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2014-08-10T08:38:40Z

KerstinTibbs:

{{DISPLAYTITLE:''abc'' conjecture}}
The '''''abc'' conjecture''' (also known as '''Oesterlé–Masser conjecture''') is a [[conjecture]] in [[number theory]], first proposed by [[Joseph Oesterlé]] and [[David Masser]] in 1985. The conjecture is stated in terms of three positive integers, ''a'', ''b'' and ''c'' (whence comes the name), which have no common factor and satisfy ''a'' + ''b'' = ''c''. If ''d'' denotes the product of the distinct [[prime factor]]s of ''abc'', the conjecture essentially states that ''d'' is rarely much smaller than ''c''.

Although there is no obvious strategy for resolving the problem, it has already become well known for the number of [[#Some consequences|interesting consequences]] it entails. Many famous conjectures and theorems in number theory would [[#Some consequences|follow immediately]] from the ''abc'' conjecture. {{harvtxt|Goldfeld|1996}} described the ''abc'' conjecture as "the most important unsolved problem in [[Diophantine analysis]]".

{{unsolved|mathematics|
Are there for every ''ε'' > 0, only finitely many triples of [[coprime]] [[positive integer]]s

<center>''a'' + ''b'' = ''c''</center>

such that

<center>''c''&ensp;>&ensp;''d''&ensp;(1+''ε'')</center>

where ''d'' denotes the product of the distinct prime factors of ''abc''?}}

==Formulations==
For a [[positive integer]] ''n'', the [[radical of an integer|radical]] of ''n'', denoted rad(''n''), is the product of the distinct [[prime factor]]s of ''n''. For example

* rad(16) = rad(24) = 2,
* rad(17) = 17,
* rad(18) = rad(2·32) = 2·3 = 6.

If ''a'', ''b'', and ''c'' are [[coprime]]<ref name="Ref_">Note that when it is given that ''a'' + ''b'' = ''c'', coprimeness of ''a'', ''b'', ''c'' implies [[pairwise coprime]]ness of ''a'', ''b'', ''c''. So in this case, it does not matter which concept we use.</ref> positive integers such that ''a'' + ''b'' = ''c'', it turns out that
"usually" ''c'' < rad(''abc''). The ''abc conjecture'' deals with the exceptions. Specifically, it states that for every ε>0 there exist only finitely many triples (''a'',''b'',''c'') of positive coprime integers with ''a'' + ''b'' = ''c'' such that
:<math>c>\operatorname{rad}(abc)^{1+\varepsilon}.</math>

An equivalent formulation states that for any ''ε'' > 0, there exists a constant ''K'' such that, for all triples of coprime positive integers (''a'', ''b'', ''c'') satisfying ''a'' + ''b'' = ''c'', the inequality

:<math>c < K \cdot \operatorname{rad}(abc)^{1+\varepsilon}</math>

holds.

A third formulation of the conjecture involves the ''quality''
''q''(''a'', ''b'', ''c'') of the
triple (''a'', ''b'', ''c''), defined by:

: <math> q(a, b, c) = \frac{ \log(c) }{ \log( \operatorname{rad}( abc ) ) }. </math>

For example

* ''q''(4, 127, 131) = log(131) / log(rad(4·127·131)) = log(131) / log(2·127·131) = 0.46820...
* ''q''(3, 125, 128) = log(128) / log(rad(3·125·128)) = log(128) / log(30) = 1.426565...

A typical triple (''a'', ''b'', ''c'') of coprime positive integers with ''a'' + ''b'' = ''c'' will have ''c'' < rad(''abc''), i.e. ''q''(''a'', ''b'', ''c'') < 1. Triples with ''q'' > 1 such as in the second example are rather special, they consist of numbers divisible by high powers of small prime numbers.

The ''abc conjecture'' states that, for any ''ε'' > 0, there exist only finitely many triples (''a'', ''b'', ''c'') of coprime positive integers with ''a'' + ''b'' = ''c'' such that ''q''(''a'', ''b'', ''c'') > 1 + ''ε''.

Whereas it is known that there are infinitely many triples (''a'', ''b'', ''c'') of coprime positive integers with ''a'' + ''b'' = ''c'' such that ''q''(''a'', ''b'', ''c'') > 1, the conjecture predicts that only finitely many of those have ''q'' > 1.01 or ''q'' > 1.001 or even ''q'' > 1.0001, etc.

==Examples of triples with small radical==
The condition that ε > 0 is necessary for the truth of the conjecture, as there exist infinitely many triples ''a'', ''b'', ''c'' with rad(''abc'') < ''c''. For instance, such a triple may be taken as
:''a'' = 1
:''b'' = 26''n'' − 1
:''c'' = 26''n''.
As ''a'' and ''c'' together contribute only a factor of two to the radical, while ''b'' is divisible by 9, rad(''abc'') < 2''c''/3 for these examples. By replacing the exponent 6''n'' by other exponents forcing ''b'' to have larger square factors, the ratio between the radical and ''c'' may be made arbitrarily large. Another triple with a particularly small radical was found by Eric Reyssat {{harv|Lando|Zvonkin|2004|p=137}}:
:''a'' = 2:
:''b'' = 310 109 = 6436341
:''c'' = 235 = 6436343
:rad(''abc'') = 15042.

==Some consequences==
The conjecture has not been proven, but it has a large number of interesting consequences. These include both known results, and conjectures for which it gives a [[conditional proof]].
* [[Thue–Siegel–Roth theorem]] on diophantine approximation of algebraic numbers
* [[Fermat's Last Theorem]] for all sufficiently large exponents (proven in general by [[Andrew Wiles]])
* The [[Mordell conjecture]] {{harv|Elkies|1991}}
* The [[Erdős–Woods number|Erdős–Woods conjecture]] except for a finite number of counterexamples {{harv|Langevin|1993}}
* The existence of infinitely many [[Wieferich prime|non-Wieferich primes]] {{harv|Silverman|1988}}
* The weak form of [[Marshall Hall's conjecture]] on the separation between squares and cubes of integers {{harv|Nitaj|1996}}
* The [[Fermat–Catalan conjecture]], a generalization of Fermat's last theorem concerning powers that are sums of powers {{harv|Pomerance|2008}}
* The [[Dirichlet L-function|L function]] ''L''(''s'',(−''d''/.)) formed with the [[Legendre symbol]], has no [[Siegel zero]] (this consequence actually requires a uniform version of the abc conjecture in number fields, not only the abc conjecture as formulated above for rational integers)
* ''P''(''x'') has only finitely many perfect powers for [[integer|integral]] ''x'' for ''P'' a [[polynomial]] with at least three simple zeros.<ref name="Ref_a">http://www.math.uu.nl/people/beukers/ABCpresentation.pdf</ref>
* A generalization of [[Tijdeman's theorem]]
* It is equivalent to the Granville–Langevin conjecture
* It is equivalent to the [[modified Szpiro conjecture]].
* {{harvtxt|Dąbrowski|1996}} has shown that the abc conjecture implies that ''n''! + ''A''= ''k''2 has only finitely many solutions for any given integer ''A''.

While the first group of these have now been proven, the abc conjecture itself remains of interest, because of its numerous links with deep questions in number theory.

== Theoretical results ==
It remains unknown whether ''c'' can be [[upper bound]]ed by a near-linear function of the radical of ''abc'', as the abc conjecture states, or even whether it can be bounded by a [[polynomial]] of rad(''abc''). However, [[exponential function|exponential]] bounds are known. Specifically, the following bounds have been proven:

:<math>c < \exp{(K_1 \operatorname{rad}(abc)^{15}) } </math> {{harv|Stewart|Tijdeman|1986}},

:<math>c < \exp{ (K_2 \operatorname{rad}(abc)^{2/3+\varepsilon}) } </math> {{harv|Stewart|Yu|1991}}, and

:<math>c < \exp{ (K_3 \operatorname{rad}(abc)^{1/3+\varepsilon}) } </math> {{harv|Stewart|Yu|2001}}.

In these bounds, ''K''1 is a [[Constant (mathematics)|constant]] that does not depend on ''a'', ''b'', or ''c'', and ''K''2 and ''K''3 are constants that depend on ε (in an [[effectively computable]] way) but not on ''a'', ''b'', or ''c''. The bounds apply to any triple for which ''c'' > 2.

==Computational results==

In 2006, the Mathematics Department of [[Leiden University]] in the Netherlands, together with the Dutch [[Kennislink]] science institute, launched the [[ABC@Home]] project, a [[grid computing]] system which aims to discover additional triples ''a'', ''b'', ''c'' with rad(''abc'') < ''c''. Although no finite set of examples or counterexamples can resolve the abc conjecture, it is hoped that patterns in the triples discovered by this project will lead to insights about the conjecture and about number theory more generally.

{| class="wikitable sortable collapsible" border="1" style="text-align:right;"
|+ Distribution of triples with ''q'' > 1<ref name="Ref_d">{{Citation |url=http://www.rekenmeemetabc.nl/?item=h_stats |title=Synthese resultaten |work=RekenMeeMetABC.nl |accessdate=January 1, 2011 }} {{nl icon}}.</ref>
|-
! scope="col" |  
! scope="col" | ''q'' > 1
! scope="col" | ''q'' > 1.05
! scope="col" | ''q'' > 1.1
! scope="col" | ''q'' > 1.2
! scope="col" | ''q'' > 1.3
! scope="col" | ''q'' > 1.4
|-
! scope="row" | ''c'' < 102
| 6 || 4 || 4 || 2 || 0 || 0
|-
! scope="row" | ''c'' < 103
| 31 || 17 || 14 || 8 || 3 || 1
|-
! scope="row" | ''c'' < 104
| 120 || 74 || 50 || 22 || 8 || 3
|-
! scope="row" | ''c'' < 105
| 418 || 240 || 152 || 51 || 13 || 6
|-
! scope="row" | ''c'' < 106
| 1.268 || 667 || 379 || 102 || 29 || 11
|-
! scope="row" | ''c'' < 107
| 3.499 || 1.669 || 856 || 210 || 60 || 17
|-
! scope="row" | ''c'' < 108
| 8.987 || 3.869 || 1.801 || 384 || 98 || 25
|-
! scope="row" | ''c'' < 109
| 22.316 || 8.742 || 3.693 || 706 || 144 || 34
|-
! scope="row" | ''c'' < 1010
| 51.677 || 18.233 || 7.035 || 1.159 || 218 || 51
|-
! scope="row" | ''c'' < 1011
| 116.978 || 37.612 || 13.266 || 1.947 || 327 || 64
|-
! scope="row" | ''c'' < 1012
| 252.856 || 73.714 || 23.773 || 3.028 || 455 || 74
|-
! scope="row" | ''c'' < 1013
| 528.275 || 139.762 || 41.438 || 4.519 || 599 || 84
|-
! scope="row" | ''c'' < 1014
| 1.075.319 || 258.168 || 70.047 || 6.665 || 769 || 98
|-
! scope="row" | ''c'' < 1015
| 2.131.671 || 463.446 || 115.041 || 9.497 || 998 || 112
|-
! scope="row" | ''c'' < 1016
| 4.119.410 || 812.499 || 184.727 || 13.118 || 1.232 || 126
|-
! scope="row" | ''c'' < 1017
| 7.801.334 || 1.396.909 || 290.965 || 17.890 || 1.530 || 143
|-
! scope="row" | ''c'' < 1018
| 14.482.059 || 2.352.105 || 449.194 || 24.013 || 1.843 || 160
|-
|}

{{As of|2012|06}}, ABC@Home has found 23.1 million triples, and its present goal is to obtain a complete list of all ABC triples (a,b,c) with c no more than 1020.<ref name="Ref_c">{{Citation |url=http://abcathome.com/data/ |title=Data collected sofar |work=ABC@Home |accessdate=June 9, 2012 }}</ref>

{| class="wikitable sortable collapsible" border="1"
|+ Highest quality triples<ref>{{cite web |url=http://www.math.leidenuniv.nl/~desmit/abc/index.php?set=1 |title=100 unbeaten triples |work=Reken mee met ABC |date=2010-11-07 }}</ref>
|-
! scope="col" |  
! scope="col" | ''q''
! scope="col" | ''a''
! scope="col" | ''b''
! scope="col" | ''c''
! scope="col" class="unsortable" | Discovered by
|-
! scope="row" | 1
| 1.6299 || 2 || 310109 || 235 || Eric Reyssat
|-
! scope="row" | 2
| 1.6260 || 112 || 325673 || 22123 || Benne de Weger
|-
! scope="row" | 3
| 1.6235 || 19·1307 || 7·292318 || 2832254 || Jerzy Browkin, Juliusz Brzezinski
|-
! scope="row" | 4
| 1.5444 || 724123113 || 111613279 || 2·33523953 || Abderrahmane Nitaj
|-
! scope="row" | 5
| 1.4805 || 52279·45949 || 321318613 || 223174251217333 || Frank Rubin
|}

where the ''quality'' ''q''(''a'', ''b'', ''c'') of the triple (''a'', ''b'', ''c''), defined by:

: <math> q(a, b, c) = \frac{ \log(c) }{ \log( \operatorname{rad}( abc ) ) }. </math>

==Refined forms and generalizations==
A stronger inequality proposed in 1996 by [[Alan Baker (mathematician)|Alan Baker]] states that in the [[inequality (mathematics)|inequality]], one can replace rad(''abc'') by

:ε−ωrad(''abc''),

where ω is the total number of distinct primes dividing ''a'', ''b'' and ''c'' {{Harv|Bombieri|Gubler|2006|p=404}}. A related conjecture of [[Andrew Granville]] states that on the [[Left-hand side and right-hand side of an equation|RHS]] we could also put

:O(rad(''abc'') Θ(rad(''abc'')))

where Θ(''n'') is the number of integers up to ''n'' divisible only by primes dividing ''n''.

{{harvtxt|Browkin|Brzeziński|1994}} formulated the ''n''-conjecture—a version of the ''abc'' conjecture involving <math>n>2</math> integers.

==See also==
*[[Mason–Stothers theorem]], an analogous statement for [[polynomial]]s.

==Notes==
{{reflist|colwidth=45em}}

== References ==
{{Refbegin}}
*{{cite book | first1=Enrico | last1=Bombieri | authorlink1=Enrico Bombieri | first2=Walter | last2=Gubler | title=Heights in Diophantine Geometry | series=New Mathematical Monographs | volume=4 | publisher=[[Cambridge University Press]] | year=2006 | isbn=978-0-521-71229-3 | zbl=1130.11034 | doi=10.2277/0521846153 }}
*{{Cite journal |authorlink=Jerzy Browkin |first=Jerzy |last=Browkin |first2=Juliusz |last2=Brzeziński | title=Some remarks on the ''abc''-conjecture | journal=Math. Comp. | volume=62 | pages=931–939 | year=1994 | doi=10.2307/2153551 | jstor=2153551 | issue=206 |ref=harv }}
* {{cite book |last=Browkin |first=Jerzy |chapter=The ''abc''-conjecture |editor1-last=Bambah |editor1-first=R. P. |editor2-last=Dumir |editor2-first=V. C. |editor3-last=Hans-Gill |editor3-first=R. J. |year=2000 |title=Number Theory |series=Trends in Mathematics |location=Basel |publisher=Birkhäuser |isbn=3-7643-6259-6 |pages=75–106 |ref=harv }}
*{{Cite journal |first=Andrzej |last=Dąbrowski |title=On the diophantine equation <math>x!+A=y^2</math> | journal=Nieuw Archief voor Wiskunde, IV. |volume=14 |pages=321–324 |year=1996 |ref=harv }}
*{{Cite journal |last=Elkies |first=N. D. |authorlink=Noam Elkies |title=ABC implies Mordell |journal=Intern. Math. Research Notices |volume=7 |year=1991 |pages=99–109 |doi=10.1155/S1073792891000144 |issue=7 |ref=harv }}
*{{cite journal |last=Goldfeld |first=Dorian |authorlink=Dorian M. Goldfeld |year=1996 |title=Beyond the last theorem |journal=[[Math Horizons]] |issue=September |pages=26–34 |ref=harv }}
* {{cite book |editor1-last=Gowers |editor1-first=Timothy |editor1-link=Timothy Gowers |editor2-last=Barrow-Green |editor2-first=June |editor3-last=Leader |editor3-first=Imre |title=[[The Princeton Companion to Mathematics]] |year=2008 |publisher=Princeton University Press |location=Princeton |isbn=978-0-691-11880-2 |pages=361–362, 681 |ref=harv }}
* {{cite book |first=Richard K. |last=Guy |authorlink=Richard K. Guy |title=Unsolved Problems in Number Theory |publisher=[[Springer-Verlag]] |location=Berlin |year=2004 |isbn=0-387-20860-7 |ref=harv }}
* {{cite book |last=Lando |first=Sergei K. |first2=Alexander K. |last2=Zvonkin |title=Graphs on Surfaces and Their Applications |publisher=Springer-Verlag |work=Encyclopaedia of Mathematical Sciences: Lower-Dimensional Topology II |volume=141 |year=2004 |isbn=3-540-00203-0 |ref=harv }}
*{{Cite journal |last=Langevin |first=M. |year=1993 |title={{lang|fr|Cas d'égalité pour le théorème de Mason et applications de la conjecture ''abc''}} |journal=Comptes rendus de l'Académie des sciences |volume=317 |issue=5 |pages=441–444 |doi= |ref=harv }} {{fr icon}}
*{{Cite journal |last=Nitaj |first=Abderrahmane |title={{lang|fr|La conjecture ''abc''}} |journal=Enseign. Math. |volume=42 |issue=1–2 |pages=3–24 |year=1996 |ref=harv }} {{fr icon}}
*{{Cite book |last=Pomerance |first=Carl |authorlink=Carl Pomerance |chapter=Computational Number Theory |title=The Princeton Companion to Mathematics |location= |publisher=Princeton University Press |year=2008 |pages=361–362 |ref=harv }}
*{{Cite journal |last=Silverman |first=Joseph H. |year=1988 |title=Wieferich's criterion and the ''abc''-conjecture |journal=Journal of Number Theory |volume=30 |issue=2 |pages=226–237 |doi=10.1016/0022-314X(88)90019-4 | zbl=0654.10019 |ref=harv }}
*{{Cite journal |last=Stewart |first=C. L. |authorlink=Cameron Leigh Stewart |last2=Tijdeman |first2=R. |authorlink2=Robert Tijdeman |year=1986 |title=On the Oesterlé-Masser conjecture |journal=Monatshefte für Mathematik |volume=102 |issue=3 |pages=251–257 |doi=10.1007/BF01294603 |ref=harv }}
*{{Cite journal |last=Stewart |first=C. L. |authorlink2=Kunrui Yu |first2=Kunrui |last2=Yu |year=1991 |title=On the ''abc'' conjecture |journal=[[Mathematische Annalen]] |volume=291 |issue=1 |pages=225–230 |doi=10.1007/BF01445201 |ref=harv }}
*{{Cite journal |last=Stewart |first=C. L. |first2=Kunrui |last2=Yu |year=2001 |title=On the ''abc'' conjecture, II |journal=[[Duke Mathematical Journal]] |volume=108 |issue=1 |pages=169–181 |doi=10.1215/S0012-7094-01-10815-6 |ref=harv }}
{{Refend}}

==External links==
* [http://abcathome.com/ ABC@home] [[Distributed Computing]] project called [[ABC@Home]].
* [http://bit-player.org/2007/easy-as-abc Easy as ABC]: Easy to follow, detailed explanation by Brian Hayes.
* {{MathWorld | urlname=abcConjecture | title=abc Conjecture}}
* Abderrahmane Nitaj's [http://www.math.unicaen.fr/~nitaj/abc.html ABC conjecture home page]
* Bart de Smit's [http://www.math.leidenuniv.nl/~desmit/abc/ ABC Triples webpage]
* http://www.math.columbia.edu/~goldfeld/ABC-Conjecture.pdf
* [http://www.maa.org/mathland/mathtrek_12_8.html The amazing ABC conjecture]
* [http://www.thehcmr.org/issue1_1/elkies.pdf The ABC's of Number Theory] by Noam D. Elkies

[[Category:Conjectures]]
[[Category:Number theory]]

[[bn:Abc অনুমান]]
[[de:Abc-Vermutung]]
[[es:Conjetura abc]]
[[fr:Conjecture abc]]
[[it:Congettura abc]]
[[ja:ABC予想]]
[[hu:Abc-sejtés]]
[[nl:ABC-vermoeden]]
[[pl:Hipoteza ABC]]
[[fi:Abc-konjektuuri]]
[[tr:Abc sanısı]]
[[vi:Giả định abc]]
[[zh:Abc猜想]]

Main Page

2014-08-10T02:11:32Z

KerstinTibbs:

{{Otheruses4|the mathematics of the chi-squared distribution|its uses in statistics|chi-squared test|the music group|Chi2 (band)}}

{{Probability distribution
| type = density
| pdf_image = [[File:chi-square pdf.svg|325px]]
| cdf_image = [[File:chi-square distributionCDF.svg|325px]]
| notation = <math>\chi^2(k)\!</math> or <math>\chi^2_k\!</math>
| parameters = <math>k \in \mathbb{N}~~</math> (known as "degrees of freedom")
| support = ''x'' ∈ [0, +∞)
| pdf = <math>\frac{1}{2^{\frac{k}{2}}\Gamma\left(\frac{k}{2}\right)}\; x^{\frac{k}{2}-1} e^{-\frac{x}{2}}\,</math>
| cdf = <math>\frac{1}{\Gamma\left(\frac{k}{2}\right)}\;\gamma\left(\frac{k}{2},\,\frac{x}{2}\right)</math>
| mean = ''k''
| median = <math>\approx k\bigg(1-\frac{2}{9k}\bigg)^3</math>
| mode = max{ ''k'' − 2, 0 }
| variance = 2''k''
| skewness = <math>\scriptstyle\sqrt{8/k}\,</math>
| kurtosis = 12 / ''k''
| entropy = <math>\frac{k}{2}\!+\!\ln(2\Gamma(k/2))\!+\!(1\!-\!k/2)\psi(k/2)</math>
| mgf = {{nowrap|(1 − 2 ''t'')−''k''/2}}   for  t  < ½
| char = {{nowrap|(1 − 2 ''i'' ''t'')−''k''/2}}      <ref>{{cite web | url=http://www.planetmathematics.com/CentralChiDistr.pdf | title=Characteristic function of the central chi-squared distribution | author=M.A. Sanders | accessdate=2009-03-06}}</ref>
}}

In [[probability theory]] and [[statistics]], the '''chi-squared distribution''' (also '''chi-square''' or {{nowrap|1='''[[chi (letter)|''χ'']]²-distribution'''}}) with ''k'' [[Degrees of freedom (statistics)|degrees of freedom]] is the distribution of a sum of the squares of ''k'' [[Independence (probability theory)|independent]] [[standard normal]] random variables. It is one of the most widely used [[probability distribution]]s in [[inferential statistics]], e.g., in [[hypothesis testing]] or in construction of [[confidence interval]]s.<ref name=abramowitz>{{Abramowitz_Stegun_ref|26|940}}</ref><ref>NIST (2006). [http://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm Engineering Statistics Handbook - Chi-Squared Distribution]</ref><ref>{{cite book
| last = Jonhson
| first = N.L.
| coauthors = S. Kotz, , N. Balakrishnan
| title = Continuous Univariate Distributions (Second Ed., Vol. 1, Chapter 18)
| publisher = John Willey and Sons
| year = 1994
| isbn = 0-471-58495-9
}}</ref><ref>{{cite book
| last = Mood
| first = Alexander
| coauthors = Franklin A. Graybill, Duane C. Boes
| title = Introduction to the Theory of Statistics (Third Edition, p. 241-246)
| publisher = McGraw-Hill
| year = 1974
| isbn = 0-07-042864-6
}}</ref> When there is a need to contrast it with the [[noncentral chi-squared distribution]], this distribution is sometimes called the '''central chi-squared distribution'''.

The chi-squared distribution is used in the common [[chi-squared test]]s for [[goodness of fit]] of an observed distribution to a theoretical one, the [[statistical independence|independence]] of two criteria of classification of [[data analysis|qualitative data]], and in [[confidence interval]] estimation for a population [[standard deviation]] of a normal distribution from a sample standard deviation. Many other statistical tests also use this distribution, like [[Friedman test|Friedman's analysis of variance by ranks]].

The chi-squared distribution is a special case of the [[gamma distribution]].

==Definition==
If ''Z''1, ..., ''Z''''k'' are [[independence (probability theory)|independent]], [[standard normal]] random variables, then the sum of their squares,
: <math>
Q\ = \sum_{i=1}^k Z_i^2 ,
</math>
is distributed according to the '''chi-squared distribution''' with ''k'' degrees of freedom. This is usually denoted as
: <math>
Q\ \sim\ \chi^2(k)\ \ \text{or}\ \ Q\ \sim\ \chi^2_k .
</math>

The chi-squared distribution has one parameter: ''k'' — a positive integer that specifies the number of [[degrees of freedom (statistics)|degrees of freedom]] (i.e. the number of ''Z''''i''’s)

==Characteristics==
Further properties of the chi-squared distribution can be found in the box at the upper right corner of this article.

===Probability density function===
The [[probability density function]] (pdf) of the chi-squared distribution is
:<math>
f(x;\,k) =
\begin{cases}
\frac{x^{(k/2)-1} e^{-x/2}}{2^{k/2} \Gamma\left(\frac{k}{2}\right)}, & x \geq 0; \\ 0, & \text{otherwise}.
\end{cases}
</math>
where Γ(''k''/2) denotes the [[Gamma function]], which has [[particular values of the Gamma function|closed-form values for odd ''k'']].

For derivations of the pdf in the cases of one and two degrees of freedom, see [[Proofs related to chi-squared distribution]].

===Cumulative distribution function===
Its [[cumulative distribution function]] is:
: <math>
F(x;\,k) = \frac{\gamma(\frac{k}{2},\,\frac{x}{2})}{\Gamma(\frac{k}{2})} = P\left(\frac{k}{2},\,\frac{x}{2}\right),
</math>
where γ(''k'',''z'') is the [[incomplete Gamma function|lower incomplete Gamma function]] and ''P''(''k'',''z'') is the [[regularized Gamma function]].

In a special case of ''k'' = 2 this function has a simple form:
: <math>
F(x;\,2) = 1 - e^{-\frac{x}{2}}.
</math>

For the cases when ''0'' < ''z'' < ''1'' (which include all of the cases when this CDF is less than half), the following [[Chernoff_bound#The_first_step_in_the_proof_of_Chernoff_bounds| Chernoff upper bound]] may be obtained:<ref>{{cite journal |last1=Dasgupta |first1=Sanjoy D. A. |last2=Gupta |first2=Anupam K. |year=2002 |title=An Elementary Proof of a Theorem of Johnson and Lindenstrauss |journal=Random Structures and Algorithms |volume=22 |issue= |pages=60-65 |publisher= |doi= |url=http://cseweb.ucsd.edu/~dasgupta/papers/jl.pdf |accessdate=2012-05-01 }}</ref>
: <math>
F(z k;\,k) \leq (z e^{1-z})^{k/2}.
</math>

The tail bound for the cases when ''z'' > ''1'' follows similarly
: <math>
1-F(z k;\,k) \leq (z e^{1-z})^{k/2}.
</math>

Tables of this cumulative distribution function are widely available and the function is included in many [[spreadsheet]]s and all [[List of statistical packages|statistical packages]]. For another [[approximation]] for the CDF modeled after the cube of a Gaussian, see [[Noncentral_chi-squared_distribution#Approximation|under Noncentral chi-squared distribution]].

===Additivity===
It follows from the definition of the chi-squared distribution that the sum of independent chi-squared variables is also chi-squared distributed. Specifically, if {''Xi''}''i''=1''n'' are independent chi-squared variables with {''ki''}''i''=1''n'' degrees of freedom, respectively, then {{nowrap|''Y {{=}} X''1 + ⋯ + ''Xn''}} is chi-squared distributed with {{nowrap|''k''1 + ⋯ + ''kn''}} degrees of freedom.

===Information entropy===
The [[information entropy]] is given by
: <math>
H = \int_{-\infty}^\infty f(x;\,k)\ln f(x;\,k) \, dx
= \frac{k}{2} + \ln\left(2\Gamma\left(\frac{k}{2}\right)\right) + \left(1-\frac{k}{2}\right) \psi\left(\frac{k}{2}\right),
</math>
where ''ψ''(''x'') is the [[Digamma function]].

The Chi-squared distribution is the [[maximum entropy probability distribution]] for a random variate ''X'' for which <math>E(X)=\nu</math> and <math>E(\ln(X))=\psi\left(\frac{1}{2}\right)+\ln(2)</math> are fixed. <ref>{{cite journal |last1=Park |first1=Sung Y. |last2=Bera |first2=Anil K. |year=2009 |title=Maximum entropy autoregressive conditional heteroskedasticity model |journal=Journal of Econometrics |volume= |issue= |pages=219–230 |publisher=Elsevier |doi= |url=http://www.wise.xmu.edu.cn/Master/Download/..%5C..%5CUploadFiles%5Cpaper-masterdownload%5C2009519932327055475115776.pdf |accessdate=2011-06-02 }}</ref>

===Noncentral moments===
The moments about zero of a chi-squared distribution with ''k'' degrees of freedom are given by<ref>[http://mathworld.wolfram.com/Chi-SquaredDistribution.html Chi-squared distribution], from [[MathWorld]], retrieved Feb. 11, 2009</ref><ref>M. K. Simon, ''Probability Distributions Involving Gaussian Random Variables'', New York: Springer, 2002, eq. (2.35), ISBN 978-0-387-34657-1</ref>
: <math>
\operatorname{E}(X^m) = k (k+2) (k+4) \cdots (k+2m-2) = 2^m \frac{\Gamma(m+\frac{k}{2})}{\Gamma(\frac{k}{2})}.
</math>

===Cumulants===
The [[cumulant]]s are readily obtained by a (formal) power series expansion of the logarithm of the characteristic function:
: <math>
\kappa_n = 2^{n-1}(n-1)!\,k
</math>

===Asymptotic properties===
By the [[central limit theorem]], because the chi-squared distribution is the sum of ''k'' independent random variables with finite mean and variance, it converges to a normal distribution for large ''k''. For many practical purposes, for ''k'' > 50 the distribution is sufficiently close to a [[normal distribution]] for the difference to be ignored.<ref>{{cite book|title=Statistics for experimenters|author=Box, Hunter and Hunter|publisher=Wiley|page=46}}</ref> Specifically, if ''X'' ~ ''χ''²(''k''), then as ''k'' tends to infinity, the distribution of <math>(X-k)/\sqrt{2k}</math> [[convergence of random variables#Convergence in distribution|tends]] to a standard normal distribution. However, convergence is slow as the [[skewness]] is <math>\sqrt{8/k}</math> and the [[excess kurtosis]] is 12/''k''. Other functions of the chi-squared distribution converge more rapidly to a normal distribution. Some examples are:
* If ''X'' ~ ''χ''²(''k'') then <math>\scriptstyle\sqrt{2X}</math> is approximately normally distributed with mean <math>\scriptstyle\sqrt{2k-1}</math> and unit variance (result credited to [[R. A. Fisher]]).
* If ''X'' ~ ''χ''²(''k'') then <math>\scriptstyle\sqrt[3]{X/k}</math> is approximately normally distributed with mean <math>\scriptstyle 1-2/(9k)</math> and variance <math>\scriptstyle 2/(9k) .</math><ref>Wilson, E.B.; Hilferty, M.M. (1931) "The distribution of chi-squared". ''Proceedings of the National Academy of Sciences, Washington'', 17, 684–688.
</ref> This is known as the Wilson-Hilferty transformation.

==Relation to other distributions==
{{Ref improve section|date=September 2011}}

[[File:Chi_on_SAS.png|thumb|right|400px|Approximate formula for median compared with numerical quantile (top) as presented in [[SAS (software) | SAS Software]]. Difference between numerical quantile and approximate formula (bottom).]]
* As <math>k\to\infty</math>, <math> (\chi^2_k-k)/\sqrt{2k} \xrightarrow{d}\ N(0,1) \,</math> ([[normal distribution]])

*<math> \chi_k^2 \sim {\chi'}^2_k(0)</math> ([[Noncentral chi-squared distribution]] with non-centrality parameter <math> \lambda = 0 </math>)

*If <math>X \sim \mathrm{F}(\nu_1, \nu_2)</math> then <math>Y = \lim_{\nu_2 \to \infty} \nu_1 X</math> has the [[chi-squared distribution]] <math>\chi^2_{\nu_{1}}</math>

*As a special case, if <math>X \sim \mathrm{F}(1, \nu_2)\,</math> then <math>Y = \lim_{\nu_2 \to \infty} X\,</math> has the [[chi-squared distribution]] <math>\chi^2_{1}</math>

*<math> \|\boldsymbol{N}_{i=1,...,k}{(0,1)}\|^2 \sim \chi^2_k </math> (The squared [[Norm (mathematics)|norm]] of '''n''' standard normally distributed variables is a chi-squared distribution with '''k''' [[degrees of freedom (statistics)|degrees of freedom]])

*If <math>X \sim {\chi}^2(\nu)\,</math> and <math>c>0 \,</math>, then <math>cX \sim {\Gamma}(k=\nu/2, \theta=2c)\,</math>. ([[gamma distribution]])

*If <math>X \sim \chi^2_k</math> then <math>\sqrt{X} \sim \chi_k</math> ([[chi distribution]])

*If <math>X \sim \mathrm{Rayleigh}(1)\,</math> ([[Rayleigh distribution]]) then <math>X^2 \sim \chi^2(2)\,</math>

*If <math>X \sim \mathrm{Maxwell}(1)\,</math> ([[Maxwell distribution]]) then <math>X^2 \sim \chi^2(3)\,</math>

*If <math>X \sim \chi^2(\nu)</math> then <math>\tfrac{1}{X} \sim \mbox{Inv-}\chi^2(\nu)\, </math> ([[Inverse-chi-squared distribution]])

*The chi-squared distribution is a special case of type 3 [[Pearson distribution]]

* If <math>X \sim \chi^2(\nu_1)\,</math> and <math>Y \sim \chi^2(\nu_2)\,</math> are independent then <math>\tfrac{X}{X+Y} \sim {\rm Beta}(\tfrac{\nu_1}{2}, \tfrac{\nu_2}{2})\,</math> ([[beta distribution]])

*If <math> X \sim {\rm U}(0,1)\, </math> ([[Uniform distribution (continuous)|uniform distribution]]) then <math> -2\log{(U)} \sim \chi^2(2)\,</math>

* <math>\chi^2(6)\,</math> is a transformation of [[Laplace distribution]]

*If <math>X_i \sim \mathrm{Laplace}(\mu,\beta)\,</math> then <math>\sum_{i=1}^n{\frac{2 |X_i-\mu|}{\beta}} \sim \chi^2(2n)\,</math>

* chi-squared distribution is a transformation of [[Pareto distribution]]

* [[Student's t-distribution]] is a transformation of chi-squared distribution

* [[Student's t-distribution]] can be obtained from chi-squared distribution and [[normal distribution]]

* [[Noncentral beta distribution]] can be obtained as a transformation of chi-squared distribution and [[Noncentral chi-squared distribution]]

* [[Noncentral t-distribution]] can be obtained from normal distribution and chi-squared distribution

A chi-squared variable with ''k'' degrees of freedom is defined as the sum of the squares of ''k'' independent [[standard normal distribution|standard normal]] random variables.

If ''Y'' is a ''k''-dimensional Gaussian random vector with mean vector ''μ'' and rank ''k'' covariance matrix ''C'', then ''X'' = (''Y''−''μ'')T''C''−1(''Y''−''μ'') is chi-squared distributed with ''k'' degrees of freedom.

The sum of squares of [[statistically independent]] unit-variance Gaussian variables which do ''not'' have mean zero yields a generalization of the chi-squared distribution called the [[noncentral chi-squared distribution]].

If ''Y'' is a vector of ''k'' [[i.i.d.]] standard normal random variables and ''A'' is a ''k×k'' [[idempotent matrix]] with [[rank (linear algebra)|rank]] ''k−n'' then the [[quadratic form]] ''YTAY'' is chi-squared distributed with ''k−n'' degrees of freedom.

The chi-squared distribution is also naturally related to other distributions arising from the Gaussian. In particular,

* ''Y'' is [[F-distribution|F-distributed]], ''Y'' ~ ''F''(''k''1,''k''2) if <math>\scriptstyle Y = \frac{X_1 / k_1}{X_2 / k_2}</math> where ''X''1 ~ ''χ''²(''k''1) and ''X''2  ~ ''χ''²(''k''2) are statistically independent.

* If ''X'' is chi-squared distributed, then <math>\scriptstyle\sqrt{X}</math> is [[chi distribution|chi distributed]].

* If {{nowrap|''X''1  ~  ''χ''2''k''1}} and {{nowrap|''X''2  ~  ''χ''2''k''2}} are statistically independent, then {{nowrap|''X''1 + ''X''2  ~ ''χ''2''k''1+''k''2}}. If ''X''1 and ''X''2 are not independent, then {{nowrap|''X''1 + ''X''2}} is not chi-squared distributed.

==Generalizations==
The chi-squared distribution is obtained as the sum of the squares of ''k'' independent, zero-mean, unit-variance Gaussian random variables. Generalizations of this distribution can be obtained by summing the squares of other types of Gaussian random variables. Several such distributions are described below.
===Chi-squared distributions===
====Noncentral chi-squared distribution====
{{Main|Noncentral chi-squared distribution}}
The noncentral chi-squared distribution is obtained from the sum of the squares of independent Gaussian random variables having unit variance and ''nonzero'' means.

====Generalized chi-squared distribution====
{{Main|Generalized chi-squared distribution}}
The generalized chi-squared distribution is obtained from the quadratic form ''z′Az'' where ''z'' is a zero-mean Gaussian vector having an arbitrary covariance matrix, and ''A'' is an arbitrary matrix.

===Gamma, exponential, and related distributions===
The chi-squared distribution ''X'' ~ ''χ''²(''k'') is a special case of the [[gamma distribution]], in that ''X'' ~ Γ(''k''/2, 1/2) (using the shape parameterization of the gamma distribution) where ''k'' is an integer.

Because the [[exponential distribution]] is also a special case of the Gamma distribution, we also have that if ''X'' ~ ''χ''²(2), then ''X'' ~ Exp(1/2) is an [[exponential distribution]].

The [[Erlang distribution]] is also a special case of the Gamma distribution and thus we also have that if ''X'' ~ ''χ''²(''k'') with even ''k'', then ''X'' is Erlang distributed with shape parameter ''k''/2 and scale parameter 1/2.

==Applications==
The chi-squared distribution has numerous applications in inferential [[statistics]], for instance in [[chi-squared test]]s and in estimating [[variance]]s. It enters the problem of estimating the mean of a normally distributed population and the problem of estimating the slope of a [[linear regression|regression]] line via its role in [[Student’s t-distribution]]. It enters all [[analysis of variance]] problems via its role in the [[F-distribution]], which is the distribution of the ratio of two independent chi-squared [[random variable]]s, each divided by their respective degrees of freedom.

Following are some of the most common situations in which the chi-squared distribution arises from a Gaussian-distributed sample.

*if ''X''1, ..., ''Xn'' are [[independent identically-distributed random variables|i.i.d.]] ''N''(''μ'', ''σ''2) [[random variable]]s, then <math>\sum_{i=1}^n(X_i - \bar X)^2 \sim \sigma^2 \chi^2_{n-1}</math> where <math>\bar X = \frac{1}{n} \sum_{i=1}^n X_i</math>.

*The box below shows probability distributions with name starting with '''chi''' for some [[statistic]]s based on {{nowrap|''Xi'' ∼ Normal(''μi'', ''σ''2''i''), ''i'' {{=}} 1, ⋯, ''k'', }} independent random variables:
<center>
{| class="wikitable" align="center"
|-
! Name !! Statistic
|-
| chi-squared distribution || <math>\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2</math>
|-
| [[noncentral chi-squared distribution]] || <math>\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2</math>
|-
| [[chi distribution]] || <math>\sqrt{\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2}</math>
|-
| [[noncentral chi distribution]] || <math>\sqrt{\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2}</math>
|}
</center>

==Table of ''χ''2 value vs p-value==
The [[p-value]] is the probability of observing a test statistic ''at least'' as extreme in a chi-squared distribution. Accordingly, since the [[cumulative distribution function]] (CDF) for the appropriate degrees of freedom ''(df)'' gives the probability of having obtained a value ''less extreme'' than this point, subtracting the CDF value from 1 gives the p-value. The table below gives a number of p-values matching to ''χ''2 for the first 10 degrees of freedom.

A p-value of 0.05 or less is usually regarded as [[Statistical significance|statistically significant]], i.e. the observed deviation from the null hypothesis is significant.

{| class="wikitable"
|-
! Degrees of freedom (df)
!colspan=11| ''χ''2 value <ref>[http://www2.lv.psu.edu/jxm57/irp/chisquar.html Chi-Squared Test] Table B.2. Dr. Jacqueline S. McLaughlin at The Pennsylvania State University. In turn citing: R.A. Fisher and F. Yates, Statistical Tables for Biological Agricultural and Medical Research, 6th ed., Table IV</ref>
|-
| <div align="center"> 1
| 0.004
| 0.02
| 0.06
| 0.15
| 0.46
| 1.07
| 1.64
| 2.71
| 3.84
| 6.64
| 10.83
|-
| <div align="center"> 2
| 0.10
| 0.21
| 0.45
| 0.71
| 1.39
| 2.41
| 3.22
| 4.60
| 5.99
| 9.21
| 13.82
|-
| <div align="center"> 3
| 0.35
| 0.58
| 1.01
| 1.42
| 2.37
| 3.66
| 4.64
| 6.25
| 7.82
| 11.34
| 16.27
|-
| <div align="center"> 4
| 0.71
| 1.06
| 1.65
| 2.20
| 3.36
| 4.88
| 5.99
| 7.78
| 9.49
| 13.28
| 18.47
|-
| <div align="center"> 5
| 1.14
| 1.61
| 2.34
| 3.00
| 4.35
| 6.06
| 7.29
| 9.24
| 11.07
| 15.09
| 20.52
|-
| <div align="center"> 6
| 1.63
| 2.20
| 3.07
| 3.83
| 5.35
| 7.23
| 8.56
| 10.64
| 12.59
| 16.81
| 22.46
|-
| <div align="center"> 7
| 2.17
| 2.83
| 3.82
| 4.67
| 6.35
| 8.38
| 9.80
| 12.02
| 14.07
| 18.48
| 24.32
|-
| <div align="center"> 8
| 2.73
| 3.49
| 4.59
| 5.53
| 7.34
| 9.52
| 11.03
| 13.36
| 15.51
| 20.09
| 26.12
|-
| <div align="center"> 9
| 3.32
| 4.17
| 5.38
| 6.39
| 8.34
| 10.66
| 12.24
| 14.68
| 16.92
| 21.67
| 27.88
|-
| <div align="center"> 10
| 3.94
| 4.86
| 6.18
| 7.27
| 9.34
| 11.78
| 13.44
| 15.99
| 18.31
| 23.21
| 29.59
|-
! <div align="right"> P value (Probability)
| style="background: #ffa2aa" | 0.95
| style="background: #efaaaa" | 0.90
| style="background: #e8b2aa" | 0.80
| style="background: #dfbaaa" | 0.70
| style="background: #d8c2aa" | 0.50
| style="background: #cfcaaa" | 0.30
| style="background: #c8d2aa" | 0.20
| style="background: #bfdaaa" | 0.10
| style="background: #b8e2aa" | 0.05
| style="background: #afeaaa" | 0.01
| style="background: #a8faaa" | 0.001
|-
|
!colspan=8 style="background: #e8b2aa" | Nonsignificant
!colspan=3 style="background: #afeaaa" | Significant
|}

==History==

This distribution was first described by the German statistician Helmert.

==See also==
{{Portal|Statistics}}
{{Colbegin}}
*[[Cochran's theorem]]
*[[Fisher's method]] for combining [[Statistical independence|independent]] tests of significance
* [[Pearson's chi-squared test]]
* [[F-distribution]]
* [[Generalized chi-squared distribution]]
* [[Gamma distribution]]
* [[Hotelling's T-squared distribution]]
* [[Student's t-distribution]]
* [[Wilks' lambda distribution]]
* [[Wishart distribution]]
{{Colend}}

==References==
{{Reflist}}

==External links==
*[http://jeff560.tripod.com/c.html Earliest Uses of Some of the Words of Mathematics: entry on Chi squared has a brief history]
*[http://www.stat.yale.edu/Courses/1997-98/101/chigf.htm Course notes on Chi-Squared Goodness of Fit Testing] from Yale University Stats 101 class.
*[http://demonstrations.wolfram.com/StatisticsAssociatedWithNormalSamples/ ''Mathematica'' demonstration showing the chi-squared sampling distribution of various statistics, e.g. Σ''x''², for a normal population]
*[http://www.jstor.org/stable/2348373 Simple algorithm for approximating cdf and inverse cdf for the chi-squared distribution with a pocket calculator]

{{ProbDistributions|continuous-semi-infinite}}
{{Common univariate probability distributions}}

{{DEFAULTSORT:Chi-Squared Distribution}}
[[Category:Continuous distributions]]
[[Category:Normal distribution]]

[[Category:Exponential family distributions]]

[[ar:توزيع خي تربيع]]
[[ca:Distribució khi quadrat]]
[[cs:Χ² rozdělení]]
[[de:Chi-Quadrat-Verteilung]]
[[es:Distribución χ²]]
[[eu:Khi-karratu banakuntza]]
[[fa:توزیع کی‌دو]]
[[fr:Loi du χ²]]
[[ko:카이제곱 분포]]
[[id:Distribusi khi-kuadrat]]
[[is:Kí-kvaðratsdreifing]]
[[it:Distribuzione chi quadrato]]
[[he:התפלגות כי בריבוע]]
[[hu:Khi-négyzet eloszlás]]
[[nl:Chi-kwadraatverdeling]]
[[ja:カイ二乗分布]]
[[no:Kjikvadratfordeling]]
[[pl:Rozkład chi kwadrat]]
[[pt:Chi-quadrado]]
[[ru:Распределение хи-квадрат]]
[[simple:Chi-square distribution]]
[[sk:Χ²-rozdelenie]]
[[sl:Porazdelitev hi-kvadrat]]
[[su:Sebaran chi-kuadrat]]
[[fi:Khii toiseen -jakauma]]
[[sv:Chitvåfördelning]]
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[[uk:Розподіл хі-квадрат]]
[[zh:卡方分佈]]
[[zh-yue:Chi-square]]

Main Page

2014-08-09T19:33:20Z

KerstinTibbs:

'''Physisorption''', also called '''physical adsorption''', is a process in which the electronic structure of the atom or molecule is barely perturbed upon [[adsorption]].<ref>{{citation|author=K. Oura et al.|title=Surface Science, An Introduction|location= Berlin|publisher= Springer|year= 2003| isbn =978-3-540-00545-2}}</ref><ref name=ConceptsinSurfacePhysics>{{citation |author=M. C. Desjonqueres et al |title=Concepts in surface physics |edition=2nd |url=http://books.google.com/?id=XW_Wvjwt5nIC&printsec=frontcover&dq=Concepts+in+surface+physics |place=New York |publisher=Springer-Verlag |date= 1996. Corrected printing 1998 |isbn=3-540-58622-9 |accessdate=29 August 2012}}</ref><ref>{{Citation|author=Hans Luth et al.|title=Surfaces and interfaces of solids|publisher= Springer-Verlag|year= 1993| isbn =978-3-540-56840-7}}</ref>

==Introduction==

The fundamental interacting force of physisorption is caused by [[van der Waals force]]. Even though the interaction energy is very weak (~10–100 meV), physisorption plays an important role in nature. For instance, the van der Waals attraction between surfaces and foot-hairs of [[gecko]]s provides the remarkable ability to climb up vertical walls.<ref>{{Citation | author=K. Autumn ''et al.''|title= Adhesive force of a single gecko foot-hair| journal=Nature| volume=405 | pages= 681–5| year=2000 | doi=10.1038/35015073 | pmid=10864324 | issue=6787}}</ref> Van der Waals forces originate from the interactions between induced, permanent or transient electric dipoles.

In comparison with [[chemisorption]], in which the electronic structure of bonding atoms or molecules is changed and covalent or ionic bonds form, physisorption, generally speaking, can only be observed in the environment of low temperature (thermal energy at room temperature ~26 meV) and the absence of the relatively strong chemisorptions. In practice, the categorisation of a particular adsorption as physisorption or chemisorption depends principally on the [[binding energy]] of the adsorbate to the substrate.

==Modeling by image charge==
[[Image:physisorption 1.jpg|thumbnail|200px|Fig. 1. Schematic illustration of an adsorbed hydrogen atom near a perfect [[electrical conductor|conductor]] interacting with its [[image charge]]s.]]

To give a simple illustration of physisorption, we can first consider an adsorbed hydrogen atom in front of a perfect conductor, as shown in Fig. 1. A nucleus with positive charge is located at '''R''' = (0, 0, ''Z''), and the position coordinate of its electron, '''r''' = (''x'', ''y'', ''z'') is given with respect to the nucleus. The adsorption process can be viewed as the interaction between this hydrogen atom and its image charges of both the nucleus and electron in the conductor. As a result, the total electrostatic energy is the sum of attraction and repulsion terms:

:<math>V = {e^2\over 4\pi\varepsilon_0}\left(\frac{-1}{|2\mathbf R|}+\frac{-1}{|2\mathbf R+\mathbf r-\mathbf r'|}+\frac{1}{|2\mathbf R-\mathbf r'|}+\frac{1}{|2\mathbf R+\mathbf r|}\right).</math>

The first term is the attractive interaction of nucleus and its image charge, and the second term is due to the interaction of the electron and its image charge. The repulsive interaction is shown in the third and fourth terms arising from the interaction of nucleus-image electron and electron-image nucleus, respectively.

By [[Taylor expansion]] in powers of |'''r'''| / |'''R'''|, this interaction energy can be further expressed as:

:<math>V = {-e^2\over 16\pi\varepsilon_0 Z^3}\left(\frac{x^2+y^2}{2}+z^2\right)+ {3e^2\over 32\pi\varepsilon_0 Z^4}\left(\frac{x^2+y^2}{2}{z}+z^3\right)+O\left(\frac{1}{Z^5}\right).</math>

One can find from the first non-vanishing term that the physisorption potential depends on the distance ''Z'' between adsorbed atom and surface as ''Z''−3, in contrast with the ''r''−6 dependence of the molecular [[van der Waals]] potential, where ''r'' is the distance between two [[dipoles]].

==Modeling by quantum-mechanical oscillator==

The [[van der Waals force|van der Waals]] binding energy can be analyzed by another simple physical picture: modeling the motion of an electron around its nucleus by a three-dimensional simple [[harmonic oscillator]] with a potential energy ''Va'':

:<math>V_a = \frac{m_e}{2}{\omega^2}(x^2+y^2+z^2),</math>

where ''me'' and ''ω'' are the mass and vibrational frequency of the electron, respectively.

As this atom approaches the surface of a metal and forms adsorption, this potential energy ''Va'' will be modified due to the image charges by additional potential terms which are quadratic in the displacements:

:<math>V_a = \frac{m_e}{2}{\omega^2}(x^2+y^2+z^2)-{e^2\over 16\pi\varepsilon_0 Z^3}\left(\frac{x^2+y^2}{2}+z^2\right)+\ldots</math> (from the Taylor expansion above.)

Assuming

:<math> m_e \omega^2>>{e^2\over 16\pi\varepsilon_0 Z^3},</math>

the potential is well approximated as

:<math>V_a \sim \frac{m_e}{2}{\omega_1^2}(x^2+y^2)+\frac{m_e}{2}{\omega_2^2}z^2</math>,

where

:<math>
\begin{align}
\omega_1 &= \omega - {e^2\over 32\pi\varepsilon_0 m_e\omega Z^3},\\
\omega_2 &= \omega - {e^2\over 16\pi\varepsilon_0 m_e\omega Z^3}.
\end{align}
</math>

If one assumes that the electron is in the ground state, then the van der Waals binding energy is essentially the change of the zero-point energy:

:<math>V_v = \frac{\hbar}{2}(2\omega_1+\omega_2-3\omega)= - {\hbar e^2\over 16\pi\varepsilon_0 m_e\omega Z^3}.</math>

This expression also shows the nature of the ''Z''−3 dependence of the van der Waals interaction.

Furthermore by introducing the atomic [[polarizability]],

:<math> \alpha= \frac {e^2} {m_e\omega^2},</math>

the van der Waals potential can be further simplified:

:<math>V_v = - {\hbar \alpha \omega\over 16\pi\varepsilon_0 Z^3}= -\frac{C_v}{Z^3},</math>

where

:<math>C_v = {\hbar \alpha \omega\over 16\pi\varepsilon_0},</math>

is the van der Waals constant which is related to the atomic polarizability.

Also, by expressing the fourth-order correction in the Taylor expansion above as (''aCvZ''0) / (Z4), where ''a'' is some constant, we can define ''Z''0 as the position of the ''dynamical image plane'' and obtain
[[Image:physisorption table.jpg|thumbnail|400px|Table 1. The van der Waals constant ''Cv'' and the position of the dynamical image plane ''Z''0 for various rare gases atoms adsorbed on noble metal surfaces obtained by the jellium model. Note that ''Cv'' is in eV/Å3 and ''Z''0 in Å.]]

:<math>V_v = - \frac{C_v}{(Z-Z_0)^3}+O\left(\frac{1}{Z^5}\right).</math>

The origin of ''Z''0 comes from the spilling of the electron wavefunction out of the surface. As a result, the position of image plane representing the reference for the space coordinate is different from the substrate surface itself and modified by ''Z''0.

Table 1 shows the [[jellium]] model calculation for van der Waals constant ''Cv'' and dynamical image plane ''Z''0 of rare gas atoms on various metal surfaces. The increasing of ''Cv'' from He to Xe for all metal substrates is caused by the larger atomic [[polarizability]] of the heavier rare gas atoms. For the position of the dynamical image plane, it decreases with increasing dielectric function and is typically on the order of 0.2 Å.

==Physisorption potential==
[[Image:physisorption 2.jpg|thumbnail|400px|Fig. 2. Calculated physisorption potential energy for He adsorbed on various [[jellium]] metal surfaces. Note that the weak van der Waals attraction forms shallow wells with energy about few meV.<ref name="Kohn"/>]]

Even though the [[van der Waals interaction]] is attractive, as the adsorbed atom moves closer to the surface the wavefunction of electron starts to overlap with that of the surface atoms. Further the energy of the system will increase due to the orthogonality of wavefunctions of the approaching atom and surface atoms.

This [[Pauli exclusion]] and repulsion are particularly strong for atoms with closed valence shells that dominate the surface interaction. As a result, the minimum energy of physisorption must be found by the balance between the long-range van der Waals attraction and short-range [[Pauli repulsion]]. For instance, by separating the total interaction of physisorption into two contributions- a short-range term depicted by [[Hartree–Fock]] theory and a long-range van der Waals attraction, the equilibrium position of physisorption for rare gases adsorbed on [[jellium]] substrate can be determined.<ref name="Kohn">{{Citation | author=E. Zaremba and W. Kohn|title= Theory of helium adsorption on simple and noble-metal surfaces| journal= Phys. Rev. B| volume=15 | issue=4 | pages= 1769| year=1977 | doi=10.1103/PhysRevB.15.1769|bibcode = 1977PhRvB..15.1769Z }}</ref> Fig. 2 shows the physisorption potential energy of He adsorbed on Ag, Cu, and Au substrates which are described by the [[jellium]] model with different densities of smear-out background positive charges. It can be found that the weak van der Waals interaction leads to shallow attractive energy wells (<10 meV). One of the experimental methods for exploring physisorption potential energy is the scattering process, for instance, inert gas atoms scattered from metal surfaces. Certain specific features of the interaction potential between scattered atoms and surface can be extracted by analyzing the experimentally determined angular distribution and cross sections of the scattered particles.

==Comparison with chemisorption==

* Physisorption is a general phenomenon and occurs in any solid/fluid or solid/gas system. [[Chemisorption]] is characterized by chemical specificity.
* In physisorption, perturbation of the electronic states of adsorbent and adsorbate is minimal. For chemisorption, changes in the electronic states may be detectable by suitable physical means.
* Typical binding energy of physisorption is about 10–100 meV. Chemisorption usually forms bonding with energy of 1–10 eV.
* The elementary step in physisorption from a gas phase does not involve an activation energy. Chemisorption often involves an activation energy.
* For physisorption, under appropriate conditions, gas phase molecules can form multilayer adsorption. In chemisorption, molecules are adsorbed on the surface by valence bonds and only form monolayer adsorption.

==See also==
*[[Adsorption]]
*[[Chemisorption]]
*[[van der Waals force]]

==References==
{{reflist}}

[[Category:Surface chemistry]]

Main Page

2014-08-09T19:32:59Z

KerstinTibbs:

In [[mathematics]], a '''pseudometric''' or '''semi-metric space'''<ref>Dmitri Burago, Yu D Burago, Sergei Ivanov, A Course in Metric Geometry, American Mathematical Society, 2001, ISBN 0-8218-2129-6.</ref> is a generalized [[metric space]] in which the distance between two distinct points can be zero. In the same way as every [[normed space]] is a [[metric space]], every [[seminormed space]] is a pseudometric space. Because of this analogy the term [[semimetric space]] (which has a different meaning in [[topology]]) is sometimes used as a synonym, especially in [[functional analysis]].

When a topology is generated using a family of pseudometrics, the space is called a [[gauge space]].

==Definition==
A pseudometric space <math>(X,d)</math> is a set <math>X</math> together with a non-negative real-valued function <math>d\colon X \times X \longrightarrow \mathbb{R}_{\geq 0}</math> (called a '''pseudometric''') such that, for every <math>x,y,z \in X</math>,

#<math>d(x,x) = 0</math>.
#<math>d(x,y) = d(y,x)</math> (''symmetry'')
#<math>d(x,z) \leq d(x,y) + d(y,z)</math> (''[[subadditivity]]''/''[[triangle inequality]]'')


Unlike a metric space, points in a pseudometric space need not be [[identity of indiscernibles|distinguishable]]; that is, one may have <math>d(x,y)=0</math> for distinct values <math>x\ne y</math>.

==Examples==
* Pseudometrics arise naturally in [[functional analysis]]. Consider the space <math>\mathcal{F}(X)</math> of real-valued functions <math>f\colon X\to\mathbb{R}</math> together with a special point <math>x_0\in X</math>. This point then induces a pseudometric on the space of functions, given by
::<math>d(f,g) = |f(x_0)-g(x_0)|</math>
:for <math>f,g\in \mathcal{F}(X)</math>

* For vector spaces <math>V</math>, a [[seminorm]] <math>p</math> induces a pseudometric on <math>V</math>, as
::<math>d(x,y)=p(x-y).</math>
:Conversely, a homogeneous, translation invariant pseudometric induces a seminorm.

* Pseudometrics also arise in the theory of hyperbolic [[complex manifold]]s: see [[Kobayashi metric]].

* Every [[measure space]] <math>(\Omega,\mathcal{A},\mu)</math> can be viewed as a complete pseudometric space by defining
::<math>d(A,B) := \mu(A\Delta B)</math>
:for all <math>A,B\in\mathcal{A}</math>.
* If <math>f:X_1 \rightarrow X_2</math> is a function and d2 is a pseudometric on X2, then <math>d_1(x,y) := d_2(f(x),f(y))</math> gives a pseudometric on X1. If d2 is a metric and f is [[Injective function|injective]], then d1 is a metric.

==Topology==
The '''pseudometric topology''' is the [[topological space|topology]] induced by the [[open balls]]

:<math>B_r(p)=\{ x\in X\mid d(p,x)<r \},</math>

which form a [[basis (topology)|basis]] for the topology.<ref>{{planetmath reference|id=6284|title=Pseudometric topology}}</ref> A topological space is said to be a '''pseudometrizable topological space''' if the space can be given a pseudometric such that the pseudometric topology coincides with the given topology on the space.

The difference between pseudometrics and metrics is entirely topological. That is, a pseudometric is a metric if and only if the topology it generates is [[T0 space|T0]] (i.e. distinct points are topologically distinguishable).

==Metric identification==
The vanishing of the pseudometric induces an [[equivalence relation]], called the '''metric identification''', that converts the pseudometric space into a full-fledged [[metric space]]. This is done by defining <math>x\sim y</math> if <math>d(x,y)=0</math>. Let <math>X^*=X/{\sim}</math> and let
:<math>d^*([x],[y])=d(x,y)</math>
Then <math>d^*</math> is a metric on <math>X^*</math> and <math>(X^*,d^*)</math> is a well-defined metric space.<ref>{{cite book|last=Howes|first=Norman R.|title=Modern Analysis and Topology|year=1995|publisher=Springer|location=New York, NY|isbn=0-387-97986-7|url=http://www.springer.com/mathematics/analysis/book/978-0-387-97986-1|accessdate=10 September 2012|page=27|quote=Let <math>(X,d)</math> be a pseudo-metric space and define an equivalence relation <math>\sim</math> in <math>X</math> by <math>x \sim y</math> if <math>d(x,y)=0</math>. Let <math>Y</math> be the quotient space <math>X/\sim</math> and <math>p\colon X\to Y</math> the canonical projection that maps each point of <math>X</math> onto the equivalence class that contains it. Define the metric <math>\rho</math> in <math>Y</math> by <math>\rho(a,b) = d(p^{-1}(a),p^{-1}(b))</math> for each pair <math>a,b \in Y</math>. It is easily shown that <math>\rho</math> is indeed a metric and <math>\rho</math> defines the quotient topology on <math>Y</math>.}}</ref>

The metric identification preserves the induced topologies. That is, a subset <math>A\subset X</math> is open (or closed) in <math>(X,d)</math> if and only if <math>\pi(A)=[A]</math> is open (or closed) in <math>(X^*,d^*)</math>. The topological identification is the [[Kolmogorov quotient]].

An example of this construction is the [[Complete_metric_space#Completion|completion of a metric space]] by its [[Cauchy sequences]].

==Notes==
{{Reflist}}

==References==
* {{cite book | title=General Topology I: Basic Concepts and Constructions Dimension Theory | last=Arkhangel'skii | first=A.V. |author2=Pontryagin, L.S. | year=1990 | isbn=3-540-18178-4 | publisher=[[Springer Science+Business Media|Springer]] | series=Encyclopaedia of Mathematical Sciences}}
* {{cite book | title=Counterexamples in Topology | last=Steen | first=Lynn Arthur |author2=Seebach, Arthur | year=1995 | origyear=1970 | isbn=0-486-68735-X | publisher=[[Dover Publications]] | edition=new edition }}
* {{PlanetMath attribution|id=6273|title=Pseudometric space}}
* {{planetmath reference|id=6275|title=Example of pseudometric space}}

{{DEFAULTSORT:Pseudometric Space}}
[[Category:Properties of topological spaces]]
[[Category:Metric geometry]]

Main Page

2014-08-09T15:56:32Z

KerstinTibbs:

{{Redirect|Sinh|the garment|sinh (clothing)}}
[[Image:Hyperbolic functions-2.svg|thumb|296px|right|A ray through the origin intercepts the [[unit hyperbola]] <math>\scriptstyle x^2\ -\ y^2\ =\ 1</math> in the point <math>\scriptstyle (\cosh\,a,\,\sinh\,a)</math>, where <math>\scriptstyle a</math> is twice the area between the ray, the hyperbola, and the <math>\scriptstyle x</math>-axis. For points on the hyperbola below the <math>\scriptstyle x</math>-axis, the area is considered negative (see [[:Image:HyperbolicAnimation.gif|animated version]] with comparison with the trigonometric (circular) functions).]]

In [[mathematics]], '''hyperbolic functions''' are analogs of the ordinary [[trigonometric function|trigonometric]], or circular, functions. The basic hyperbolic functions are the '''hyperbolic sine''' "sinh" ({{IPAc-en|ˈ|s|ɪ|n|tʃ}} or {{IPAc-en|ˈ|ʃ|aɪ|n}}),<ref>(1999) ''Collins Concise Dictionary'', 4th edition, HarperCollins, Glasgow, ISBN 0 00 472257 4, p.1386</ref> and the '''hyperbolic cosine''' "cosh" ({{IPAc-en|ˈ|k|ɒ|ʃ}}),<ref>''Collins Concise Dictionary'', p.328</ref> from which are derived the '''hyperbolic tangent''' "tanh" ({{IPAc-en|ˈ|t|æ|n|tʃ}} or {{IPAc-en|ˈ|θ|æ|n}}),<ref>''Collins Concise Dictionary'', p.1520</ref> '''hyperbolic cosecant''' "csch" or "cosech" ({{IPAc-en|ˈ|k|oʊ|ʃ|ɛ|k}}<ref>''Collins Concise Dictionary'', p.328</ref> or {{IPAc-en|ˈ|k|oʊ|s|ɛ|tʃ}}), '''hyperbolic secant''' "sech" ({{IPAc-en|ˈ|ʃ|ɛ|k}} or {{IPAc-en|ˈ|s|ɛ|tʃ}}),<ref>''Collins Concise Dictionary'', p.1340</ref> and '''hyperbolic cotangent''' "coth" ({{IPAc-en|ˈ|k|oʊ|θ}} or {{IPAc-en|ˈ|k|ɒ|θ}}),<ref>''Collins Concise Dictionary'', p.329</ref><ref>[http://www.mathcentre.ac.uk/resources/workbooks/mathcentre/hyperbolicfunctions.pdf tanh]</ref> corresponding to the derived trigonometric functions. The [[inverse hyperbolic function]]s are the '''area hyperbolic sine''' "arsinh" (also called "asinh" or sometimes "arcsinh")<ref>[http://www.google.com/books?q=arcsinh+-library Some examples of using '''arcsinh'''] found in [[Google Books]].</ref> and so on.

Just as the points (cos ''t'', sin ''t'') form a circle with a unit radius, the points (cosh ''t'', sinh ''t'') form the right half of the equilateral [[hyperbola]]. The hyperbolic functions take a [[Real number|real argument]] called a [[hyperbolic angle]]. The size of a hyperbolic angle is the area of its [[hyperbolic sector]]. The hyperbolic functions may be defined in terms of the [[hyperbolic sector#Hyperbolic triangle|legs of a right triangle]] covering this sector.

Hyperbolic functions occur in the solutions of some important linear [[differential equation]]s, for example the equation defining a [[catenary]], of some [[Cubic function#Trigonometric (and hyperbolic) method|cubic equations]], and of [[Laplace's equation]] in [[Cartesian coordinates]]. The latter is important in many areas of [[physics]], including [[electromagnetic theory]], [[heat transfer]], [[fluid dynamics]], and [[special relativity]].

In [[complex analysis]], the hyperbolic functions arise as the imaginary parts of sine and cosine. When considered defined by a complex variable, the hyperbolic functions are [[rational function]]s of [[exponential function|exponentials]], and are hence [[meromorphic function|meromorphic]].

Hyperbolic functions were introduced in the 1760s independently by [[Vincenzo Riccati]] and [[Johann Heinrich Lambert]].<ref>Robert E. Bradley, Lawrence A. D'Antonio, Charles Edward Sandifer. ''Euler at 300: an appreciation.'' Mathematical Association of America, 2007. Page 100.</ref> Riccati used ''Sc.'' and ''Cc.'' (''[co]sinus circulare'') to refer to circular functions and ''Sh.'' and ''Ch.'' (''[co]sinus hyperbolico'') to refer to hyperbolic functions. Lambert adopted the names but altered the abbreviations to what they are today.<ref>Georg F. Becker. ''Hyperbolic functions.'' Read Books, 1931. Page xlviii.</ref> The abbreviations ''sh'' and ''ch'' are still used in some other languages, like French and Russian.

==Standard algebraic expressions==
[[Image:sinh cosh tanh.svg|256px|thumb|sinh, cosh and tanh]]
[[Image:csch sech coth.svg|256px|thumb|csch, sech and coth]]
{{multiple image
| direction = vertical
| width = 225
| footer = Hyperbolic functions (a) cosh and (b) sinh obtained using exponential functions <math>e^x</math> and <math>e^{-x}</math>
| image1 = Hyperbolic and exponential; cosh.svg
| caption1 = (a) cosh(''x'') is the [[Arithmetic mean|average]] of ''ex''and ''e−x''
| alt1 = (a) cosh(''x'') is the [[Arithmetic mean|average]] of ''ex''and ''e−x''
| image2 = Hyperbolic and exponential; sinh.svg
| caption2 = (b) sinh(''x'') is half the [[Subtraction|difference]] of ''ex'' and ''e−x''
| alt2 = (b) sinh(''x'') is half the [[Subtraction|difference]] of ''ex'' and ''e−x''
}}
The hyperbolic functions are:

* Hyperbolic sine:
::<math>\sinh x = \frac {e^x - e^{-x}} {2} = \frac {e^{2x} - 1} {2e^x} = \frac {1 - e^{-2x}} {2e^{-x}}</math>

* Hyperbolic cosine:
::<math>\cosh x = \frac {e^x + e^{-x}} {2} = \frac {e^{2x} + 1} {2e^x} = \frac {1 + e^{-2x}} {2e^{-x}}</math>

* Hyperbolic tangent:
::<math>\tanh x = \frac{\sinh x}{\cosh x} = \frac {e^x - e^{-x}} {e^x + e^{-x}} = \frac{e^{2x} - 1} {e^{2x} + 1} = \frac{1 - e^{-2x}} {1 + e^{-2x}}</math>

* Hyperbolic cotangent:
::<math>\coth x = \frac{\cosh x}{\sinh x} = \frac {e^x + e^{-x}} {e^x - e^{-x}} = \frac{e^{2x} + 1} {e^{2x} - 1} = \frac{1 + e^{-2x}} {1 - e^{-2x}}</math>

* Hyperbolic secant:
::<math>\operatorname{sech}\,x = \left(\cosh x\right)^{-1} = \frac {2} {e^x + e^{-x}} = \frac{2e^x} {e^{2x} + 1} = \frac{2e^{-x}} {1 + e^{-2x}}</math>

* Hyperbolic cosecant:
::<math>\operatorname{csch}\,x = \left(\sinh x\right)^{-1} = \frac {2} {e^x - e^{-x}} = \frac{2e^x} {e^{2x} - 1} = \frac{2e^{-x}} {1 - e^{-2x}}</math>

Hyperbolic functions can be introduced via [[hyperbolic angle#Imaginary circular angle|imaginary circular angles]]:

* Hyperbolic sine:
::<math>\sinh x = -i \sin (i x)</math>

* Hyperbolic cosine:
::<math>\cosh x = \cos (i x)</math>

* Hyperbolic tangent:
::<math>\tanh x = -i \tan (i x)</math>

* Hyperbolic cotangent:
::<math>\coth x = i \cot (i x)</math>

* Hyperbolic secant:
::<math>\operatorname{sech} x = \sec (i x)</math>

* Hyperbolic cosecant:
::<math>\operatorname{csch} x = i \csc (i x)</math>

where ''i'' is the [[imaginary unit]] defined by ''i''2 = −1.

The [[complex number|complex]] forms in the definitions above derive from [[Euler's formula]].

==Useful relations==
Odd and even functions:
:<math>\begin{align}
\sinh (-x) &= -\sinh x \\
\cosh (-x) &= \cosh x
\end{align}</math>

Hence:
:<math>\begin{align}
\tanh (-x) &= -\tanh x \\
\coth (-x) &= -\coth x \\
\operatorname{sech} (-x) &= \operatorname{sech} x \\
\operatorname{csch} (-x) &= -\operatorname{csch} x
\end{align}</math>

It can be seen that cosh ''x'' and sech ''x'' are [[even function]]s; the others are [[odd functions]].

:<math>\begin{align}
\operatorname{arsech} x &= \operatorname{arcosh} \frac{1}{x} \\
\operatorname{arcsch} x &= \operatorname{arsinh} \frac{1}{x} \\
\operatorname{arcoth} x &= \operatorname{artanh} \frac{1}{x}
\end{align}</math>

Hyperbolic sine and cosine satisfy the identity
:<math>\cosh^2 x - \sinh^2 x = 1 </math>

which is similar to the [[Pythagorean trigonometric identity]]. One also has
:<math>\begin{align}
\operatorname{sech} ^{2} x &= 1 - \tanh^{2} x \\
\operatorname{csch} ^{2} x &= \coth^{2} x - 1
\end{align}</math>

for the other functions.

The hyperbolic tangent is the solution to the [[differential equation]] <math>f'=1-f^2</math> with f(0)=0 and the [[nonlinear]] [[boundary value problem]]:<ref>
{{cite web
| url = http://mathworld.wolfram.com/HyperbolicTangent.html
| title = Hyperbolic Tangent
| author = [[Eric W. Weisstein]]
| publisher = [[MathWorld]]
| date =
| accessdate = 2008-10-20
}}</ref>

:<math>\frac{1}{2} f'' = f^3 - f ; \quad f(0) = f'(\infty) = 0</math>

It can be shown that the area under the curve of cosh (''x'') over a finite interval is always equal to the arc length corresponding to that interval:<ref>
{{cite book
|title=Golden Integral Calculus
|first1=Bali
|last1=N.P.
|publisher=Firewall Media
|year=2005
|isbn=81-7008-169-6
|page=472
|url=http://books.google.com/books?id=hfi2bn2Ly4cC}}, [http://books.google.com/books?id=hfi2bn2Ly4cC&pg=PA472 Extract of page 472]
</ref>
:<math>\text{area} = \int_a^b{ \cosh{(x)} } \ dx = \int_a^b\sqrt{1 + \left(\frac{d}{dx} \cosh{(x)}\right)^2} \ dx = \text{arc length}</math>

Sums of arguments:
:<math>\begin{align}
\sinh(x + y) &= \sinh (x) \cosh (y) + \cosh (x) \sinh (y) \\
\cosh(x + y) &= \cosh (x) \cosh (y) + \sinh (x) \sinh (y) \\

\end{align}</math>
particularly
:<math>\begin{align}
\cosh (2x) &= \sinh^2{x} + \cosh^2{x} = 2\sinh^2 x + 1 = 2\cosh^2 x - 1\\
\sinh (2x) &= 2\sinh x \cosh x
\end{align}</math>

Sum and difference of cosh and sinh:
:<math>\begin{align}
\cosh x + \sinh x &= e^x \\
\cosh x - \sinh x &= e^{-x}
\end{align}</math>

==Inverse functions as logarithms==
:<math>\begin{align}
\operatorname {arsinh} (x) &= \ln \left(x + \sqrt{x^{2} + 1} \right) \\

\operatorname {arcosh} (x) &= \ln \left(x + \sqrt{x^{2} - 1} \right); x \ge 1 \\

\operatorname {artanh} (x) &= \frac{1}{2}\ln \left( \frac{1 + x}{1 - x} \right); \left| x \right| < 1 \\

\operatorname {arcoth} (x) &= \frac{1}{2}\ln \left( \frac{x + 1}{x - 1} \right); \left| x \right| > 1 \\

\operatorname {arsech} (x) &= \ln \left( \frac{1}{x} + \frac{\sqrt{1 - x^{2}}}{x} \right); 0 < x \le 1 \\

\operatorname {arcsch} (x) &= \ln \left( \frac{1}{x} + \frac{\sqrt{1 + x^{2}}}{\left| x \right|} \right); x \ne 0
\end{align}</math>

==Derivatives==
:<math> \frac{d}{dx}\sinh x = \cosh x \,</math>

:<math> \frac{d}{dx}\cosh x = \sinh x \,</math>

:<math> \frac{d}{dx}\tanh x = 1 - \tanh^2 x = \operatorname{sech}^2 x = 1/\cosh^2 x \,</math> 

:<math> \frac{d}{dx}\coth x = 1 - \coth^2 x = -\operatorname{csch}^2 x = -1/\sinh^2 x \,</math>

:<math> \frac{d}{dx}\ \operatorname{sech}\,x = - \tanh x \ \operatorname{sech}\,x \,</math>

:<math> \frac{d}{dx}\ \operatorname{csch}\,x = - \coth x \ \operatorname{csch}\,x \,</math>

:<math>\frac{d}{dx}\, \operatorname{arsinh}\,x =\frac{1}{\sqrt{x^{2}+1}}</math>

:<math>\frac{d}{dx}\, \operatorname{arcosh}\,x =\frac{1}{\sqrt{x^{2}-1}}</math>

:<math>\frac{d}{dx}\, \operatorname{artanh}\,x =\frac{1}{1-x^{2}}</math>

:<math>\frac{d}{dx}\, \operatorname{arcoth}\,x =\frac{1}{1-x^{2}}</math>

:<math>\frac{d}{dx}\, \operatorname{arsech}\,x =-\frac{1}{x\sqrt{1-x^{2}}}</math>

:<math>\frac{d}{dx}\, \operatorname{arcsch}\,x =-\frac{1}{\left| x \right|\sqrt{1+x^{2}}}</math>

==Standard integrals==
{{For|a full list|list of integrals of hyperbolic functions}}

<math>\begin{align}
\int \sinh (ax)\,dx &= a^{-1} \cosh (ax) + C \\
\int \cosh (ax)\,dx &= a^{-1} \sinh (ax) + C \\
\int \tanh (ax)\,dx &= a^{-1} \ln (\cosh (ax)) + C \\
\int \coth (ax)\,dx &= a^{-1} \ln (\sinh (ax)) + C \\
\int \operatorname{sech} (ax)\,dx &= a^{-1} \arctan (\sinh (ax)) + C \\
\int \operatorname{csch} (ax)\,dx &= a^{-1} \ln \left( \tanh \left( \frac{ax}{2} \right) \right) + C &= a^{-1} \ln\left|\operatorname{csch} (ax) - \coth (ax)\right| + C
\end{align}</math>

The following integrals can be proved using [[hyperbolic substitution]].

<math>\begin{align}
\int {\frac{du}{\sqrt{a^2 + u^2}}} & = \operatorname{arsinh} \left( \frac{u}{a} \right) + C \\
\int {\frac{du}{\sqrt{u^2 - a^2}}} &= \operatorname{arcosh} \left( \frac{u}{a} \right) + C \\
\int {\frac{du}{a^2 - u^2}} & = a^{-1}\operatorname{artanh} \left( \frac{u}{a} \right) + C; u^2 < a^2 \\
\int {\frac{du}{a^2 - u^2}} & = a^{-1}\operatorname{arcoth} \left( \frac{u}{a} \right) + C; u^2 > a^2 \\
\int {\frac{du}{u\sqrt{a^2 - u^2}}} & = -a^{-1}\operatorname{arsech}\left( \frac{u}{a} \right) + C \\
\int {\frac{du}{u\sqrt{a^2 + u^2}}} & = -a^{-1}\operatorname{arcsch}\left| \frac{u}{a} \right| + C
\end{align}</math>

where ''C'' is the [[constant of integration]].

==Taylor series expressions==
It is possible to express the above functions as [[Taylor series]]:

:<math>\sinh x = x + \frac {x^3} {3!} + \frac {x^5} {5!} + \frac {x^7} {7!} +\cdots = \sum_{n=0}^\infty \frac{x^{2n+1}}{(2n+1)!}</math>
The function sinh ''x'' has a Taylor series expression with only odd exponents for ''x''. Thus it is an [[odd function]], that is, −sinh ''x'' = sinh(−''x''), and sinh 0 = 0.

:<math>\cosh x = 1 + \frac {x^2} {2!} + \frac {x^4} {4!} + \frac {x^6} {6!} + \cdots = \sum_{n=0}^\infty \frac{x^{2n}}{(2n)!}</math>

The function cosh ''x'' has a Taylor series expression with only even exponents for ''x''. Thus it is an [[even function]], that is, symmetric with respect to the ''y''-axis. The sum of the sinh and cosh series is the [[infinite series]] expression of the [[exponential function]].

:<math>\begin{align}

\tanh x &= x - \frac {x^3} {3} + \frac {2x^5} {15} - \frac {17x^7} {315} + \cdots = \sum_{n=1}^\infty \frac{2^{2n}(2^{2n}-1)B_{2n} x^{2n-1}}{(2n)!}, \left |x \right | < \frac {\pi} {2} \\

\coth x &= x^{-1} + \frac {x} {3} - \frac {x^3} {45} + \frac {2x^5} {945} + \cdots = x^{-1} + \sum_{n=1}^\infty \frac{2^{2n} B_{2n} x^{2n-1}} {(2n)!}, 0 < \left |x \right | < \pi \\

\operatorname {sech}\, x &= 1 - \frac {x^2} {2} + \frac {5x^4} {24} - \frac {61x^6} {720} + \cdots = \sum_{n=0}^\infty \frac{E_{2 n} x^{2n}}{(2n)!} , \left |x \right | < \frac {\pi} {2} \\

\operatorname {csch}\, x &= x^{-1} - \frac {x} {6} +\frac {7x^3} {360} -\frac {31x^5} {15120} + \cdots = x^{-1} + \sum_{n=1}^\infty \frac{ 2 (1-2^{2n-1}) B_{2n} x^{2n-1}}{(2n)!} , 0 < \left |x \right | < \pi

\end{align}</math>

where

:<math>B_n \,</math> is the ''n''th [[Bernoulli number]]
:<math>E_n \,</math> is the ''n''th [[Euler number]]

==Comparison with circular functions==

[[File:Circular and hyperbolic angle.svg|right|250px|thumb|Circle and hyperbola tangent at (1,1) display geometry of circular functions in terms of [[sector of a circle|circular sector]] area ''u'' and hyperbolic functions depending on [[hyperbolic sector]] area ''u''.]]
The hyperbolic functions represent an expansion of [[trigonometry]] beyond the [[circular function]]s. Both types depend on an [[argument of a function|argument]], either [[angle|circular angle]] or [[hyperbolic angle]].

Since the [[circular sector#Area|area of a circular sector]] is <math>\frac {r^2 u} {2} ,</math> it will be equal to ''u'' when ''r'' = [[square root of 2]]. In the diagram such a circle is tangent to the hyperbola ''x y'' = 1 at (1,1). The yellow sector depicts an area and angle magnitude. Similarly, the red augmentation depicts an area and magnitude as hyperbolic angle.

The legs of the two [[right triangle]]s with hypotenuse on the ray defining the angles are of length √2 times the circular and hyperbolic functions.

Mellon Haskell of [[University of California, Berkeley]] described the basis of hyperbolic functions in areas of [[hyperbolic sector]]s in an 1895 article in [[Bulletin of the American Mathematical Society]] (see External links). He refers to the hyperbolic angle as an [[invariant measure]] with respect to the [[squeeze mapping]] just as circular angle is invariant under rotation.

==Identities==
The hyperbolic functions satisfy many identities, all of them similar in form to the [[trigonometric identity|trigonometric identities]]. In fact, '''Osborn's rule'''<ref>G. Osborn, [http://links.jstor.org/sici?sici=0025-5572(190207)2%3A2%3A34%3C189%3A1MFHF%3E2.0.CO%3B2-Z Mnemonic for hyperbolic formulae], The Mathematical Gazette, p. 189, volume 2, issue 34, July 1902</ref> states that one can convert any trigonometric identity into a hyperbolic identity by expanding it completely in terms of integral powers of sines and cosines, changing sine to sinh and cosine to cosh, and switching the sign of every term which contains a product of 2, 6, 10, 14, ... sinhs. This yields for example the addition theorems
:<math>\begin{align}
\sinh(x + y) &= \sinh (x) \cosh (y) + \cosh (x) \sinh (y) \\
\cosh(x + y) &= \cosh (x) \cosh (y) + \sinh (x) \sinh (y) \\
\tanh(x + y) &= \frac{\tanh (x) + \tanh (y)}{1 + \tanh (x) \tanh (y)}
\end{align}</math>

the "double argument formulas"
:<math>\begin{align}
\sinh 2x &= 2\sinh x \cosh x \\
\cosh 2x &= \cosh^2 x + \sinh^2 x = 2\cosh^2 x - 1 = 2\sinh^2 x + 1 \\
\tanh 2x &= \frac{2\tanh x}{1 + \tanh^2 x}\\
\sinh 2x &= \frac{2\tanh x}{1-\tanh^2 x}\\
\cosh 2x &= \frac{1+ \tanh^2 x}{1-\tanh^2 x}
\end{align}</math>

and the "half-argument formulas"<ref>
{{cite book
|title=Technical mathematics with calculus
|edition=3rd
|first1=John Charles
|last1=Peterson
|publisher=Cengage Learning
|year=2003
|isbn=0-7668-6189-9
|page=1155
|url=http://books.google.com/books?id=PGuSDjHvircC}}, [http://books.google.com/books?id=PGuSDjHvircC&pg=PA1155 Chapter 26, page 1155]
</ref>
:<math>\sinh \frac{x}{2} = \sqrt{ \frac{1}{2}(\cosh x - 1)} \,</math>    Note: This is equivalent to its circular counterpart multiplied by −1.
:<math>\cosh \frac{x}{2} = \sqrt{ \frac{1}{2}(\cosh x + 1)} \,</math>    Note: This corresponds to its circular counterpart.
:<math> \tanh \frac{x}{2} = \sqrt \frac{\cosh x - 1}{\cosh x + 1} = \frac{\sinh x}{\cosh x + 1} = \frac{\cosh x - 1}{\sinh x} = \coth x - \operatorname{csch}x.</math>
:<math> \coth \frac{x}{2} = \coth x + \operatorname{csch}x.</math>

The [[derivative]] of sinh ''x'' is cosh ''x'' and the derivative of cosh ''x'' is sinh ''x''; this is similar to trigonometric functions, albeit the sign is different (i.e., the derivative of cos ''x'' is −sin ''x'').

The [[Gudermannian function]] gives a direct relationship between the circular functions and the hyperbolic ones that does not involve complex numbers.

The graph of the function ''a'' cosh(''x''/''a'') is the [[catenary]], the curve formed by a uniform flexible chain hanging freely between two fixed points under uniform gravity.

==Relationship to the exponential function==
From the definitions of the hyperbolic sine and cosine, we can derive the following identities:

:<math>e^x = \cosh x + \sinh x</math>

and

:<math>e^{-x} = \cosh x - \sinh x</math>

These expressions are analogous to the expressions for sine and cosine, based on [[Euler's formula]], as sums of complex exponentials.

==Hyperbolic functions for complex numbers==
Since the [[exponential function]] can be defined for any [[complex number|complex]] argument, we can extend the definitions of the hyperbolic functions also to complex arguments. The functions sinh ''z'' and cosh ''z'' are then [[Holomorphic function|holomorphic]].

Relationships to ordinary trigonometric functions are given by [[Euler's formula]] for complex numbers:

:<math>\begin{align}
e^{i x} &= \cos x + i \;\sin x \\
e^{-i x} &= \cos x - i \;\sin x
\end{align}</math>

so:

:<math>\begin{align}
\cosh ix &= \frac{1}{2} \left(e^{i x} + e^{-i x}\right) = \cos x \\
\sinh ix &= \frac{1}{2} \left(e^{i x} - e^{-i x}\right) = i \sin x \\
\cosh(x+iy) &= \cosh(x) \cos(y) + i \sinh(x) \sin(y) \\
\sinh(x+iy) &= \sinh(x) \cos(y) + i \cosh(x) \sin(y) \\
\tanh ix &= i \tan x \\
\cosh x &= \cos ix \\
\sinh x &= - i \sin ix \\
\tanh x &= - i \tan ix
\end{align}</math>

Thus, hyperbolic functions are [[periodic function|periodic]] with respect to the imaginary component, with period <math>2 \pi i</math> (<math>\pi i</math> for hyperbolic tangent and cotangent).

{| style="text-align:center"
|+ Hyperbolic functions in the complex plane
|[[Image:Complex Sinh.jpg|1000x140px|none]]
|[[Image:Complex Cosh.jpg|1000x140px|none]]
|[[Image:Complex Tanh.jpg|1000x140px|none]]
|[[Image:Complex Coth.jpg|1000x140px|none]]
|[[Image:Complex Sech.jpg|1000x140px|none]]
|[[Image:Complex Csch.jpg|1000x140px|none]]
|-
|<math>\operatorname{sinh}(z)</math>
|<math>\operatorname{cosh}(z)</math>
|<math>\operatorname{tanh}(z)</math>
|<math>\operatorname{coth}(z)</math>
|<math>\operatorname{sech}(z)</math>
|<math>\operatorname{csch}(z)</math>
|}

==See also==
{{Commons category|Hyperbolic functions}}
* [[e (mathematical constant)]]
* [[Equal incircles theorem]], based on sinh
* [[Inverse hyperbolic function]]s
* [[List of integrals of hyperbolic functions]]
* [[Poinsot's spirals]]
* [[Sigmoid function]]
* [[Trigonometric functions]]
* [[Modified hyperbolic tangent]]

==References==
{{Reflist}}

==External links==
* Mellon W. Haskell (1895) [http://www.ams.org/journals/bull/1895-01-06/S0002-9904-1895-00266-9/S0002-9904-1895-00266-9.pdf On the introduction of the notion of hyperbolic functions] [[Bulletin of the American Mathematical Society]] 1(6):155–9.
*{{springer|title=Hyperbolic functions|id=p/h048250}}
*[http://planetmath.org/hyperbolicfunctions Hyperbolic functions] on [[PlanetMath]]
*[http://mathworld.wolfram.com/HyperbolicFunctions.html Hyperbolic functions] entry at [[MathWorld]]
*[http://glab.trixon.se/ GonioLab]: Visualization of the unit circle, trigonometric and hyperbolic functions ([[Java Web Start]])
*[http://www.calctool.org/CALC/math/trigonometry/hyperbolic Web-based calculator of hyperbolic functions]

{{DEFAULTSORT:Hyperbolic Function}}
[[Category:Elementary special functions]]
[[Category:Exponentials]]
[[Category:Hyperbolic geometry]]
[[Category:Analytic functions]]