Xi (letter)

Xi (/zaɪ/ ZY or /(k)saɪ/ (K)SY;^[1]^[2] uppercase Ξ, lowercase ξ; Greek: ξι) is the fourteenth letter of the Greek alphabet, representing the voiceless consonant cluster [ks]. Its name is pronounced [ksi] in Modern Greek. In the system of Greek numerals, it has a value of 60. Xi was derived from the Phoenician letter samekh .

Xi is distinct from the letter chi, which gave its form to the Latin letter X.

Greek

A joined variant of Ξ (New Athena Unicode font)

Both in classical Ancient Greek and in Modern Greek, the letter Ξ represents the consonant cluster /ks/. In some archaic local variants of the Greek alphabet, this letter was missing. Instead, especially in the dialects of most of the Greek mainland and Euboea, the cluster /ks/ was represented by Χ (which in classical Greek is chi, used for /kʰ/).

Because this variant of the Greek alphabet was used in Magna Graecia (the Greek colonies in Sicily and the southern part of the Italian peninsula), the Latin alphabet borrowed Χ rather than Ξ as the Latin letter that represented the /ks/ cluster that was also present in Latin.

Cyrillic

The Xi was adopted into the early Cyrillic alphabet, as the letter ksi (Ѯ, ѯ).

Mathematics and science

Uppercase

The uppercase letter Ξ is used as a symbol in various contexts.

Pure mathematics

Harish-Chandra's Ξ function in harmonic analysis and representation theory
The Riemann Xi function in analytic number theory and complex analysis

Physics

The "cascade particles" in particle physics
The partition function under the grand canonical ensemble in statistical mechanics

Other uses

Indicating "no change of state" in Z notation in computing
Used as the Latin letter E in some logos.
Monetary units of the cryptocurrencies Ether (and less commonly ETC), equal to 10¹⁸ Wei

Lowercase

The lowercase letter ξ is used as a symbol for:

Pure mathematics

Random variables
A parameter in a generalized Pareto distribution
The symmetric function equation of the Riemann zeta function in mathematics, also known as the Riemann Xi function
A universal set in set theory
A number used in the remainder term of Taylor's theorem that falls between the limits a and b
A number used in error approximations for formulas that are applications of Taylor's theorem, such as Newton–Cotes formulas

Physics and astronomy

In fluid dynamics, the Iribarren parameter.
The initial mass function in astronomy.
The correlation function in astronomy.
Spatial frequency;^[3] also sometimes temporal frequency.
A small displacement in MHD plasma stability theory
The x-coordinate of computational space as used in computational fluid dynamics
Potential difference in physics (in volts)
The radial integral in the spin-orbit matrix operator in atomic physics.
The Killing vector in general relativity.
Average logarithmic energy decrement per collision (neutron calculations in nuclear physics)
Pippard's cohesion length in superconductors
The diameter of a crystal nucleus in nucleation theory
Microturbulence velocity in a stellar atmosphere
The dimensionless longitudinal momentum loss of a beam particle after a two-body interaction in accelerator physics.
Dimensionless distance variable used in the Lane–Emden equation

Other uses

Propositional variables in some philosophical works, first found in Wittgenstein's Tractatus
Extent of reaction, a concept in physical chemistry^[4] used often in chemical engineering kinetics and thermochemistry
Unknown stereochemistry or stereocentre configuration in a planar ring system in organic chemistry,^[5] as well as uppercase Xi for unknown R/S/E/Z configuration in general^[6]
One of the two different polypeptide chains of the human embryonic hemoglobin types Hb-Portland (ξ₂γ₂) and Hb-Gower I (ξ₂ε₂)
A parameter denoted as warped time used to derive the equations for homogeneous azeotropic distillation
State Price Density in mathematical finance
The information vector in the Information Filter, GraphSLAM, and a number of other algorithms used for robot localization and robotic mapping.
Used in Support Vector machines in cases where the data is not linearly separable.
Used in Microelectronics to represent the distance from a p-n junction to a point in the depletion region where the electric field is strongest.

Unicode

Unicode Code Charts: Greek and Coptic (Range: 0370-03FF)

U+039E Ξ GREEK CAPITAL LETTER XI (Ξ) (\Xi in TeX)
U+03BE ξ GREEK SMALL LETTER XI (ξ) (\xi in TeX)
U+2C9C Ⲝ COPTIC CAPITAL LETTER KSI
U+2C9D ⲝ COPTIC SMALL LETTER KSI

The following characters are used only as mathematical symbols. Stylized Greek text should be encoded using the normal Greek letters, with markup and formatting to indicate text style:

U+1D6B5 𝚵 MATHEMATICAL BOLD CAPITAL XI
U+1D6CF 𝛏 MATHEMATICAL BOLD SMALL XI
U+1D6EF 𝛯 MATHEMATICAL ITALIC CAPITAL XI
U+1D709 𝜉 MATHEMATICAL ITALIC SMALL XI
U+1D729 𝜩 MATHEMATICAL BOLD ITALIC CAPITAL XI
U+1D743 𝝃 MATHEMATICAL BOLD ITALIC SMALL XI
U+1D763 𝝣 MATHEMATICAL SANS-SERIF BOLD CAPITAL XI
U+1D77D 𝝽 MATHEMATICAL SANS-SERIF BOLD SMALL XI
U+1D79D 𝞝 MATHEMATICAL SANS-SERIF BOLD ITALIC CAPITAL XI
U+1D7B7 𝞷 MATHEMATICAL SANS-SERIF BOLD ITALIC SMALL XI

Other uses

Ξ for E in a commercial logo

Media related to Xi (letter) at Wikimedia Commons

Uppercase Ξ is used as an 'E' to stylise company names/logos like Razer (styled as RΛZΞR), Tesla (styled as TΞSLA), musician Banners (styled as BANNΞRS), and in South Korean boy group ZE:A's newest logo (styled as "ZΞA") (Compare: Heavy Metal umlaut; Faux Cyrillic)

References

^ "xi". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
^ "xi". New Oxford American Dictionary, 2nd Edition.
^ SPIE Optipedia article: "Spatial Frequency"
^ IUPAC Gold Book Entry: "Extent of Reaction" Archived June 29, 2012, at the Wayback Machine
^ Buckingham, John; Cooper, Caroline M.; Purchase, Rupert (18 November 2015). Natural Products Desk Reference. CRC Press. pp. 53–. ISBN 978-1-4398-7362-5.
^ Nomenclature of Organic Chemistry: IUPAC Recommendations and Preferred Names. 2014. pp. 1159–1160. ISBN 9780854041824. Retrieved 11 September 2017.

External links

Media related to Xi (letter) at Wikimedia Commons

[1] "xi". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)

[2] "xi". New Oxford American Dictionary, 2nd Edition.

[3] SPIE Optipedia article: "Spatial Frequency"

[4] IUPAC Gold Book Entry: "Extent of Reaction" Archived June 29, 2012, at the Wayback Machine

[BuckinghamCooper2015-5] Buckingham, John; Cooper, Caroline M.; Purchase, Rupert (18 November 2015). Natural Products Desk Reference. CRC Press. pp. 53–. ISBN 978-1-4398-7362-5.

[6] Nomenclature of Organic Chemistry: IUPAC Recommendations and Preferred Names. 2014. pp. 1159–1160. ISBN 9780854041824. Retrieved 11 September 2017.

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