities associated with the logarithm can be leveraged to exploit this:

:{|
|-
|<math>\ln(123.456)\!</math>
|<math>= \ln(1.23456 \times 10^2) \,\!</math>
|-
|
|<math>= \ln(1.23456) + \ln(10^2) \,\!</math>
|-
|
|<math>= \ln(1.23456) + 2 \times \ln(10) \,\!</math>
|-
|
|<math>\approx \ln(1.23456) + 2 \times 2.3025851. \,\!</math>
|}

Such techniques were used before calculators, by referring to numerical tables and performing manipulations such as those above.

===Natural logarithm of 10===
The natural logarithm of 10 ({{OEIS2C|A002392}}) plays a role for example in computation of natural logarithms of numbers represented in the [[scientific notation]], a mantissa multiplied by a power of 10:

: <math>\ln (a\times 10^n) = \ln a + n \ln 10.</math>

By this scaling, the algorithm may reduce the logarithm of all positive real numbers to an algorithm for natural logarithms in the range <math>1\le a< 10</math>.

===High precision===
To compute the natural logarithm with many digits of precision, the Taylor series approach is not efficient since the convergence is slow. If ''x'' is near 1, an alternative is to use [[Newton's method]] to invert the exponential function, whose series converges more quickly. For an optimal function, the iteration simplifies to
:<math> y_{n+1} = y_n + 2 \cdot \frac{ x - \exp ( y_n ) }{ x + \exp ( y_n ) } \,</math>
which has [[cubic convergence]] to ln(''x'').

Another alternative for extremely high precision calculation is the formula<ref>{{cite journal |first1=T. |last1=Sasaki |first2=Y. |last2=Kanada |title=Practically fast multiple-precision evaluation of log(x) |journal=Journal of Information Processing |volume=5|issue=4 |pages=247–250 |year=1982 | url=http://ci.nii.ac.jp/naid/110002673332 | accessdate=30 March 2011}}</ref>
<ref>
{{Cite journal
|first1=Timm
|last1=Ahrendt
|title=Stacs 99
|series=Lecture notes in computer science
|doi=10.1007/3-540-49116-3_28
|volume=1564
|year=1999
|pages=302–312
|postscript=
|chapter=Fast Computations of the Exponential Function
|isbn=978-3-540-65691-3
}}
</ref>

:<math>\ln x \approx \frac{\pi}{2 M(1,4/s)} - m \ln 2,</math>

where ''M'' denotes the [[arithmetic-geometric mean]] of 1 and 4/s, and

:<math>s = x \,2^m > 2^{p/2},</math>

with ''m'' chosen so that ''p'' bits of precision is attained. (For most purposes, the value of 8 for m is sufficient.) In fact, if this method is used, Newton inversion of the natural logarithm may conversely be used to calculate the exponential function efficiently. (The constants ln 2 and [[pi|π]] can be pre-computed to the desired precision using any of several known quickly converging series.)

===Computational complexity===
{{main|Computational complexity of mathematical operations}}
The [[computational complexity]] of computing the natural logarithm (using the arithmetic-geometric mean) is O(''M''(''n'') ln ''n''). Here ''n'' is the number of digits of precision at which the natural logarithm is to be evaluated and ''M''(''n'') is the computational complexity of multiplying two ''n''-digit numbers.

==Continued fractions==
While no simple [[continued fraction]]s are available, several [[generalized continued fraction]]s are, including:

:<math>
\ln (1+x)=\frac{x^1}{1}-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\frac{x^5}{5}-\cdots=
\cfrac{x}{1-0x+\cfrac{1^2x}{2-1x+\cfrac{2^2x}{3-2x+\cfrac{3^2x}{4-3x+\cfrac{4^2x}{5-4x+\ddots}}}}}
</math>

:<math>
\ln \left( 1+\frac{x}{y} \right) = \cfrac{x} {y+\cfrac{1x} {2+\cfrac{1x} {3y+\cfrac{2x} {2+\cfrac{2x} {5y+\cfrac{3x} {2+\ddots}}}}}}
= \cfrac{2x} {2y+x-\cfrac{(1x)^2} {3(2y+x)-\cfrac{(2x)^2} {5(2y+x)-\cfrac{(3x)^2} {7(2y+x)-\ddots}}}}
</math>

These continued fractions—particularly the last—converge rapidly for values close to 1. However, the natural logarithms of much larger numbers can easily be computed by repeatedly adding those of smaller numbers, with similarly rapid convergence.

For example, since 2 = 1.253 × 1.024, the [[natural logarithm of 2]] can be computed as:
:<math>
\ln 2 = 3 \ln \left( 1+\frac{1}{4} \right) + \ln \left( 1+\frac{3}{125} \right)
= \cfrac{6} {9-\cfrac{1^2} {27-\cfrac{2^2} {45-\cfrac{3^2} {63-\ddots}}}}
+ \cfrac{6} {253-\cfrac{3^2} {759-\cfrac{6^2} {1265-\cfrac{9^2} {1771-\ddots}}}}.
</math>

Furthermore, since 10 = 1.2510 × 1.0243, even the natural logarithm of 10 similarly can be computed as:
:<math>
\ln 10 = 10 \ln \left( 1+\frac{1}{4} \right) + 3\ln \left( 1+\frac{3}{125} \right)
= \cfrac{20} {9-\cfrac{1^2} {27-\cfrac{2^2} {45-\cfrac{3^2} {63-\ddots}}}}
+ \cfrac{18} {253-\cfrac{3^2} {759-\cfrac{6^2} {1265-\cfrac{9^2} {1771-\ddots}}}}.
</math>

==Complex logarithms==
{{Main|Complex logarithm}}
The exponential function can be extended to a function which gives a [[complex number]] as ''e''''x'' for any arbitrary complex number ''x''; simply use the infinite series with ''x'' complex. This exponential function can be inverted to form a complex logarithm that exhibits most of the properties of the ordinary logarithm. There are two difficulties involved: no ''x'' has ''e''''x'' = 0; and it turns out that ''e''2''πi'' = 1 = ''e''0. Since the multiplicative property still works for the complex exponential function, ''e''''z'' = ''e''''z''+2''nπi'', for all complex ''z'' and integers ''n''.

So the logarithm cannot be defined for the whole [[complex plane]], and even then it is multi-valued – any complex logarithm can be changed into an "equivalent" logarithm by adding any integer multiple of 2''πi'' at will. The complex logarithm can only be single-valued on the [[complex plane#Cutting the plane|cut plane]]. For example, ln ''i'' = 1/2 ''πi'' or 5/2 ''πi'' or −3/2 ''πi'', etc.; and although ''i''4 = 1, 4 log ''i'' can be defined as 2''πi'', or 10''πi'' or −6 ''πi'', and so on.

<gallery mode=packed caption="Plots of the natural logarithm function on the complex plane ([[principal branch]])">
Image:Natural Logarithm Re.svg| ''z'' = Re(ln(x+iy))
Image:Natural Logarithm Im Abs.svg| ''z'' = |Im(ln(x+iy))|
Image:Natural Logarithm Abs.svg| ''z'' = |ln(x+iy)|
Image:Natural Logarithm All.svg| Superposition of the previous 3 graphs
</gallery>

==See also==
* [[John Napier]] – inventor of logarithms
* [[Logarithm of a matrix]]
* [[Logarithmic integral function]]
* [[Nicholas Mercator]] – first to use the term natural log
* [[Polylogarithm]]
* [[Von Mangoldt function]]
* [[e (mathematical constant)|The number ''e'']]

==References==
{{Reflist|2}}

==External links==
* [http://betterexplained.com/articles/demystifying-the-natural-logarithm-ln/ Demystifying the Natural Logarithm (ln) | BetterExplained]

{{DEFAULTSORT:Natural Logarithm}}
[[Category:Logarithms]]
[[Category:Elementary special functions]]
[[Category:E (mathematical constant)]]

[[de:Logarithmus#Natürlicher Logarithmus]]

Surface-wave-sustained mode

2012-12-11T10:02:03Z

112.198.64.75:

In [[mathematics]], a '''Galois module''' is a [[G-module|''G''-module]] where ''G'' is the [[Galois group]] of some [[field extension|extension]] of [[Field (mathematics)|fields]]. The term '''Galois representation''' is frequently used when the ''G''-module is a [[vector space]] over a [[field (mathematics)|field]] or a [[free module]] over a [[ring (mathematics)|ring]], but can also be used as a synonym for ''G''-module. The study of Galois modules for extensions of [[local field|local]] or [[global field]]s is an important tool in [[number theory]].

==Examples==
*Given a field ''K'', the [[unit group|multiplicative group]] (''Ks'')× of a [[separable closure]] of ''K'' is a Galois module for the [[absolute Galois group]]. Its second cohomology group is [[isomorphic]] to the [[Brauer group]] of ''K'' (by [[Hilbert's theorem 90]], its first [[group cohomology|cohomology group]] is zero).

*If ''X'' is a [[smooth morphism|smooth]] [[proper morphism|proper]] scheme over a field ''K'' then the [[l-adic cohomology|ℓ-adic cohomology]] groups of its [[geometric fibre]] are Galois modules for the absolute Galois group of ''K''.

===Ramification theory===
Let ''K'' be a [[valued field]] (with valuation denoted ''v'') and let ''L''/''K'' be a [[finite extension|finite]] [[Galois extension]] with Galois group ''G''. For an [[extension of a valuation|extension]] ''w'' of ''v'' to ''L'', let ''Iw'' denote its [[inertia group of an extension of valuations|inertia group]]. A Galois module ρ : ''G'' → Aut(''V'') is said to be '''unramified''' if ρ(''Iw'') = {1}.

==Galois module structure of algebraic integers==
In classical [[algebraic number theory]], let ''L'' be a Galois extension of a field ''K'', and let ''G'' be the corresponding Galois group. Then the ring ''O''''L'' of [[algebraic integer]]s of ''L'' can be considered as an ''O''''K''[''G'']-module, and one can ask what its structure is. This is an arithmetic question, in that by the [[normal basis theorem]] one knows that ''L'' is a free ''K''[''G'']-module of rank 1. If the same is true for the integers, that is equivalent to the existence of a '''normal integral basis''', i.e. of α in ''OL'' such that its [[conjugate element]]s under ''G'' give a free basis for ''OL'' over ''O''''K''. This is an interesting question even (perhaps especially) when ''K'' is the [[rational number]] field '''Q'''.

For example, if ''L'' = '''Q'''(√-3), is there a normal integral basis? The answer is yes, as one sees by identifying it with '''Q'''(ζ) where

:ζ = exp(2πi/3).

In fact all the subfields of the [[cyclotomic field]]s for ''p''-th [[roots of unity]] for ''p'' a ''prime number'' have normal integral bases (over '''Z'''), as can be deduced from the theory of [[Gaussian period]]s (the [[Hilbert–Speiser theorem]]). On the other hand the [[Gaussian rational|Gaussian field]] does not. This is an example of a ''necessary'' condition found by [[Emmy Noether]] (''perhaps known earlier?''). What matters here is ''tame'' [[ramification]]. In terms of the [[discriminant of an algebraic number field|discriminant]] ''D'' of ''L'', and taking still ''K'' = '''Q''', no prime ''p'' must divide ''D'' to the power ''p''. Then Noether's theorem states that tame ramification is necessary and sufficient for ''OL'' to be a [[projective module]] over '''Z'''[''G'']. It is certainly therefore necessary for it to be a ''free'' module. It leaves the question of the gap between free and projective, for which a large theory has now been built up.

A classical result, based on a result of [[David Hilbert]], is that a tamely ramified [[abelian number field]] has a normal integral basis. This may be seen by using the [[Kronecker–Weber theorem]] to embed the abelian field into a cyclotomic field.<ref name=F8>Fröhlich (1983) p.8</ref>

==Galois representations in number theory==
Many objects that arise in number theory are naturally Galois representations. For example, if ''L'' is a [[Galois extension]] of a [[number field]] ''K'', the [[ring of integers]] ''OL'' of ''L'' is a Galois module over ''OK'' for the Galois group of ''L''/''K'' (see [[Hilbert–Speiser theorem]]). If ''K'' is a local field, the multiplicative group of its separable closure is a module for the absolute Galois group of ''K'' and its study leads to [[local class field theory]]. For [[global class field theory]], the union of the [[idele class group]]s of all finite [[separable extension]]s of ''K'' is used instead.

There are also Galois representations that arise from auxiliary objects and can be used to study Galois groups. An important family of examples are the [[Tate module|ℓ-adic Tate modules]] of [[abelian variety|abelian varieties]].

===Artin representations===
Let ''K'' be a number field. [[Emil Artin]] introduced a class of Galois representations of the absolute Galois group ''GK'' of ''K'', now called '''Artin representations'''. These are the [[continuous function|continuous]] finite-dimensional linear representations of ''GK'' on [[complex vector space]]s. Artin's study of these representations led him to formulate the [[Artin reciprocity law]] and conjecture what is now called the [[Artin conjecture (L-functions)|Artin conjecture]] concerning the [[holomorphy]] of [[Artin L-function|Artin ''L''-functions]].

Because of the incompatibility of the [[profinite topology]] on ''GK'' and the usual (Euclidean) topology on complex vector spaces, the [[image (mathematics)|image]] of an Artin representation is always finite.

===ℓ-adic representations===
Let ℓ be a [[prime number]]. An '''ℓ-adic representation''' of ''GK'' is a continuous [[group homomorphism]] {{nowrap|ρ : ''GK'' → Aut(''M'')}} where ''M'' is either a finite-dimensional vector space over {{overline|'''Q'''}}ℓ (the algebraic closure of the [[p-adic number|ℓ-adic numbers]] '''Q'''ℓ) or a [[finitely generated module|finitely generated]] {{overline|'''Z'''}}ℓ-module (where {{overline|'''Z'''}}ℓ is the [[integral closure]] of '''Z'''ℓ in {{overline|'''Q'''}}ℓ). The first examples to arise were the [[p-adic cyclotomic character|ℓ-adic cyclotomic character]] and the ℓ-adic Tate modules of abelian varieties over ''K''. Other examples come from the Galois representations of modular forms and automorphic forms, and the Galois representations on ℓ-adic cohomology groups of algebraic varieties.

Unlike Artin representations, ℓ-adic representations can have infinite image. For example, the image of ''G'''''Q''' under the ℓ-adic cyclotomic character is <math>\mathbf{Z}_\ell^\times</math>. ℓ-adic representations with finite image are often called Artin representations. Via an isomorphism of {{overline|'''Q'''}}ℓ with '''C''' they can be identified with ''bona fide'' Artin representations.

===Mod ℓ representations===

These are representations over a finite field of characteristic ℓ. They often arise as the reduction mod ℓ of an ℓ-adic representation.

===Local conditions on representations===

There are numerous conditions on representations given by some property of the representation restricted to a decomposition group of some prime. The terminology for these conditions is somewhat chaotic, with different authors inventing different names for the same condition and using the same name with different meanings. Some of these conditions include:

*Abelian representations. This means that the image of the Galois group in the representations is abelian.
*Absolutely irreducible representations. These remain irreducible over an algebraic closure of the field.
*Barsotti–Tate representations. These are similar to finite flat representations.
*Crystalline representations.
*de Rham representations.
*Finite flat representations. (This name is a little misleading, as they are really profinite rather than finite.) These can be constructed as a projective limit of representations of the Galois group on a finite flat group scheme.
*Good representations. These are similar to finite flat representations.
*Hodge–Tate representations.
*Irreducible representations. These are irreducible in the sense that the only subrepresentation is the whole space or zero.
*Minimally ramified representations.
*Modular representations. These are representations coming from a modular form.
*Ordinary representations. These are 2-dimensional representations that are reducible with a 1-dimensional subrepresentation, such that the inertia group acts in a certain way on the submodule and the quotient. The exact condition depends on the author; for example it might act trivially on the quotient and by the character ε on the submodule.
*Potentially something representations. This means that the representations restricted to an open subgroup of finite index has some property.
*Reducible representations. These have a proper non-zero sub-representation.
*Semistable representations. These are two dimensional representations related to the representations coming from semistable elliptic curves.
*Tamely ramified representations. These are trivial on the (first) ramification group.
*Unramified representations. These are trivial on the inertia group.
*Wildly ramified representations. These are non-trivial on the (first) ramification group.

==Representations of the Weil group==
If ''K'' is a local or global field, the theory of [[class formation]]s attaches to ''K'' its [[Weil group of a class formation|Weil group]] ''WK'', a continuous group homomorphism {{nowrap|φ : ''WK'' → ''GK''}}, and an [[isomorphism]] of [[topological group]]s
:<math>r_K:C_K\tilde{\rightarrow}W_K^{\text{ab}}</math>
where ''CK'' is ''K''× or the [[idele class group]] ''IK''/''K''× (depending on whether ''K'' is local or global) and {{SubSup|''W''|''K''|ab}} is the [[abelianization]] of the Weil group of ''K''. Via φ, any representation of ''GK'' can be considered as a representation of ''WK''. However, ''WK'' can have strictly more representations than ''GK''. For example, via ''rK'' the continuous complex characters of ''WK'' are in bijection with those of ''CK''. Thus, the absolute value character on ''CK'' yields a character of ''WK'' whose image is infinite and therefore is not a character of ''GK'' (as all such have finite image).

An ℓ-adic representation of ''WK'' is defined in the same way as for ''GK''. These arise naturally from geometry: if ''X'' is a smooth projective variety over ''K'', then the ℓ-adic cohomology of the geometric fibre of ''X'' is an ℓ-adic representation of ''GK'' which, via φ, induces an ℓ-adic representation of ''WK''. If ''K'' is a local field of residue characteristic ''p'' ≠ ℓ, then it is simpler to study the so-called Weil–Deligne representations of ''WK''.

===Weil–Deligne representations===
Let ''K'' be a local field. Let ''E'' be a field of characteristic zero. A '''Weil–Deligne representation''' over ''E'' of ''WK'' (or simply of ''K'') is a pair (''r'', ''N'') consisting of
* a continuous group homomorphism {{nowrap|''r'' : ''WK'' → Aut''E''(''V'')}}, where ''V'' is a finite-dimensional vector space over ''E'' equipped with the [[discrete topology]],
* a [[nilpotent]] [[endomorphism]] {{nowrap|''N'' : ''V'' → ''V''}} such that ''r''(''w'')N''r''(''w'')−1= ||''w''||''N'' for all ''w'' ∈ ''WK''.<ref>Here ||''w''|| is given by {{SubSup|''q''|''K''|''v''(''w'')}} where ''qK'' is the size of the residue field of ''K'' and ''v''(''w'') is such that ''w'' is equivalent to the −''v''(''w'')th power of the (arithmetic) Frobenius of ''WK''.</ref>
These representations are the same as the representations over ''E'' of the [[Weil–Deligne group]] of ''K''.

If the residue characteristic of ''K'' is different from ℓ, [[Grothendieck]]'s [[ℓ-adic monodromy theorem]] sets up a bijection between ℓ-adic representations of ''WK'' (over {{overline|'''Q'''}}ℓ) and Weil–Deligne representations of ''WK'' over {{overline|'''Q'''}}ℓ (or equivalently over '''C'''). These latter have the nice feature that the continuity of ''r'' is only with respect to the discrete topology on ''V'', thus making the situation more algebraic in flavor.

==See also==
*[[Compatible system of ℓ-adic representations]]

==Notes==
{{reflist}}

==References==
*{{Citation
|last=Kudla
|first=Stephen S.
|contribution=The local Langlands correspondence: the non-archimedean case
|title=Motives, Part 2
|pages=365–392
|series=Proc. Sympos. Pure Math.
|volume=55
|publisher=Amer. Math. Soc.
|publication-place=Providence, R.I.
|year=1994
|isbn=978-0-8218-1635-6
}}
*{{Neukirch et al. CNF}}
*{{Citation
|last=Tate
|first=John
|author-link=John Tate
|contribution=Number theoretic background
|url=http://www.ams.org/online_bks/pspum332/
|title=Automorphic forms, representations, and L-functions, Part 2
|pages=3–26
|series=Proc. Sympos. Pure Math.
|volume=33
|publisher=Amer. Math. Soc.
|publication-place=Providence, R.I.
|year=1979
|isbn=978-0-8218-1437-6
}}

==Further reading==
* {{citation | last=Snaith | first=Victor P. | title=Galois module structure | series=Fields Insitute monographs | location=Providence, RI | publisher=[[American Mathematical Society]] | year=1994 | isbn=0-8218-0264-X | zbl=0830.11042 }}
* {{citation | last=Fröhlich | first=Albrecht | authorlink=Albrecht Fröhlich | title=Galois module structure of algebraic integers | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge | volume=1 | location=Berlin-Heidelberg-New York-Tokyo | publisher=[[Springer-Verlag]] | year=1983 | isbn=3-540-11920-5 | zbl=0501.12012 }}

[[Category:Algebraic number theory]]
[[Category:Galois theory]]